Geometry

Table of Contents

Notation

  • geometry_1856818ca7b3668f91084773f82f10e083cbf6ca.png denotes the space of all functions which have continuous derivatives to the k-th order
  • smooth means geometry_f89c92349319aa0b3df52f1e86b5d9fa1c767c5f.png, i.e. infitively differentiable, more specificely, geometry_c4b25af310d56b47f0019171f3018064a1a6893c.png means all infitively differentiable functions with domain geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png
  • Maps geometry_aef116116dce35045ca7a102590f15b1bbd586a4.png are assumed to be smooth unless stated otherwise, i.e. partial derivatives of every order exist and are continuous on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png
  • Euclidean space geometry_c1f165fcdce7308f429c2f50922989344d6c5e8c.png as the set geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png together with its natural vector space operations and the standard inner product
  • sub-scripts (e.g. basis vectors geometry_157ddbf7c321d84b32ee2ac5b95a5be98704fffb.png for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and coeffs. geometry_9cae7a6af894098b8437d0b6cb97edf27209d672.png for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png ) are co-variant
  • super-scripts (e.g. basis vectors of geometry_2b73fbf94bf864bde510ca460454eea6ab004431.png for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png and coeffs geometry_19450effc4fd9e2d9788856a06873dd3c6421e5b.png for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png ) are contra-variant
  • geometry_f2ff52e9422e6ca34545963e0f7907632d5e1765.png where geometry_9dc54cb072169a34d0110a86ac53e24ce75ba419.png are manifolds, uses the geometry_230220721c63703fdbe33a5ead0093180ae7e5c8.png to refer to a linear map from geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png to geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png

Stuff

Curves

Examples

Helix

The helix in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png is defined by

geometry_989b3c2d57add85a759fbdddcddb86cdb8a5a046.png

Constructing a sphere in Euclidean space

I suggest having a look at this page in the notes

Surface of a sphere in the Euclidean space is defiend as:

geometry_6ff8ad259d62480f93a3e243870490db4b6d129d.png

As it turns out, geometry_3aaafda7ec86d0a499dd894ae5fc2a3327983130.png is not a vector space. How do you define vectors on this spherical surface geometry_3aaafda7ec86d0a499dd894ae5fc2a3327983130.png ?

Defining vectors on spherical surface

At each point on geometry_3aaafda7ec86d0a499dd894ae5fc2a3327983130.png, construct a whole plan which is tangent to the sphere, called the tangent plane.

This plane geometry_e4f375c26796781f71b7ae3026445db617a6e78b.png is the two dimensional vector space of lines tangent to the sphere at the given point, called tangent vectors.

Each point on the sphere geometry_3aaafda7ec86d0a499dd894ae5fc2a3327983130.png defines a different tangent plane. This leads to the notion of a vector field which is: a rule for smoothly assigning a tangent vector to each point on geometry_3aaafda7ec86d0a499dd894ae5fc2a3327983130.png.

The above description of a vector space on geometry_3aaafda7ec86d0a499dd894ae5fc2a3327983130.png is valid everywhere, and so we refer to it as a global description.

Usually, we don't have this luxury. Then we we parametrise a number of "patches" of the surface using coordinates, in such a way that the patches cover the whole surface. We refer to this as a local description.

Motivation

The tangent-space geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png at some point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png on the 2-sphere is a function of the point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.

The issue with the 2-sphere is that we cannot obtain a (smooth) basis for the surface. We therefore want to think about the operations which do not depend on having a basis. geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png gives a way of doing this, since each of the derivatives are linear inpendent.

Ricci calculus and Einstein summation

This is reason why we're using superscript to index our coordinates, geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png.

Suppose that I have a vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and dual space geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png. A choice of basis geometry_b0729161b21ce453199a3932bf041d650a10d766.png for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png induces a dual basis geometry_eed9e16d0ec5bd224074d2d9b3fc12593770a57f.png on the dual vector space geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png, determined by the rule

geometry_f603d5aa97f7738ba79f5e27a5f413df05e995b1.png

where geometry_1520eb3ab6b273c653603e426bc9c01db9b9a03e.png is the Kronecker delta.

Any element geometry_c81bd11ade4642e0df8424b5610ed47006b210a6.png of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png or geometry_fe0aeda2eb51952e6329b759383f4bcb448137f3.png of geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png can be written as lin. comb. of these basis vectors:

geometry_a2fe674c31fc5c285c7ddbdf380378e6a0cc91ad.png

and we have geometry_f7343032054e02b7667c164cfb3a94bc8850fa2d.png.

If we do a change of basis of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, which induces a change of basis for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png, then the coefficients of a vector in geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png transform in the same way as the basis of vectors geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png and vice versa, the coefficients of a vector geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png transform in the same way as the basis vectors of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png.

Suppose a new basis for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is given by geometry_293837c026991dc334c2df24c20f541ae01dad3b.png, with

geometry_0ce63c0ad6f40cc0658cd28c2760c1fdf762a3e2.png

where the geometry_55a828fae9776b1610eeec6157b4f13bed101cf7.png are the coefficients of the invertible change-of-basis matrix, and geometry_cf27692af162daac8a30ee836c1925568604429b.png are the coefficients of its inverse (i.e. geometry_fdc5a8a02c1c60bb952fe44583c4fd53af5b4107.png). If we denote the new induced dual basis for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png by geometry_4ae439100abd1f2930b8aa637d1789817680ab3e.png, we have

geometry_ddc10aa5a2748ce07b27a9a1eb8d314561aa02e7.png

Moreover, for any elements of geometry_c81bd11ade4642e0df8424b5610ed47006b210a6.png of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and geometry_fe0aeda2eb51952e6329b759383f4bcb448137f3.png of geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png which we can write as

geometry_1ccf100542b246a1e8eba05d1e5ecbf3a4595975.png

we have

geometry_747e2d24cb77a18e38856383e134297381d378c8.png

See how the order of the indices are different?

The entities geometry_9cae7a6af894098b8437d0b6cb97edf27209d672.png and geometry_157ddbf7c321d84b32ee2ac5b95a5be98704fffb.png are co-variant .

The entities geometry_19450effc4fd9e2d9788856a06873dd3c6421e5b.png and geometry_2b73fbf94bf864bde510ca460454eea6ab004431.png are contra-variant .

One-forms are sometimes referred to as co-vectors , because their coefficients transform in a co-variant way.

The notation then goes:

  • sub-scripts (e.g. basis vectors geometry_157ddbf7c321d84b32ee2ac5b95a5be98704fffb.png for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and coeffs. geometry_9cae7a6af894098b8437d0b6cb97edf27209d672.png for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png ) are co-variant
  • super-scripts (e.g. basis vectors of geometry_2b73fbf94bf864bde510ca460454eea6ab004431.png for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png and coeffs geometry_19450effc4fd9e2d9788856a06873dd3c6421e5b.png for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png ) are contra-variant

Very important: "super-script indicies in the denominator" are understood to be lower indices, i.e. co-variant in denominator equals contravariant.

Now, consider this notation for our definition of tangent space and dual space:

If you choose coordinates geometry_f286aa2998c7cebf0281f9dc1ff1993cfdb35ea5.png on an open set geometry_cff58b5e2ba3dcd0ff3faecbb2b90e6f93f91495.png containing a point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, then you get a basis for the tangent space geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png

geometry_f7c27b934bc97aa7d8c723824af1daab007309ff.png

which have super-script in denominator, indicating a co-variant entity (see note).

Similarily we get a basis for the cotangent space geometry_78b31cc40bdecc809136e4d37bc4e05c37a073ee.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png

geometry_bea979e0ef433793cd60d393f05d88fa36386dc0.png

which have super-script indices, indicating a contra-variant entity.

Why did we decide the first case is the co-variant (co- and contra- are of course relative)?

Because in differential geometry the co-variant entities transform like the coordinates do, and we choose the coordinates to be our "relative thingy".

Differential forms

Differential forms are an approach to multivariable calculus which is independent of coordinates.

Surfaces

Notation

  • geometry_e51648a490a46ef05bcb78c489b7982ec76973fd.png is the domain in the plane whose Cartesian coordinates will be denoted geometry_d614fd37d126b9f336aa989f8238ec95182f41fa.png unless otherwise stated
  • geometry_a79479c1dfbd160ea2cd4268283c36a017abdca0.png, unless otherwise stated
  • geometry_6b2b3670e3907ea4b71c206039510af32d636f30.png denotes the image of the smooth, injective map geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png

Regular surfaces

A local surface in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png is smooth, injective map geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png with a continuous inverse of geometry_dd31246d9ac9f38533b335ad3d8bfcb7db51c341.png. Sometimes we denote the image geometry_6b2b3670e3907ea4b71c206039510af32d636f30.png by geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

The assumptation that geometry_40fb973cd997849a029e392c385d32d4b8c40196.png is injective means that points in the image geometry_6b2b3670e3907ea4b71c206039510af32d636f30.png are uniquely labelled by points in geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Given a local surface we define

geometry_573499b889433fc38de4d2612748990ef5343c4e.png

For every point geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png, these are vectors in geometry_b997efa188f98732ecb42b516ae46455779b77c1.png, which we will identify with geometry_455d05a6d8b058d1856814e9128d0f920d211517.png itself. We say that a local surface geometry_40fb973cd997849a029e392c385d32d4b8c40196.png is regular at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png if geometry_42196cd2126d2ba9ad1ef3b9a4aa1981960f737b.png and geometry_1d67202790d01738b0035a9fd1baf993ec166861.png are linearly independent. A local surface is regular if it is regular at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png for all geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png.

This gives rise to the differential form geometry_5e5bc7de28420995f3a68575c283c15130b7899e.png:

geometry_3bb7ab4c71458a0761a42a6323070d2f97c50713.png

Here is a quick example of evaluating the differential form induced by the definition of a regular local surface:

geometry_05ce9ef3c1ef006e8b45a3b5eb52f3da9e560716.png

geometry_db1d082aeb448dfa52fb4b00a601ad1fc796bf67.png is a regular surface if for each geometry_7be7959bb936fa0c3b64c670e333a8270bc69b2d.png there exists a regular local surface geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png such that geometry_6e356db8b2e3c7d9ce8557b2cc515eef74e4c18b.png and geometry_6105eda58b1032310eb1d441ea3bb0003845a997.png for some open set geometry_05a4ad5a50af6275daff3f7e7b4c7ec13c3924cd.png.

In other words, if for each point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png on the surface we can construct a regular local surface, then the entire surface is said to be regular.

A map geometry_40fb973cd997849a029e392c385d32d4b8c40196.png defines a local surface which is part of some surface geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png, is sometimes called a coordinate chart on geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png.

Thus, if the surface geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png is a regular surface (not just locally regular) we can "define" geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png from a set of all these coordinate charts .

At a regular point on a local surface, the plane spanned by geometry_42196cd2126d2ba9ad1ef3b9a4aa1981960f737b.png and geometry_689b2e62e5bc9fa57f183cd688742d49b15949d2.png is the tangent plane to the surface at geometry_13fbd2569a688d8a3cfaae9d3a63a006253d8de7.png, which we denote by geometry_90dbc53efbfb8de98ec012c927b0fd3ac4036bf6.png. At a regular point, the unit normal to the surface is

geometry_58d94d1303b27ecf348e0f04e87592e8255900f5.png

Clearly, geometry_fb75fd92928ac80f03b061a1c284542343541bfa.png is orthogonal to the tangent plane geometry_90dbc53efbfb8de98ec012c927b0fd3ac4036bf6.png.

Given a local surface the map geometry_431d0f1f41723fc58d0a4da23f4ae913566a1424.png is a smooth function whose image lies in a unit sphere geometry_d912447739957da50faf997705c46270df0b091b.png. The map geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png is called the local Gauss map.

Standard Surfaces

Let geometry_bc71c9deea1eef5f64923f761abe89a31f28fafe.png be a smooth function. The graph of geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png is the local surface defined by

geometry_38483604d15e18ca2b2faf84a384f35ab9b77f83.png

An implicitly defined surface geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png is zero set of a smooth function geometry_e6e60d241fc7b1be2e5dfc8e2e88a11397ec281c.png, i.e.

geometry_21b1fdfeb8f3c815cd02cbc944377e3dd79b719e.png

Note that geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png is a mapping from geometry_455d05a6d8b058d1856814e9128d0f920d211517.png, and we're saying that the inverse of this function defines a surface, where it's also important to note the smooth requirement, as this implies that geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png is differentiable.

An implicitly defined surface geometry_4e33a3826722654138923ad9ca114f88cb3381aa.png, such that geometry_3cc38871215b8ed6be4ced468760901a17fe70ea.png everywhere on geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png, is a regular surface.

This is due to the fact that if there is a point geometry_7be7959bb936fa0c3b64c670e333a8270bc69b2d.png such that geometry_c4d4bee19d611eb624b5adc01752563801c109e2.png, then that implies that geometry_f8a758cd45d4b3f5d0114ff7045d42baba8c196a.png and geometry_d19738d989ce45d471335c1d84ba6f3e45b3b368.png are linearly dependent, hence not a regular surface.

A surface of revolution with profile curve geometry_273588ab14f17d60f2102166e980e9c935e0a10b.png is a local surface of the form

geometry_8636d7ea8415988acada80453d932e20d9048bcf.png

A surface of revolution can be constructed by rotation a curve geometry_9304298dfb0f32d9a20602aab650b78b89c4fdb2.png around the geometry_c6d015c821a1ec771af672f15f207610c2e56121.png axis in geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png. It thus has cylindrical symmetry.

A ruled surface is a surface of the form

geometry_2db3efe04b5cc4ca5c78d08959ccb85396049208.png

Notice that curves of constant geometry_8b0e7a5ea4d56f5afdac3c2658ca87bd50b0ad6f.png are straight lines in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png through geometry_550b37f95768bb7373ff8a753daa497da84a6e03.png in the direction geometry_4204a2cc73fae3201bd6a08f1611d00ae328be1f.png.

Examples of surfaces

Quadratic surfaces

Quadratic surfaces are the graphs of any equation that can be put into the general form:

geometry_4f4df0d12b083adf4d0e070c4437b4ed3ca536b0.png

The general equation for a cone

geometry_cb0a0ff7e4f0ae7fab81870482fd43135fbdc2b0.png

The general equation for a hyperboloid of one sheet

geometry_e1d146c13ec11edb20f41030aedfefc974dad168.png

hyperboloid-one-sheet.gif

The general equation for a hyperboloid of two sheets

geometry_5e97927a8e1ffa8036cf983a69169e6f128510cc.png

hyperboloid-two-sheet.gif

The general equation for an ellipsoid

geometry_110e808e5921202947b2e68bcce3442c8b6d07f2.png

with geometry_17bdcab30d5cbe3ee4bd1ae8b1721ded36fb8d6d.png being a sphere.

General equation for an elliptic paraboloid

geometry_186f446f6bd276c034d344053e444c3d1aaefa40.png

elliptic-paraboloid.gif

General equation for an hyperbolic paraboloid

geometry_2e5f84909e911f8ebdb21f03853d67d82615d951.png

hyperbolic-paraboloid.gif

Fundamental forms

Symmetric tensors

Notation
  • geometry_c6229db7b8ec8292287aca9c874abf03cc6409d4.png
  • geometry_c4d3c7120327c8936c0875e3eaf9e50899e2554b.png are coordinates
Definitions

A symmetric bilinear form on geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png is a bilinear map geometry_34aaeb46ec0c7173e03ac9e0b75d117bad2871d5.png such that geometry_ac176dc5d0362234cd350b497eb87810ad580c7f.png.

Given two 1-forms geometry_8e5e29091f3d01e42203c44c1336fe194fa7113a.png at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png define a symmtric bilinear form geometry_4b20de16243383474a0d7436a5d7fd07eee99ba5.png on geometry_97b71c3e983d01db83e7ed91e43ef5ea808ad713.png by

geometry_dca32859270ea176da48f75b91d9fa57683d9733.png

where geometry_6b1cf8bb8f1f34113192c283df010427e5193496.png.

Note that geometry_5688ce492cf7f5c73a164679c340e2211282138e.png and we denote geometry_fc7312a061b0e98a1f6d4a3a210570467cf427ee.png.

A symmetric tensor on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a map which assigns to each geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png a symmetric bilinear form on geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png; it can be written as

geometry_e645b4acd7327bac9649503bc9f78a0fab3e6dae.png

where geometry_7571496c805a61027ff4bdf6e3faa86f033e6f8d.png are smooth functions on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Remember, we're using Einstein notation.

A (Riemannian) metric on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a symmetric tensor geometry_cd28d2b5aed69d0f8a1a35f4a4ce0eb568598996.png which is positive definite at each point; geometry_65f387165e11e2af0b800eb54f837b19c9c4f9bc.png, with equality if and only if geometry_c2965754872d5f83d04a502439d473f6a58b6878.png.

Equivalently, it is a choice for each geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png of an inner product on geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png

First fundamental form

Notation
  • geometry_e51648a490a46ef05bcb78c489b7982ec76973fd.png and geometry_9662bfe0f36ea321426caa7a52ebe77bc5d8b3a9.png are our coordinates
Stuff

Consider a regular local surface defined by geometry_c93cfd770d8c9416fccbff9a757d2df30ea557df.png. The linear map

geometry_0d2ad157c63e66e52bd965b0e1fbed0975d7d8e8.png

is a bijection.

This bijectivity can be used to give a coordinate free definition of regularity of a local surface.

Given a regular local surface geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png, the first fundamental form is defined by

geometry_067653e5a839651d260c515ad8f20a951ea8f9cd.png

where we have introduced the notation geometry_9825b2bac448b6e2c1d1581b83ed94cb7109e812.png.

The first fundamental form of a local surface geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png is

geometry_2bd2a2f24ee24dbbe9dfc45464572999047b175b.png

where geometry_047680cb48dadbd55ff71cc69c5286e25faa880a.png are functions on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png given by

geometry_2320209c83d8ff6d5a27e2022d199ab76c00b842.png

The first fundamental form is a metric on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Problems
  • Prove bijectivity of linear map from regular local surface

    Let geometry_c5cd2e111c21decdec253f54b39b5789e0778151.png be a vector field on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png,

    geometry_8a0a29a5d0429db6b27aa6e7930e1c2b64f602c9.png

    Also, we have the geometry_455d05a6d8b058d1856814e9128d0f920d211517.png valued 1-form on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png:

    geometry_bc801ad5b212ee115fa6825615b1d985cdedb829.png

    Then,

    geometry_c8c0e39a874c4dee82bcc82db5366d02a409823a.png

    Evaluating this at each geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png it is clear that this is onto geometry_320bff20f086149e4307fe7c301fac21eed6a23b.png. Further, since the surface is geometry_9069c175f78b19fb9bad6f0f87346dc71b6ce13b.png implies geometry_2220fe559d602fd05e712d51918fd03ca83e7fba.png, so the map is one-to-one.

Second Fundamental form

Notation
  • geometry_e51648a490a46ef05bcb78c489b7982ec76973fd.png and geometry_9662bfe0f36ea321426caa7a52ebe77bc5d8b3a9.png are our coordinates
  • geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png is the normal of the surface (if my understanding is correct)
Stuff

Given a local surface geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png, the second fundamental form is defined by

geometry_026ca24f76976e08853886452ea9d31c593513e3.png

with the dot product interpreted as usual.

The geometry_455d05a6d8b058d1856814e9128d0f920d211517.png valued 1-form geometry_4a2f255ec1fdf3deaeb958898cc89134561f8ce0.png is linear map which may have a non-trivial kernel. It is convenient to use the isomorphism geometry_5de84cb1ed5fd97750c3bcc49998fc0843158c3e.png to rewrite the map geometry_4a2f255ec1fdf3deaeb958898cc89134561f8ce0.png as a symmetric form.

Since geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png is unit normalised it follows that geometry_7735073296b4c10915eb78e227834e8cb52a4c96.png (by differentiating geometry_0a91c0618b981934aeba617f44d85d950fcde843.png by geometry_cea522a8f76cdecb8449f6047ee1cd80db7a7238.png and geometry_1e1b1d1508fbc2ea2236e8d348e6595424e78726.png respectively).

Hence, geometry_1cd7e82c9c3213e18e08a1b70a0c280046733976.png and geometry_b39ce3483c5b3b68c4a2b3526aab63d27e0cb91c.png must belong to the tangent plane geometry_320bff20f086149e4307fe7c301fac21eed6a23b.png. In other words, geometry_ea812a0d63f28a27aee7af206742b2eb110c1007.png.

The second fundamental form is given by

geometry_bfac4378f67fe6c11a02bd8225798ed8bf077566.png

where geometry_af06225e02c9fa1d21613f7672b8cad526701c56.png are continuous functions on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png given by

geometry_6ca6bf1771541c332bc6078cbc7ec4e63a3d23a0.png

Which can also be written as

geometry_498d08f6f81b699ff95d721f79efb307d362f1e3.png

Q & A
  • DONE What do we mean by a 1-form having a "non-trivial kernel"?

    In Group-theory we have the following definition of a kernel :

    geometry_da8bd1b8e13c0a3653b48c3b8043fd3cb4414590.png

    where geometry_29fe8b279f2106e506971c5d4d2e9f6975ef9b13.png is a homomorphism.

    When we say the mapping geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png has a non-trivial kernel, we mean that there are more elements in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png than just the identity element which is being mapped to the identity-element in geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png, i.e.

    geometry_8851fa999671731f5744063311f877c071de6300.png

    Hence, in the case of the some 1-form geometry_4a2f255ec1fdf3deaeb958898cc89134561f8ce0.png, we have mean

    geometry_51d271a197e0ec21d54d8553903669b76f55c231.png

    i.e. non-trivial kernel refers to the 1-form mapping more than just the zero-vector to the zero-vector in the target vector-space.

  • DONE What do we mean when we write dx from TpD to Tx(p) S?

    What do we mean when we write the following:

    geometry_0d2ad157c63e66e52bd965b0e1fbed0975d7d8e8.png

    where:

    • geometry_34f254620ddb5e312d39fcc7debc15add5f84314.png is some surface in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png
    • geometry_e51648a490a46ef05bcb78c489b7982ec76973fd.png is the domain of our "coordinates"
    • geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png is a smooth map
    • Answer

      We're saying that the differential 1-form geometry_5de84cb1ed5fd97750c3bcc49998fc0843158c3e.png maps from the vector-fields defined on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to the vector-fields defined on the point geometry_13fbd2569a688d8a3cfaae9d3a63a006253d8de7.png on the surface geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

Curvature

Notation

Bilinear algebra

The eigenvalues of geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png wrt. geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png are roots of the polynomial

geometry_f8221f9ac5af3075f3788f6e053b181496835428.png

where geometry_69974818922827f27c1c62a8e437195629c77d35.png are represented by symmetric geometry_a7d4035621523aa82e431c26baa2ea61bbbac2b2.png matrices.

If geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is positive definite (i.e. geometry_eca0a50a9798952ebf86680d8e2ce20752ea57fd.png defines an inner product) there exists a basis geometry_b3ada46e8f9e21ec7633ffab214ee0be46cd95c4.png of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png such that:

  1. geometry_b3ada46e8f9e21ec7633ffab214ee0be46cd95c4.png is orthonormal wrt. geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  2. each geometry_2ce73aaf3979a944b1f3cada28e0297e8973cba7.png is an eigenvector of geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png wrt. geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png with a real eigenvalue

Gauss and mean curvatures

geometry_d0ab7d011f4da15c4c4d1c4bc23dc8737ae92004.png have 2 symmetric bilinear forms on geometry_8cb2f4aa03cb95924781a775db3359ef0ff7acfa.png, geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png and geometry_eecec026c4e5cae1ca9621df12c8d8976987cfac.png look for eigenvalues & eigenvectors of geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png and geometry_eecec026c4e5cae1ca9621df12c8d8976987cfac.png.

The eigenvalues geometry_25de83b7dcccb0f3e7f6502314196eef90b86c48.png of geometry_eecec026c4e5cae1ca9621df12c8d8976987cfac.png wrt. geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png are the principal curvatures of the surface. The corresponding eigenvectors are the principal directions of the surface. Hence the principal curvatures are the roots of the polynomial geometry_447a9b72d9c616b99251c3244dd1c4817dbc8eb4.png.

The principal curvatures may vary with position and so are (smooth) functions on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

The product of the principal curvatures is the Gauss curvature :

geometry_5050b5b72d28bbed7fb1afbfc4f9ee24472a9bf6.png

Average of the principal curvatures is the Mean curvature :

geometry_6d55ba3bec163e6a7a8f7ce92c52c2d7421cda48.png

If geometry_e547f1ac2d7de8ad4bbd44e410d55e4e06fbd2ed.png we have that all directions are principal.

geometry_eddc4df08d97725c2c38ba5ee633a0fd925a3efd.png

where all variables are as given by the first and second fundamental forms.

We get the elegant basis independent expressions

geometry_ea079c06cf9c80b79b23151402e6d814f8c291f0.png

Thus, the Gauss curvature is positive if and only if geometry_eecec026c4e5cae1ca9621df12c8d8976987cfac.png is positive definite.

Meaning of curvature

Notation

Curves on surfaces

The composition

geometry_76addf557a5d60dd04d49716e57499b7188c4c3f.png

describes a curve in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png lying on the surface.

geometry_95dbc02b2abe37e6e4edf2834f2bc65f8cb5dc10.png

and

geometry_717a5db183d5550a7277da70b70a112286979d94.png

The arclength of the curve geometry_4129d2299a9efd500766873e4198c243b86d7beb.png, lying on the surface is

geometry_3de89b929e716c167a20c21adf207e96a8e7d3a1.png

For a curve lying on a surface,

geometry_fbb276f5d6eba0ffffd8b754074d2ea331e4a9a6.png

where geometry_eecec026c4e5cae1ca9621df12c8d8976987cfac.png is the second fundamental form of the surface.

Invariance under Euclidean motions

Let geometry_c93cfd770d8c9416fccbff9a757d2df30ea557df.png and geometry_6ee275147ad54f3b327b0d02aac2f3eb16c712e1.png be two surfaces related by a Euclidean motion, so

geometry_6a628d727644abc5c32c2c25228c1e4b03e3fb80.png

where geometry_5621442eb84314df5b6d4f90193ccefd86434bb0.png is a orthogonal matrix with geometry_e2be6800cfd38ef7af2bc183bddcfd8c67ca4812.png and geometry_00739e0bb04483f309fe6eebb9130b71df9e3ba2.png.

Then,

geometry_e8542d2b5dc41c5a86c9fc9aec04e584c52f7d53.png

and hence, in particular,

geometry_c888a3d69eebebed0c8e5fa4eb119b9df1b874b5.png

The first fundamental form and second fundamental form determine the surface (up to Euclidean motions).

Taylor series

Let geometry_047db2d0b4954e71c57b9a7e43d535a8e702e113.png be a point on a regular local surface. By Euclidean motion, choose geometry_71e3e78e19ee2e47dfebe55e0ae6c7b74493ac36.png to be at the origin, and the unit normal at that point to be along the positive geometry_c6d015c821a1ec771af672f15f207610c2e56121.png axis so geometry_2fa58f2e12cde1a1a893d05ff68b42b556728b60.png is the geometry_0d2a8ed650790eb89576e18c6fa8b2be990c2da0.png plane.

Near geometry_71e3e78e19ee2e47dfebe55e0ae6c7b74493ac36.png we can parametrise the surface as a graph:

geometry_122d1d065072f4a5bea6566b6387bfdb5440d882.png

where at the origin

geometry_0585865ad36eb86cbbc5b360b9f00225f1131628.png

Using the above parametrization, and observing that geometry_2fd06887be9ed95457b7dd7c7cc386a169d123bd.png and geometry_6d9483b918ffc3c1bac1b5d280bfc22a44b6194d.png span geometry_5c7074300a037569d740b3a2f4ffa93e2ee43178.png, which is the plane orthogonal to geometry_007837e9bc23fe6265c1a4b413357ae464b19989.png, we see that geometry_33cd839d2b08772104d666748b4c8ff6f02f9a06.png and geometry_b18f3898a566651770df5ffc218a6b5ae081763d.png.

Further, supposing the geometry_e47e3e65f567df7a2f863216d28db0dee22d9bf2.png axes correspond to the principal directions, then the Taylor series of the surface near the origin is

geometry_0fbe56977d0ae0e21b1cebee09ff841871b7814f.png

where geometry_25de83b7dcccb0f3e7f6502314196eef90b86c48.png are the principal curvatures at geometry_71e3e78e19ee2e47dfebe55e0ae6c7b74493ac36.png.

Umbilical points

Let geometry_61401459f815c2d2f52750b5c9c70693a2690c6d.png be a regular local surface.

We then say a point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is a umbilical if and only if

geometry_68aa847346197d3a4ee4b05a0d2aa81c33f3a8d4.png

or equivalently,

geometry_90b9b6aadbcb1abc84cfdc86756915a376eb2118.png

i.e. all directions are pricipal directions.

An umbilical point is part of a sphere.

We can see the "being a part of a sphere" from the fact that a point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png on a sphere can be written as

geometry_b076c2e8b56db4e61da0813a2d47bb3f80d18d58.png

where geometry_8d9c686dae2d472ee5f6b06baa6112bbbf72adb8.png corresponds to geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png pointing inwards, while geometry_f78bca3b730bf20318289606beade020150086be.png is geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png pointing outwards. In this case, we have

geometry_9c7dd3f1812f9c43936b113237d9c1081653c698.png

hence,

geometry_6e82b9b8729c0b52364b7051d542ab6a1dc4eebd.png

Conversely, if geometry_39a08e1a2dcef6699f832abaff6af6badee20502.png then

geometry_7c249753274c32f1da3d4ca4a59ba5c4d7b38c7c.png

Which tells us that

geometry_58fb543bed72f89d31289f778174dbd5d5611453.png

Thus,

geometry_0528a479d6ef0d6b6292b26f9df1bd4007e8a2d7.png

where geometry_bae4bae48eb3ea91a3404b3c1514d97719620616.png is just some constant. Then,

geometry_396fbe20d3ae05e904c774c477dd8b9bbd6a775e.png

A regular local surface has geometry_1f5144a3cf7ac8e8b6bbbdd4b74d986cc4ebe621.png if and only if it is (a piece of) a plane.

The statement that geometry_af324e1341456e8811e2283b76f342a8e44930a9.png or geometry_e8a7e9bb0a566d5f003447248255dc4b5fe9c85b.png is equivalent of saying that geometry_40fb973cd997849a029e392c385d32d4b8c40196.png is part of a plane, since the tangents of the map geometry_40fb973cd997849a029e392c385d32d4b8c40196.png are perpendicular to the normal.

Every point is umbilical if and only if the surface is a plane or a sphere.

If geometry_d940d956e178876b5452a9006bd38e16a4cb9426.png for some smooth function geometry_49363d298fc352e263506bac8f76f95ea4788619.png, then

geometry_b798297731d22482d678daa8df45b1323822d3f2.png

(here we have geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png as a function, thus the exterior derivative of geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png gives us a 1-form).

And since

geometry_23f1d5e78cd51169a20c223289ed3a9eb7140e0c.png

and hence by regularity of the surface geometry_b17b5a9b3ef15cfcc3ccc627548a83d205f3abc1.png. Thus geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is a constant function on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png which implies geometry_16728185bcec03a0df5c8abbfbace7a87e2b895b.png.

This is because we've already stated that if geometry_8441b3a29b827eb54f5941a74cd06a22158873f3.png geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is part of a plane (thm:second-fundamental-form-zero-everywhere-on-surface) and if geometry_1f195de9aaaf5254222f55b0f111d2808775ac5d.png and constant we have geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png to be part of a sphere (thm:all-points-umbilical-surface-is-sphere-or-plane).

Problems

Lemma 9.1 (in the notes)

geometry_34563bf00184a22f4eab36845f947fcc08d23765.png

geometry_e3fa6105dd7e84fded13d539326df6aaad1394d2.png

geometry_21b58a6ffd6743f62a01292cad476219f71fd98d.png

Moving frames in Euclidean space

Notation

  • geometry_331e515ea0328eeb752bb30dc028ead8c10a9e15.png is a smooth map
  • geometry_c6229db7b8ec8292287aca9c874abf03cc6409d4.png
  • geometry_669ff4ca2dba2c3ed0fece7f72c6bd675910a879.png denotes the coordinates on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png
  • geometry_6bb3cf17352a002c4d5dd7a9fd7fb0941d7b7f3b.png
  • moving frame denotes a collection of maps geometry_dd1d1d2e4b17814a13ed272a8b5c3a3eabd80bcb.png for geometry_f18f134631da618f90a84c30f035af0eb198a6f4.png such that these geometry_157ddbf7c321d84b32ee2ac5b95a5be98704fffb.png form a oriented orthonormal basis of geometry_455d05a6d8b058d1856814e9128d0f920d211517.png
  • oriented means that geometry_e67b27ecf9619d2f29db0bedbe67c8a0b847a12c.png
  • geometry_80e066572fdb4d61f3bc7cd32ef0992d51936503.png, which, because the frame is oriented, we have geometry_9b2e2649e44d716934e61bcb58df55b6fe36719b.png, i.e. it's a rotation matrix

Stuff

A moving frame for geometry_455d05a6d8b058d1856814e9128d0f920d211517.png on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a collection of maps geometry_dd1d1d2e4b17814a13ed272a8b5c3a3eabd80bcb.png for geometry_f18f134631da618f90a84c30f035af0eb198a6f4.png such that for all geometry_46db0427b3179123c8e4a84cabd6424a01f1b08c.png the geometry_9844db1e628c990c99ea9ca94fae2fde4e70bcb0.png form an oriented orthonormal basis of geometry_455d05a6d8b058d1856814e9128d0f920d211517.png.

Oriented means that geometry_e67b27ecf9619d2f29db0bedbe67c8a0b847a12c.png.

This definition uses the notation of orientedness in three dimensions. For general geometry_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png there is a different definition of a oriented frame.

If geometry_e4b7dbe21a73adc83a1c6c49e30024d70921381d.png, given by

geometry_5d5223ece3855a47d8eb37049a92d4526162bdef.png

we write geometry_e5cab9216cdcfa6c7ae8c8d42bc36a7bc310b57e.png for its entry by entry exterior derivative:

geometry_81da857379e162bbaeba2c4db03661a41c8c127b.png

Thus, geometry_e5cab9216cdcfa6c7ae8c8d42bc36a7bc310b57e.png takes vector fields in geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png and spits out vectors in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png.

Connection forms and the structure equations

Since geometry_e80edbeef3f67751505880cecbe4c76ffb60409a.png is an orthonormal basis for geometry_455d05a6d8b058d1856814e9128d0f920d211517.png, any vector geometry_a0bad3442c91c78dc0a25fbf9bcd73061c0b1092.png can be expanded as geometry_4c6b993db7622e0627f377332d224a91036ab8d4.png in the moving frame, and the same applies to a vector-valued 1-form, e.g. geometry_5de84cb1ed5fd97750c3bcc49998fc0843158c3e.png.

Therefore we define 1-forms geometry_cb37e6b1b785c3f038c290f0abbdbd50b57c6626.png by

geometry_2b81541b6d8efba5f1424c3a0ebc647686e0d3cb.png

The 1-forms geometry_5c84b5c05cb7fe79c975228943cd08d2dc70a1b6.png are called the connection 1-forms and by definition satisfy

geometry_490c301ba1da73645b2da05f8842eb7a9f528e24.png

Each geometry_038f9d403f9c0278e9ce8551512222585bff120c.png are in this case a 1-form.

The connection 1-forms geometry_038f9d403f9c0278e9ce8551512222585bff120c.png are related by the antisymmetry property:

geometry_59af3bf95c0941428709bdb38b48f34d3d1d1eea.png

for all geometry_311cdc40f925249f9cf66d1ce9d8343b6e4b7e7d.png. In particular geometry_31fb15d8ea194a4a40d2b2a0acaf948c46ef5b61.png for all geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png.

We can now write the structure equations for a surface using matrix-notation:

geometry_6d25d707f66bf0e5ba837b21bbec4737c2b4e724.png

We can also write

geometry_3dcc6a701bef37ede10d236fbced97e0b0be733d.png

We will also write

geometry_db9597a8ffe9e0ad6e0ea6dd36318feae8c33cbe.png

The first structure equations are

geometry_1198156a8de566af942fce596bcd4f9e20ef42b2.png

where the wedge product between the vectors are taken as

geometry_b1ae24a623153c4526e603d13077661a5ca38d7d.png

The second structure equations are

geometry_6cb5bcd29eeb0988c3ed799e8028740283ef73e4.png

Definition of connection 1-forms and second structure equations only requires the existence of a moving frame and not a map geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png.

The structure equations exist in the more general context of Riemannian geometry, where geometry_0327567e23254148c5692f28edaf24183e8ec617.png is the Riemann curvature, which in general is non-vanishing. In our case it's zero because our moving frame is in geometry_2652a738c5d2139ff507b9d58cb2facc3d29ed44.png.

Structure equations for surfaces

Notation

  • geometry_c93cfd770d8c9416fccbff9a757d2df30ea557df.png
  • geometry_0f80394f771c40eeb2b966f6e280e5056f07b729.png are 1-forms
  • geometry_62f35da5343fd5a09bfaf666cba6fcf1efdc101e.png are "connection" 1-forms

Adapted frames and the structure equations

A moving frame geometry_eb3c884848d9e30669af1bf8c8c3ad53b5c1acb4.png for geometry_455d05a6d8b058d1856814e9128d0f920d211517.png on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is said to be adapted to the surface if geometry_04f545a159f5d9b61c7d7ea6fb4cba2fb27c7635.png.

I.e. it's adapted to the surface if we orient the basis such that geometry_50ca160ff9d4572f7106c4928ad82e854be712af.png corresponds to the normal of the surface.

The first and second structure equations for a local surface wrt. to an adapted frame, give the structure equations for a surface:

First structure equations:

geometry_b4d0816e83a7cf3ffcd3f6f7353b4ca39e28b3f9.png

Symmetry equation:

geometry_f4e2d088c1ed5c436e4d371a6679dbf8e69e44d7.png

Gauss equation:

geometry_18d648973b4bd98beec0697bfc78825caec7be9d.png

Codazzi equations:

geometry_f65165c73f36b80dba45f2aa72bc51c309bd2409.png

Notice how geometry_6db27f28141c53968e315dc52c7636dc01d5cbdc.png has just vanished if you compared to in a moving frame, which comes from the fact that in an adapted moving frame we have geometry_04f545a159f5d9b61c7d7ea6fb4cba2fb27c7635.png.

The Gauss equation above is equivalent to

geometry_052bed2c85055c372a087caede3cec63b3679dde.png

This shows that the Gauss curvature can be computed simply from a knowledge of geometry_ada209414f10e6505347b677d21c6cc9da55b721.png and geometry_3f1c4f040bdfd2b1728d6069f6ef0777a4855939.png without reference to the local description of the surface geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png.

Let geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png be a local surface with the first fundamental form geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png and geometry_c3dc4d39945f7a187520be31a0f6361dd7187561.png be the 1-forms on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png such that

geometry_c55625729f9f3e1e7f1b439aa705389bd8a12c94.png

Then there exists a unique adapted frame such that geometry_73297e2e9bc947870e9a35f0565586c4b206ceb2.png and geometry_9fa0521e5f83500a566af0dfc2c1b8ed18ff2901.png.

We say a 1-form is degenerate if wrt. any basis , the matrix representing the 1-form has geometry_303c60e01ca92b79d56b20aa4eb72119966777ce.png.

Two local surfaces geometry_c93cfd770d8c9416fccbff9a757d2df30ea557df.png and geometry_900235a9227a2afeb3507bda66e89cda3c2a7b61.png are isometric if and only if geometry_6d1fc65972a554ea4b620880aa9cce8f70fff0a4.png.

Isometric surfaces have the same Gauss curvature. More specifically,

If geometry_563f489ee3eb6d6bad91f3e1daaf7ba3d1aca38f.png are two isometric surfaces, then

geometry_3a4a5ed9236a98924d13b9a50c24c37559aa9dbc.png

The Guass curvature is an instrinsic invariant of a surface!

The first fundamental form geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png of a surface actually then turns out to determine the following properties:

  • distance
  • angles
  • area

Geodesics

Notation

  • geometry_08825fa8e15f4f4162ef921e28b503b27c11986e.png which defines the map geometry_7cae14f5956c3ecfbe5bca6fd1794dc4111a2758.png, and has unit speed geometry_8adea001f367f8023e639f07ac4d8d6a26ff300e.png joining two points geometry_b55b03e8c7c204b6509a8ce8d6d917d82c10a608.png

Stuff

Consider a 1-parameter family of nearby curves

geometry_bbb1236ab1ea8e6193f38214296c462e9d686dee.png

where geometry_c3f7ff197ce0833c983dbd77fe58a0cac16befcb.png and geometry_c0fbb0ccee68e576030c66400baf9650d691a214.png so that all curves in the family join geometry_03ff94c0c918b127cb6afcb7301d2794b26e0f69.png to geometry_ee2995f3facc4fb1e9e26d9f325844b2d058c0ad.png. We refer to geometry_17e9f52549b4b54d9708d6330ea265cc1a8a030c.png as a connecting vector.

It's very important that geometry_c3f7ff197ce0833c983dbd77fe58a0cac16befcb.png, because if geometry_17e9f52549b4b54d9708d6330ea265cc1a8a030c.png has a component along geometry_84e1ac79714032450446238575325326ada5acc3.png we could remove the shared component by reparametrising geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png.

We say a unit speed curve geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png as above has stationary length if the length of the nearby curves geometry_0c89c8f1f4950955cb6ea38a5a2edcd6b45681b4.png satisfies

geometry_75330dc9ef65ea21d615c74877b90dfd04cc95fc.png

for all connecting vector geometry_4ecca346b2ac6d4b2ec7d5f1bd60c048a001a17d.png.

A unit speed curve geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png in Euclidean space has stationary length if and only if it is the straight line joining the two points.

Let geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png be a unit speed curve in Euclidean space. We then have to prove the following:

  1. geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is a straight line, then it has stationary length
  2. geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png has stationary length then it's a straight line

Remember, stationary length is equivalent of

geometry_0cc579701c97b584f6a794abf543799901d986b8.png

First, suppose that geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is in fact a straight line, then

geometry_dd341503217061e65a59b9d453e2a14d778082dc.png

Now, taking the square root and the derivative wrt. geometry_224ae917d74dc2133d4403064c971bf562d4db50.png we have

geometry_81614baa1f68593446c22eee23d932b2a7f391e8.png

Remembering that geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is a unit-speed curve, i.e. geometry_8adea001f367f8023e639f07ac4d8d6a26ff300e.png, thus

geometry_6b1b91032ff48080d2bd054b75e3869314a9031e.png

Now, substituting this into the expression for geometry_d982072475d4a01071da18a5dcf7cb6a538e32e7.png, and observing that interchanging the integral wrt. geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png and derivative wrt. geometry_224ae917d74dc2133d4403064c971bf562d4db50.png is alright to do, we get

geometry_838c3ce342bb976bce78d98f0b7823500e3743aa.png

since geometry_c0fbb0ccee68e576030c66400baf9650d691a214.png by definition of connecting vectors. The final integral is zero if and only if geometry_2dbcc277006b9ca6ad1c76c29d6a183a8c65f5e8.png, which is equivalent of saying that geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is linear in geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png and thus is a straigt line, concluding the first part of our proof.

Now, for the second part, we suppose that geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png has stationary length

We again perform exactly the same computation and end up with the same integral as we got previously (since we did not use any of our assumptions until the very end), i.e.

geometry_02603d49b66f7a9b8c3255c87b7b8b8fc422ba59.png

And since geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is assumed to have stationary length,

geometry_6c1cc43ec316d366116a7e8f54a1412e6f7781a2.png

which is true if and only if geometry_bbadbc81ec4d0767764c551714af4ee629274045.png, hence by the same argument as above, geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is the straight line between the two points geometry_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and geometry_f8ba0bcc022c6d477cbfde325f315454468205bb.png.

Notice the "calculus of variations" spirit of the proof! Marvelous, innit?!

Geodesics on surfaces

A unit-speed curve geometry_3e6a04f5bc14930a295f24eb271d26621b15de67.png lying in a surface is a geodesic if its acceleration is everywhere normal to the surface, that is,

geometry_f24bbc84510538293ec40bc6764b41803e4fbbbd.png

where geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png is the unit normal to the surface and geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is some function along the curve.

This means that for a geodesic the acceleration in the direction tangent to the surface vanishes thus generalising the concept of a straight line in a plane.

You can see this from looking at the proof of stationary length in Euclidean space being equivalent to the curve being the straight line: in the final integral we have a dot-product between geometry_a6c63891c017a5692eb2ff08dc4255bd74f0692b.png and geometry_17e9f52549b4b54d9708d6330ea265cc1a8a030c.png,

geometry_5ea81bd6b8e0b7047837775d3421c9f80fc23e87.png

But, all geometry_5f21da7adb7eee4f3ff6a7cca1a646041b2614f4.png defined in the definition of a connecting vector / nearby curves also lies on the surface, hence geometry_5f21da7adb7eee4f3ff6a7cca1a646041b2614f4.png cannot have a component in the direction perpendicular to surface. Neither can geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png since this is also on the surface, which implies geometry_4ecca346b2ac6d4b2ec7d5f1bd60c048a001a17d.png also cannot have a component normal to the surface. Thus,

geometry_7c38b0ab0581ad2af0c78ebf3b28aebe2e2ab262.png

Finally implying

geometry_ec13fed97e159c57fd341755510a2bf37028b3c8.png

A curve lying in a surface has stationary length (among nearby curves on the surface joining the same endpoints) if and only if it's a geodesic.

A curve geometry_3e6a04f5bc14930a295f24eb271d26621b15de67.png lying in a surface is a geodesic if and only if, in an adapted moving frame it obeys the geodesic equations

geometry_0337d55524ce50c8e7b370e04a6b65d9c04e3359.png

and the energy equation

geometry_6b87e4b4a06b3cd164e60c796672752c391ba908.png

Given a point geometry_71e3e78e19ee2e47dfebe55e0ae6c7b74493ac36.png on a surface and a unit tangent vector geometry_a0bad3442c91c78dc0a25fbf9bcd73061c0b1092.png to the surface at geometry_71e3e78e19ee2e47dfebe55e0ae6c7b74493ac36.png, there exists a unique geodesic on the surface geometry_c9150c969e59669b94dfded5ea71ad1ccd3e178b.png for geometry_1c1eb06ea01920ee790c07e0a292fc0954319597.png (with geometry_224ae917d74dc2133d4403064c971bf562d4db50.png sufficiently small), such that geometry_480f62dfe2c736ff0901194a363df5c3f375b417.png and geometry_9a7f67f57a8a3f3d5f3211cd0d643c080bd155a1.png.

The geodesic equations only depend on the first fundamental form of a surface. Hence they are partof the intrinsic geometry of a surface and isometric surefaces have the same geodesics!

Two-dimensional hyperbolic space is the upper half plane

geometry_96e724a3134cbb2aa38ce5835d32dc9becf5591d.png

equipped with the first fundamental form given by

geometry_ffa6577afc17c8d9b87a215676f98bce978da49c.png

Integration over surfaces

Notation

  • geometry_481b8770cef94a03c92c0467374a884074414656.png defines a local map , where we drop the bold-face notation due to not anymore using the Euclidean structure
  • geometry_7f964b045c5f8a94b6d5c34dfe4356a6649be260.png denotes the pull-back of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png by the map geometry_7a84c9a383f9772338016d101ccc096be06af784.png

Integration of 2-forms over surfaces

Let geometry_f7b8c44fd5d6ead7d2b3dc08eff61643367d3d1a.png define a local surface

geometry_fcdbc312e00dbcfae5320a6093a5a89e0985fc65.png

Note we do not write the map defining the surface in bold here, to emphasise we are not going to use the Euclidean structure).

Let

geometry_3340e0fedbffc830ad71219bda7332f611387061.png

be a 2-form on geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png. We define the pull-back geometry_7f964b045c5f8a94b6d5c34dfe4356a6649be260.png of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png by the map geometry_7a84c9a383f9772338016d101ccc096be06af784.png to be the 2-form on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png given by

geometry_7c0f340aa2e05c674a1f5c7219c988549edc0017.png

IMPORTANT: where here geometry_90cf6f4ca8f960adac239def3ee709cf5fbdd917.png is the exterior derivative of geometry_4fb9ff609793873a6944f52375010b1cae6df7ed.png, i.e.

geometry_b87b6632262b51c2e66e19fe36041758e6e84496.png

Let geometry_f7b8c44fd5d6ead7d2b3dc08eff61643367d3d1a.png be a local surface and let geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png be a 2-form on geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png. We define the integral of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png over the local surface to be

geometry_b3d6c1b979dbc295e9a1a93dddaff98ca14feaca.png

So, we're defining the integral of the 2-form geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png over the map geometry_f7b8c44fd5d6ead7d2b3dc08eff61643367d3d1a.png as the integral over the pull-back of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png over the domain geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Why is this useful? It's useful because we can integrate some 2-form in the "target" manifold geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png over the "input" domain geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png.

Let geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png be a k-dimensional oriented closed and bounded submanifold in geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png with boundary geometry_ccf892cfc4b3712ce22908b81d4d03f4c8a23d1d.png given the induced orientation and geometry_d5a15cdc80bcc3fc6c22cc181926789847940e0f.png. Then

geometry_1596dccafc75eb4cd0a45d84da5b5c8c34aff302.png

The Stokes' and divergence of vector calculus are the geometry_4d6ea68a2d93a31ec6e79a2c3b75563e0ac05daf.png and geometry_96bf8094c7cbcc6cdac5ab7292c456c3f155691d.png special cases respectively.

Integration of functions over surfaces

For a local surface, we have

geometry_b9b7839da7965a5f6480339ad7e77f65de0e7877.png

Hence, we obtain an alternate expression for the area

geometry_616efa26672fd80d21966097b2f0a300028e1ca3.png

Thus the are depends only on geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png, hence it's an intrinsic property of the surface.

For a local surface geometry_8d0a10885b2a108c6f6e549a2f7bce774391ad67.png with an adapted frame,

geometry_a0075365e4b003564202ab9dff7fae0ca6f4c164.png

Let geometry_32ac151b17c861b0b9a8362b9dc21aa7048127be.png be a local surface and geometry_e6e60d241fc7b1be2e5dfc8e2e88a11397ec281c.png be a function.

Then the integral over geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png over the surface is given by

geometry_e98e0f6d06eb5a186f0f3fd979aecb3f57370cef.png

In particular,

geometry_8e05c0909a446ec4dc77b57d56c73323ae26580a.png

gives the are of the local surface. The 2-form geometry_fffaeb7e39407dc40868c3b370954a5e042a7364.png is called the area form.

Definitions

Words

space-curves
curves in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png
plane curves
curves in geometry_6d12ae337f449317180cc3e9acc5bdd4d829a1fd.png
canonically
"independent of the choice"
rigid motion / euclidean motion
motion which does not change the "structure", i.e. translation or rotation

Regular curves

A curve geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is regular if its velocity (or tangent) vector geometry_88b230a69aab21188508bb5c8d9c44465481cff4.png.

The tangent line to a regular curve geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png at geometry_c6018c7c4ae69e4e2d4e71d6c1c4a2d0b7824d09.png is the line geometry_4cc5a2514203ead6a387c814e95bf015da5eb504.png.

A unit-speed curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png is biregular if geometry_c7dcf60dd2f55dcaf87356a389793147bc0a87a0.png, where geometry_b992266287b82994936475115099d107b16c51ca.png denotes the curvature.

(Note that a unit-speed curve is necessarily regular.)

The principal normal along a unit-speed biregular curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png is

geometry_210b569ff363657fbafbb91c780592efe4b14069.png

The binormal vector field along geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png is

geometry_e657969a95fb3e9bd2b619533aa3d2a2030dd7ba.png

The norm of the velocity

geometry_fa05346b1ee028bc94f20f4e058a3fc7c0206935.png

is the speed of th curve at geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png.

A parametrisation of a regular curve geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png s.t. geometry_e2a7ac36c43f98b73cfc42f7ae7a98abe4195418.png is called a unit-speed parametrisation.

Level set

The level set of a real-valued function geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png of geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png variables is the set of the form

geometry_23cee36eaeb6fabdb74d491408ea44e56f6d18f6.png

Arc-length

The arg-length of a regular curve geometry_6089cbac303507d8b344620173af8d2ad88685ca.png from geometry_c6018c7c4ae69e4e2d4e71d6c1c4a2d0b7824d09.png to geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is

geometry_3437bf7549cad1549ea82b8563ad70e13b536bf9.png

For a unit-speed parametrisation we have geometry_795ace79f8fa8e2172b0f005cc62d8807f5721ef.png, hence it is also called an arc-length parametrisation.

As we can see in the notes, there's a theorem which says that for any regular curve, there exists a reparametrisation of which is unit-speed.

Most reparametrisations are difficult to compute, and thus it's mostly used as a theoretical tool.

Example: Helix

The helix in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png is defined by

geometry_925757a617fe972a8d7f595380f2c2f179da4d49.png

which is an arc-length parametrisation

Curvature

The unit tangent vector field along a regular curve is geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is

geometry_d08c8f3c525395d90580ee6b51d40159546674d7.png

Thus, for a unit-speed curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png it is simply geometry_fb2585ee7f2a9a0672ec73c28e26eb49b26cd72b.png.

For a unit-speed curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png the curvature geometry_071ae907a116ba204861855b392b6ab769d9b38a.png is defined by

geometry_47fe8b5a622d943250e2591b2c2b5195225f15c6.png

Torsion

The torsion of a biregular unit-speed curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png is defined by

geometry_eef4f30ff6bf8e16fb92d223dfa55fb0eaf9bba7.png

or equivalently geometry_57938c928c6a88f6228c23887d3ce8c7bd36feb2.png.

The oscillating plane at a point on a curve is the plane spanned by geometry_fc9b3740d8cab303a8debd959c24a0b239392d4e.png and geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png. The torsion measure how fast the curve is twisting out of this plane.

Isometry

An isometry of geometry_455d05a6d8b058d1856814e9128d0f920d211517.png is a map geometry_3f7f2e53a46d7a559f97f0b265de34b41fefc7f4.png given by

geometry_da18d70cea189cf186424fd4a9d1f16f4b6adfbb.png

where geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is an orthogonal matrix and geometry_aabcdeaa6145e8c3ac7b75b8d4d742eac45fdbf7.png is a fixed vector.

If geometry_90ead7ba62b2fe619bbc11d447dbb42e5d9f6226.png, so that geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is a rotation matrix, then the isometry is said to be Euclidean motion or a rigid motion.

If geometry_1f20750876042ef1017016270683d1fca54181d0.png the isometry is orientation-reversing.

By definition, an isometry preserves the Euclidean distance between two points geometry_c90f9bb90f05c5e889dca54a595e14930d65444e.png.

Tangent spaces

geometry_d4687565334c271f8c8a9109874015589f9c297e.png we define the tangent space geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png to geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png as the set of all derivative operators at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, called tangent vectors at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png

geometry_eff4ddceabe911587f3c4add433c0729ac3c04e8.png

and thus we have

geometry_6028f04764b9edad94fcbe31068cf8cd90b4a692.png

in the notation we love so much.

Vector fields are directional derivatives.

A vector field is defined by the tangent at each point geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png for all geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png in the domain of the vector field.

It's important to remember that these geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png are curves which are parametrised arbitrarily, and thus describe any potential curve not just the geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png you are "used" to seeing.

In words

  • Tangent space of a manifold facilitiates the generalization of vectors from affine spaces to general manifolds

Tangent vector

There are different ways to view a tangent vector:

  • embedded, i.e. with the manifold where we want to define the tangent vector embedded in a surrounding space, so that we can refer to the tangent vector as "sticking out" of the manifold
  • intrinsically, i.e. without having to refer to some surrounding space
Physists view

Basically considers the tangent vector as a directional derivative

A tangent vector to geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is determined by an n-tuple

geometry_bb31853a3a557fa97b05f1f5c08019af27861bd6.png

for each choice of coordinates geometry_f286aa2998c7cebf0281f9dc1ff1993cfdb35ea5.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, such that, geometry_6a2d97e7ec47eb33487babf93170d2c4d2a0cd6c.png is the set of coordinates, we have

geometry_aaeaed7c56561539b8b79bd1d3c9e383f0f018c7.png

In your "normal" vector spaces we're used to thinking about direction and derivatives as two different concepts (which they are) which can exist independently of each other.

Now, in differential geometry, we only consider these concepts together ! That is, the direction is defined by the basis which the tangent vectors ("derivative" operators) defines.

"Geometric" view

This is a more "intuitive" way of looking at tangent vectors, which directly generalises the concept used in Euclidean space.

A (regular) curve in geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a (smooth) map geometry_4aa1a71b761731fec400ad7d6fad152dd4d98639.png, given by

geometry_10b24df795fe71e9a0a9dc6d060b50857b53b0a3.png

where each geometry_7b798c93e89b1e3d56ce5ffc43d1da6bccf46e63.png is a smooth function, such that its velocity

geometry_9e8654c33abe97d25a1aad2422125cffbfbb5cba.png

is non-vanishing, geometry_e6961481572245e3655de951459dcdb88dbe0857.png, (as an element of geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png) for all geometry_4d4cca99c803ab67fbfd2ec8de1b07f918b5324c.png. We say that a curve geometry_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png passes through geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png if, say geometry_5e7ba3f1ad1b7052568fc7bc8cb0fbdb8fe3a407.png (without loss of generality one can always take the parameter value at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to be 0).

geometry_7d86acb9bac1ebf8fd6489d35ff02806469b0a9e.png means a map from the open range geometry_4dbac29c08ad38dc09a53b1d71656a5e05b88888.png to geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png, NOT a map which "takes two arguments", duh…

Let geometry_4aa1a71b761731fec400ad7d6fad152dd4d98639.png be a curve that passes through geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png. There exists a unique geometry_cdc39182095e0bace513530c18658c605a67472e.png such that for any smooth function geometry_f1fe965404fa488b718ff7b556200389f7551bb5.png

geometry_a582f374e85bc01630756136e8983ea91f28d216.png

There is a one-to-one correspondence between velocities of curves that pass through geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png and tangent vectors in geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png. By (standard) abuse of notation sometimes we denote geometry_fadcae417be9302a1ba4e908c910a38051218a35.png by the corresponding velocity geometry_f2518d27acd9f367c19eca929034d97278d9ca4b.png.

Tangent vector of smooth curves

This approach is quite similar to the geometric view of tangent vectors described above, but I prefer this one.

As of right now, you should have a look at the section about Tangent space and manifolds, as I'm not entirely sure whether or not this can be confusing together with the different notation and all. Nonetheless, the other section is more interesting as it's talking about tangent vectors and general manifolds rather than the more "specific" cases we've been looking at above.

Let geometry_a88a2e630ab8c66fe77a987fd661f2a01ea4134f.png be a smooth curve and geometry_a723da8733b4a023b680b6f08e752e7b95887974.png (wlog).

The tangent vector to curve geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is a linear map

geometry_a64f291038f8d72fe94df90d3379750c20f73bbe.png

where

geometry_a5a435b70d828f0ded4f0c64ba3bf5b89b1af542.png

where geometry_e9c73642da9cde17ad6dc40f2363e602ee59c448.png is a chart map.

Often denote geometry_157dff168e895c5151648a60490f89c4dc313376.png by geometry_b7f26129b700b046c2e8d6b071501e70ab3278cd.png.

Tangent bundle

The tangent bundle of a differential manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a manifold geometry_260fb19d62cd1add754a80212011f7883ed38dbe.png, which assembles all the tangent vectors in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png. As a set it's given by the disjoint union of the tangent spaces of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, i.e.

geometry_1ccfd3d8d61c8c1a2300cf80ad536d5d6d1f5c28.png

Thus, an element in geometry_260fb19d62cd1add754a80212011f7883ed38dbe.png can be thought of as a pair geometry_bcaca16349a57fa4db86decededc11f33eed0a02.png, where geometry_7a84c9a383f9772338016d101ccc096be06af784.png is a point in the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_eb83d466c7d035356e9f39998f357cee73da1e26.png is a tangent vector to geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png at the point geometry_7a84c9a383f9772338016d101ccc096be06af784.png.

Let geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png be a smooth manifold. Then the tangent bundle is the set

geometry_d63f03ad503daac5af5f761c0f1b04132ce39cd3.png

and further we define the bundle projection:

geometry_f6ae7a6b63e5b4c8f013a460d3526ce49bceb4cd.png

where geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is the point for which geometry_81f7f869f08e2faebdacb41f455fde6c446b913c.png. This gives us a set bundle; now we just have to show that the fibres are indeed isomorphic, and thus we've obtained a fibre bundle.

Idea: construct a smooth atlas on geometry_3a4f2fb627140752c305aea2fffee0b63aa4284a.png from a given smooth atlas on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

  • Take some chart geometry_05c36f17b2c6959d103f894804047463ba3e4d99.png
  • Construct

    geometry_603f8c5ab383962a5567c5d92fe9aac6df2a5e3e.png

    where we define geometry_42b688f7ebfaa0e3791010e7725767cadf2ad420.png as

    geometry_485f7f8c5d4b5750fe169bac7162ea52ca0bf938.png

    where

    • First geometry_c0fb46b67fdfc1039ddcfbd1c29a947362b9e76b.png coordinates we observe is projecting the tangent at some point geometry_157dff168e895c5151648a60490f89c4dc313376.png onto the point itself geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, i.e. geometry_c4b006e1852087f86d1501ec62fb20a017c6a99d.png (we don't write geometry_157dff168e895c5151648a60490f89c4dc313376.png in the above because we can do this for any point in the manifold)
    • Second geometry_c0fb46b67fdfc1039ddcfbd1c29a947362b9e76b.png coordinates account of the direction and magnitude of the tangent geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, i.e. we choose the coefficients of geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png in the tangent space at that point!

      geometry_c952dd8edca797abb392ec39852cdf5b861b6628.png

  • Finally we need to ensure that this map geometry_42b688f7ebfaa0e3791010e7725767cadf2ad420.png is indeed smooth : We start by considering the total space, which is the space of all sections geometry_de32b094129e4e03f562577e6b8b51603d8ca54e.png, i.e.

    geometry_a3093d68e82d7874b2dbf3ef570c49e816d0f9ca.png

    equipped with the two operations:

    geometry_cd6f67d1e79df3bdee02ccf73fa8228a5928b43d.png

    and multiplication:

    geometry_b016d106bde33be46f897c3b720af856b6684a4d.png

Dual space

Let geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png be a vectorspace over geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png. Then the dual space of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png denoted as geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png, is given by

geometry_3c4d7851a9fb63b65b9e02576f97d234a0cc9153.png

Properties

Dual Basis

Honestly, "automatically" is a bit weird. What is actually happening as follows:

Suppose that we have a basis in geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png defined by the set of vectors geometry_03fea797efc3968e5db0f27def373b7d55462b53.png. then we can construct a basis in the dual space geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png, called the dual basis. This dual basis is defined by the set geometry_4e940a0d447a682ad66d04f273db17fd3e8d408a.png of linear functions / 1-forms on geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, defined by the relation

geometry_9d5fa42269b3fa49539497ecd34c98a453f18236.png

for any choice of coefficients geometry_f023642c032b65e1410ca05ab1ce94819d541684.png in the field we're working in (which is usally geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png).

In particular, letting each of these coefficients be equal to 1 and the rest equal zero, we get the following set of equations

geometry_218feefd27ee6f978e8131e416b2438178a4f028.png

which defines a basis.

If geometry_958ab5acc7396b7c6e38cb4b74dd8a15e9737c54.png is a basis for geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, we automatically get a dual basis geometry_9fc44a5f1463d9c63132f1eed8b43c9fc42d4ac6.png for geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png, defined by

geometry_dba78b177f529f022f4d5918f1497d4003a7dabd.png

If geometry_9b6f157e9607f2c0e4a5c4e3330548c1365a8469.png (is finite), then geometry_bf813d84f92f106e0439218e438caead2ae53b3a.png

Dual of the dual

If geometry_9b6f157e9607f2c0e4a5c4e3330548c1365a8469.png

geometry_5869c54687397417cbc76843b7619680aa2e16ec.png

Map between duals

If geometry_f7fb6d83d79b2ccae4b70d555d430baca66b39a3.png in a linear map between (dual) vector spaces geometry_85a48dfbdbafa5eb610a72628a4edeb4106be807.png get canonically a dual map :

geometry_b437ddc1a1cebbe7ee3f0762c0fef53d8ef2298f.png

1-forms

A 1-form at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png is a linear map geometry_5215dfad03e0041f785cfc5ccfd85dd2ac8a2480.png. This means, for all geometry_6b1cf8bb8f1f34113192c283df010427e5193496.png and geometry_0827479c67fd95151b167e70d3c09a2f3f8d41f9.png,

geometry_60a93d11ae8d68aff9de664e3b9002d6de8ab9f6.png

1-forms is equivalent to linear functionals

The set of 1-forms at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png, denoted by geometry_78b31cc40bdecc809136e4d37bc4e05c37a073ee.png, is called the dual vector space of geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png

We define 1-forms geometry_9723ce951a62ded24b606511830b5005115a4bfd.png at each geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png by their action on the basis geometry_bd375ae963092b70f6563ecfc06bd990664d446f.png:

geometry_86af8d9105907e533fb6dd76fb1992e91f165f6e.png

Or equivalently, geometry_23591f126cac82f3acef0666f9b4018975c43268.png are defined by their action on an arbitrary tangent vector geometry_708be933ec7cfb40858bd2693af47ca2060138a6.png :

geometry_0a96506291e500a5a3bb555989c9ecca28422df1.png

Differential 1-form

A differential 1-form on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a smooth map geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png which assigns to each geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png a 1-form in geometry_78b31cc40bdecc809136e4d37bc4e05c37a073ee.png ; it can be written as:

geometry_4434c3cad21228334ed4ba457194305d7aa8f3db.png

where geometry_208de733b15fb6d83e3a297eaf86a7d29f149858.png are smooth functions.

Line integrals

Let geometry_20c9b0762efc75c37958763b3241ae2e2ba56216.png be a curve (the end points are included to ensure the integrals exist) and geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png on the 1-form on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png. The integral of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png over the curve geometry_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png is

geometry_64c60462a8700c681a53f5049de358b6f38a66ad.png

where geometry_1052805dd34dc1c943770c3f972d85494f5d2cae.png is the tangent vector field to the curve.

Working in coordinates, the result of applying the 1-form geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png on geometry_1052805dd34dc1c943770c3f972d85494f5d2cae.png gives the expression

geometry_210e820da2c159d3c0831157b234a0d47c489795.png

i.e. the derivative of geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png wrt. geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png times the evaluation of geometry_6f37ae5fdf61922cea53181a192dd95471012eb2.png at geometry_314a194d88bbad8393b5da7ade9eaed1fcda278b.png, where geometry_6f37ae5fdf61922cea53181a192dd95471012eb2.png denotes the evaluation of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png along geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png.

Example
  • Question

    Consider the parametrized graph of a function geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png:

    geometry_0622c3abcfae2b8acc1dcde1c30b18cd88267ab8.png

    as a curve in the plane. Show that geometry_9dc552b1b573a4d2824a8c371a0134f82c21b659.png is just the usual integral geometry_3a962d30bcd13c33c135ab9f4a0292bf9a358a42.png.

  • Answer

    geometry_c2a5617a682f902808e26d28c124e1bc19a1e7cd.png

    for any 1-form geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png over the curve geometry_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png. We then simply let

    geometry_5b3e1928caf445b4cc68b602dbfd8bbe13722eb6.png

    Then,

    geometry_832a4ebcc3ea674250f32eb4555aac35860ff25a.png

    Finally giving us the integral

    geometry_8a6e8977873de5374e0658a3a0391295bd2735ec.png

    Where we can obtain the wanted form by noting that geometry_d3c5ffefcc39afb989eec2deb1bdf5100b9dee54.png.

k-form

A 2-form at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png is a map geometry_07c97c88a0be0b9eecaae4f322f529c88bc76751.png which is linear in each argument and alternating

geometry_e79e6e3508f81638b412adf9af84bbfc3a62e039.png

More generally, a k-form at geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png is a map of geometry_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png vectors in geometry_2700f1d5a30bafb6bc657f852140a90f37efcc4b.png to geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png which is multilinear (linear in each argument) and alternating (changes sign under a swap of any two arguments).

And even more general, on the vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png with geometry_754a9aaae571c9f845f78451ec73ba00ac533185.png, a k-form (geometry_52a0d0ec2298a66bcf74cf5c4369c357a720ddff.png) is a geometry_b1a8bbaa1bfddb1e1cede3a075a91dbc5de0b838.png tensor that is anti-symmetric, e.g. for a 2-form geometry_078e72d3836697b7f5b4f7d8ad4503d844f2b1dd.png

geometry_9a8d9ae07bf36d82d428a4d4c619d37bdee3dcbf.png

In the case of a k-form, if geometry_fe2ca4f9e3cb29c126543e51ce4aae43a4654394.png, where geometry_bc2d67f2aed1038ba193fd35366c5226cda4abde.png, then geometry_94de3b0de8a044aa0bd3773e5b9f1454490f91e3.png are top forms, both non-vanishing:

geometry_63175ef66f188992650c1e6841ad833a58f44d67.png

i.e. any two top-forms are equal up to a constant factor.

Further, the definition of a volume on some d-dimensional vector space, completely depends on your choice of top-form.

Wedge product

The wedge product or exterior product geometry_c8444504d8602fc45c596f91957e3642eb68a7ff.png of 1-forms geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png and geometry_81c5f489718d4ff0ca6962103920c3133c0daea4.png is a 2-form defined by the following bilinear (linear in both arguments) and alternating map

geometry_8d819d5827a2da25fe23874004c9d0ae454cea5f.png

More generally, the wedge product of geometry_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png 1-forms, geometry_43d5b290d7ff5fa4dd89d004ef31acdd47f3dd05.png can be defined as a map acting on geometry_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png vectors geometry_1c75bf3a1ef7f0d0e43f174f1561d10fb5b21520.png

geometry_d42e3d9cbf4b4109a5e80bdd5b5742399e3ccca2.png

From the properties of the determinant it follows that the resulting map is linear in each vector sperarately an changes sign if any pair of vectors is exchanged (this corresponds to exchanging two columns in the determinant). Hence it defines a k-form.

Wedge product between different forms

We extend geometry_879b987f908d17a36565fbdec6d1ee317b72da24.png linearly in order to define the wedge product of a geometry_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png -form geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png and an geometry_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png -form geometry_81c5f489718d4ff0ca6962103920c3133c0daea4.png. Explicitly,

geometry_8221c02ff43a299df2c1c49185e52ab275fe495f.png

Here the sum is happening over all multi-indices geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png and geometry_d9c991e572f71067e0f8c2e2021ee9a8771ac2f1.png with geometry_626f300faa98d0fa26d63aa0fb85f5ad2800fcfd.png and geometry_0d7897b248fab36fa8ed3aad3b416aedb8414bb7.png.

Now two things can happen:

  • geometry_4776c4e70825d410177707a90ce5a2fa2bf4b635.png, in which case geometry_3718e127d01a2141783ed71e2457f6e90467500c.png since there will be a repeated index
  • geometry_83ba617698cb9603fae24819781680543517d905.png, in which chase geometry_1a32e6fd05b4aff6b6e299d55159103dc3c705f1.png, for some muli-index K of length geometry_8ecf6d9eb07a44d199000b33f4f8bf38d16c4cc4.png. The sign is due to having to reorder them to be increasing.

Therefore, the wedge product defines a (bilinear) map

geometry_f8ee74b8dc4b45e5e476fb3bea228b78098530d3.png

Multi-index

Useful as more "compact" notation.

By a multi-index geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png of length geometry_626f300faa98d0fa26d63aa0fb85f5ad2800fcfd.png we shall mean an increasing sequence geometry_0556f135c65f6bd065f33a86f88c938ce62d4be0.png of integers geometry_a3f8a9b31c969fc6bdf8ecb15292d4cf291293c5.png. We will write

geometry_b48b6c211532a643073bb9182e06d2af7a0c1696.png

The set of k-forms at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is a vector space of dimension geometry_4d20dd5a387993f82c1137ff71e630819ef14ff6.png for geometry_7859eb8c4fde3bc1fd6a4fd7234ba3fd35cad60d.png with basis geometry_789b78ba236d5a9dadbfaf044c4cd544441e744d.png.

Here geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png denotes the maximum number of dimensions. So we're just saying that we're taking the wedge-product between some geometry_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png indicies of the 1-forms we're considering.

Differential k-form

A differential k-form or a differential form of degree k on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a smooth map geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png which assigns to each geometry_73107cbf4601839e95b576a97f87b50ec146c8e3.png a k-form at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png; it can be written as

geometry_f9cf78103e1fe290a5d1c77c3ddc169d13802e10.png

where geometry_e442ac31669e2cf6553a186a9905f2f2ec318ce0.png are smooth functions, and the sum happens over all multi-indices geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png with geometry_626f300faa98d0fa26d63aa0fb85f5ad2800fcfd.png.

Given two differential k-forms geometry_8e5e29091f3d01e42203c44c1336fe194fa7113a.png and a function geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png the differential k-forms geometry_9735aa1d04e8959e50ab07895abf7c98c0592875.png and geometry_40d14b003fcb083477a7215b98b67efee803949d.png are

geometry_14531f31cfa0af7c4ca8660838a3125b2b627aef.png

The set of k-forms on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is denoted geometry_f42ae55a69f54eb11e6324991daafc33cc6e9ace.png.

By convention, a zero-form is a function. If geometry_06f10602e8c0f0aa4448066a9fcdb52f0f9f20af.png then geometry_dab4b2380d411a1b84bd220c6e76beb8ad089211.png (for every form has a repeated index).

To make the notation used a bit more apparent, we can expand geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png for geometry_154e1fd7b7ac5b2182c8936be8ecb1dfcc82e459.png in for a vector-space in geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png, i.e. geometry_aa8807f668cddfdd36b57b0fca43b2d35fa14eea.png, defined above as follows:

geometry_76a71640c7e070f004bdb19d0c0ee2a59e21e343.png

where we've used the fact that geometry_926dcb6d874c8bde12cd8916fe195926e4bfc0d0.png. and just combined the "common" wedge-products. It's very important to remember that the geometry_6f37ae5fdf61922cea53181a192dd95471012eb2.png here represents a 0-form / smooth function. The actual definition of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png is as a sum of all possible geometry_60bac3550988dd9d9eaaacd46b836347d44f6ecd.png but the above definition is just encoding the fact that geometry_f57f9b49605fdcfea32cb23a6897e736f33eb4ca.png.

A form geometry_8df40f1591134fbd3445a3490809f98dbafe1fc4.png is said to be closed if geometry_bfcc88a37de74719aba14a7a8548419cfa19a9ad.png.

A form geometry_39fe59b6e7c04e21b6f602b9bcddea351f9d3c1b.png is said to be exact if

geometry_f865e621a9abea3106b24b8a350ff1a829dbb52e.png

for some geometry_02b2a04d98a192900f7027cbd9a84b64caa3dcbe.png.

If a k-form is closed on geometry_e04952b75c7e1b1f43954670321ab89bd8d2e93b.png, then it is also exact.

Exterior derivative

Given a smooth function geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png, its exterior derivative (or differential) is the 1-form geometry_fdfa2f5baeb6202e4cf1d632d54217b4711da72f.png defined by

geometry_f2ea0286a15325c262bc0490217bb2c4183a72b6.png

for any vector field geometry_d198c0d06b0da5525f9a966222601066509ac01d.png. Equivalently

geometry_a61eebb65e3ac2ac87d0238e07fc8333ac0ddb0b.png

Let geometry_fbdc03bb4a5be7df8a46f84ef16c5802838f120e.png be a smooth function, i.e. geometry_3e0899e199377146636e3d5d488146b5ba732c6f.png.

As it turns out, in this particular case, the push-forward of geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png, denoted geometry_5f1c3bce8ea70057c93d98e17ad522fda87c936f.png is equivalent to the exterior derivative!

If geometry_1738794a5d39f79be93bf15fe71f681a24584bde.png, then its exterior derivative geometry_d5a6ce83e441ab98ac90eccfe5e20a8717afe3e5.png is

geometry_e7bcb1593bee9ebe07fb9c22106bc9d42a8d0f49.png

where geometry_399f16cb279fdba7a602b1fbac452e404b9414c2.png denotes the exterior derivative of the function geometry_7543735db021a534fb50472fcfa7f8e853207523.png (which we defined earlier).

More explicitly, take the example of the exterior derivative of a 1-form, i.e. geometry_8902e64f87094056cf989ba09715eecaffff8c2f.png:

geometry_a04fdec3e8dd8492d37b410320a645e42c82b3d1.png

from the the definition of geometry_fdfa2f5baeb6202e4cf1d632d54217b4711da72f.png where geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png is a function (0-form), and geometry_626089d8c9d719c72d7146ada58fd6fa8df0fa61.png.

Theorems

The exterior derivative geometry_190f1394bb21aff863c8d73d13ee3739fb44e7f5.png is a linear map satisfying the following properites

  1. geometry_f4e5566de1c75b7700b562b91a80d2308cc66d12.png obeys the graded derivation property, for any geometry_8df40f1591134fbd3445a3490809f98dbafe1fc4.png

geometry_6d6c126fcea5fd6fb8cbd9a999733c56cfc1ea1f.png

  1. geometry_0f0f741e4f6c8ac92542cbc436959e6610b32240.png for any geometry_8df40f1591134fbd3445a3490809f98dbafe1fc4.png, or more compactly, geometry_42777dcd4fbddf82000c9672c7a274847f00c282.png

Example problems

Handin 2

Let geometry_14968f11358d50dc8146534b75001258ff2313d6.png be the helix geometry_0687ed8e35d1db16aec6d68d5b6a2ee7958add58.png and consider the 1-form on geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png

geometry_56bb0d3541e1e933fe5ff60603f19c80dac3b4eb.png

  1. Find the tangent geometry_1052805dd34dc1c943770c3f972d85494f5d2cae.png at each point along curve. Hence evaluate the line integral of the 1-form geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png along the curve geometry_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png.

    geometry_c504ec1a3eb0444f2d764dcd5f60f59a38fe38b4.png

    Hence the integral is

    The tangent plane at some point geometry_0dfe6a8522da7ac1f861a4f409988d243c863360.png along the curve geometry_314a194d88bbad8393b5da7ade9eaed1fcda278b.png for a specified geometry_35c79051c6fa657562ee4834e573cd97e4fda36d.png is given by

    geometry_e7937c99f495c6ef65696d7d24546f51645373d1.png

    which in this case is equivalent of

    geometry_c504ec1a3eb0444f2d764dcd5f60f59a38fe38b4.png

    Concluding the first part of the claim.

    For the integral, we know that

    geometry_5905d29a300da2ca2302761733ac0207f3ff9c85.png

    for the boundaries geometry_bee9257137b257a2c871c637faee1ef4a7f93f96.png and geometry_931d7a0787f751e067ad21611d255b4016f3496b.png. Computing geometry_571ab32957c89b8d7fccd992989f1fdfa69e080e.png we get

    geometry_c133d908508cc10e5bfd980620944cd19ae24836.png

    geometry_5f57765a737e32189b4237ded439ca5d0a5b8b07.png

  2. Show that geometry_bfcc88a37de74719aba14a7a8548419cfa19a9ad.png. Now find a smooth function geometry_66cccdaf50f1bc86876910867f60549d56750d8e.png such that geometry_94d0f4e90790295944138fbe1c48661b03417207.png. Hence evaluate the above line integral without explicit integration.

Integration in Rn

The standard orientation (which we always assume) is defined by

geometry_fcc9baa3fe3ad5318543ac6dc3b654868b273ece.png

Coordinates geometry_2544c14ccb25c5b43b42db381394c8824f650296.png (an ordered set) are said to be oriented on geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png if and only if geometry_f0b180fefcfab64db2d2356d5dfd750ef3dd1442.png is a positive multiple of geometry_52c91ca1fd57ba310d74a60937a03b3491c5b94f.png for all geometry_4b98ed14805f0de0caca613e9179ca93eb8b74b5.png.

Observe that this induces an orientation on geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png, since we simply apply geometry_52c91ca1fd57ba310d74a60937a03b3491c5b94f.png to the coordinates geometry_f286aa2998c7cebf0281f9dc1ff1993cfdb35ea5.png, thus returning a geometry_ce90845a3935abdbf95113974191d5f35b358844.png or geometry_6165ffc9c2e9b35f28e7ecbd3e9fb38b5cda2636.png dependening on whether or not the surface is oriented.

Let geometry_f286aa2998c7cebf0281f9dc1ff1993cfdb35ea5.png be oriented coordinates for geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png. Let geometry_2544c14ccb25c5b43b42db381394c8824f650296.png be smooth functions on geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png. Then

geometry_8f7bff0a60539aeabdc4c5f6759ee85c06e1fd63.png

where the factor on the RHS is hte Jacobian of the coordinate transformation (i.e. the determinant of the matrix whose geometry_2b4ff9f6273d4d816034dae75f24680cd1eb70da.png component is geometry_fe9b04c480cfbfb1ee8480c3aaf2636764dd5018.png.

Let geometry_f286aa2998c7cebf0281f9dc1ff1993cfdb35ea5.png be oriented coordinates on geometry_c6229db7b8ec8292287aca9c874abf03cc6409d4.png and write

geometry_379207de4d7338daa3da152095ebf04b180235dc.png

Then the integral of geometry_e29af05574032ace665d996d46b3280fc49866ef.png over geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is defined by

geometry_2b9d1516c95b501e9cf37832211313e4631c747f.png

where the RHS is now the usual multi-integral of several variable caculus (provided it exists).

Topological space

A topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

Or more rigorously, let geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set. A topology on geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a collection geometry_547dc2264649729065dc2513e00cb7127f3b9977.png of subsets of geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, called open subsets, satisfying:

  • geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and geometry_2d52d323d5349e9589bfd27e9910b1ca222e825e.png are open
  • The union of any family of open subsets is open
  • The intersection of any finite family of open subsets is open

A topological space is then a pair geometry_5b9ed3fd9fa8144ceab31a866900adbf2b57638a.png consisting of a set geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png together with a topology geometry_547dc2264649729065dc2513e00cb7127f3b9977.png on geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

The definition of a topological space relies only upon set theory and is the most general notion of mathematical space that allows for the definition of concepts such as:

  • continuity
  • connectedness
  • convergence

A topology is a way of constructing a set of subsets of geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png such that theese subsets are open and satisfy the properties described above.

Atlases & coordinate charts

A chart for a topological space geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png (also called a coordinate chart, coordinate patch, coordinate map, or local frame ) is a homeomorphism geometry_95f39b94e0a511387cb1827142b73011a27c8d8c.png, where geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png is an open subset of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png. The chart is traditionally denoted as the ordered pair geometry_ba3d6f641815dc9d0603311be937ad46e6a0365c.png.

An atlas for a topological space geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a collection geometry_0c0af349a9d03530e9a911c43b7c53550962288d.png, indexed by the set geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, of charts on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png s.t. geometry_8bef9529b7bf284cdda441d31060ff354c1facc4.png.

If the codomain of each chart is the n-dimensional Euclidean space, then geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is said to be n-dimensional manifold.

Two atlases geometry_a8c0689744bce0e604beea14514b92d862739199.png and geometry_7c2baf122dabc61b74ab11fc18f461db5ead002d.png on geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png are compatible if their union is also an atlas.

So we need to check the following properties:

  1. The following are open in geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png for all geometry_13ac77bee96f71b2538ceab06ff6bc6a9621063d.png and all geometry_42e63bbe30112de0ee2508d52d340a0469f4fca9.png

    geometry_e82970631837fa09f977bbf6eae0baecab14612c.png

  2. geometry_0cc476ea286fa309832664af34f6c6a944e1d2db.png and geometry_3ccac7c43b7c5181b1ab49d512ae37c92070302b.png are geometry_142e99eaf45f1872ae84813baea3d90f924b5333.png for all geometry_13ac77bee96f71b2538ceab06ff6bc6a9621063d.png and all geometry_42e63bbe30112de0ee2508d52d340a0469f4fca9.png.

Compatibility of atlases define a equivalence relation of atlases.

A differentiable structure on geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is an equivalence class of compatible atlases.

Often one defines differentiable structure with a "maximal atlas" instead of an equivalence class. The "maximal atlas" is obtained by simply taking the union of all atlases in the equivalence class.

A transition map is a composition of one chart with the inverse of another chart, which defines a homeomorphism of an open subset of the geometry_c1f165fcdce7308f429c2f50922989344d6c5e8c.png onto another open subset of the geometry_c1f165fcdce7308f429c2f50922989344d6c5e8c.png..

Suppose we have the following two charts on some manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png

geometry_b7c7266eb0c222643fbf8ef95a6594ec865fcd3a.png

such that

geometry_fe964b2c98e111d5f7712b955a22e3c9d09bfa59.png

The transition map is defined

geometry_a3b5477522974333b7ca02ff8853b1cf78fc2f8c.png

where we've used the notation

geometry_f993609bffd8f1881d04cfaec368110cac4bdf5f.png

to denote that the function geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png is restricted to the domain geometry_a2863e3f5a27f563a5f73109fa5800fb39825e84.png, i.e. the statement is only true on that domain.

A differentiable manifold is a topological manifold equipped with an equivalence class of atlases whose transition map are all differentiable.

More generally, a geometry_1856818ca7b3668f91084773f82f10e083cbf6ca.png manifold is a topological manifold for which all the transition maps are all k-times differentiable.

A smooth manifold or geometry_f89c92349319aa0b3df52f1e86b5d9fa1c767c5f.png manifold is a differentiable manifold for which all the transition map are smooth.

To prove geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a smooth manifold if suffices to find one atlas due to the compatibility of atlases being an equivalence relation.

A complex manifold is a topological space modeled on the Euclidean space over the complex field and for which all the transition maps are holomorphic.

When talking about "some-property-manifold", it's important to remember that the "some-property" part is specifying properties of the atlas which we have equipped the manifold with.

geometry_fbdc03bb4a5be7df8a46f84ef16c5802838f120e.png is smooth if for any chart geometry_64a61d03e1ad7117d594e7feab83a9f16ed1576e.png, the function

geometry_737ce4527e4cf2233ddb6388ae72e7419f97116d.png

is smooth.

Observe that if geometry_9f6769996358f409a799b947273edcaa0eec0794.png is smooth for a chart geometry_7944ef9a2630d634c40734c538ac4ca5c90b4002.png, then we can transition between patches to get a smooth map everywhere.

Examples

Real projective space

geometry_5fa47384312bae778edfaeeed13037a767ae1f43.png

If geometry_3182008ac8d51d4418e99caea597671e2d720621.png spans a 1d subspace (up to multiplication by real numbers). So, for each geometry_6c067936f66120d2f09a8879d4f815e9d7738897.png, we let

geometry_d763ade08b15b82c658cc310ff61a98ae4141134.png

Then

geometry_94a2458263996f08b591c423b1fe04d06ff7aef3.png

and we further let

geometry_24ea046c5dc1d45458bdba8cd773fb7f1c1c69d8.png

Manifolds

A topological space that locally resembles the Euclidean space near each point.

More precisely, each n-dimensional manifold has a neighbourhood that is homomorphic to the Euclidean space of dimension geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png .

Immersed and embedded submanifolds

An immersed submanifold in a manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a subset geometry_97abe8419f56207a115b207a4bce3122fb2eacce.png with a structure of a manifold (not necessarily the one inherited from geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png!) such that the inclusion map geometry_0b7e5e91faf2279c1dd31b58bd1dcec06617a3c7.png is an immersion.

Note that the manifold structure on geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png s part of the data, thus, in general, it is not unique.

Note that for any point geometry_dbbdce097e87ba6fa4c6c5bac3563052928c96ec.png, the tangent space to geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png is naturally a subspace of the tangent space to geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, i.e. geometry_cf95f23361dd9a404a9345ffda095ddb771d4448.png.

An embedded submanifold geometry_97abe8419f56207a115b207a4bce3122fb2eacce.png is an immersed manifold such that the inclusion map geometry_0b7e5e91faf2279c1dd31b58bd1dcec06617a3c7.png is a homeomorphism, i.e. geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png is an embedding.

In this case the smooth structure on geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png is uniquely determined by the smooth structure on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

Examples
  • Figure 8 loop in geometry_e4f375c26796781f71b7ae3026445db617a6e78b.png
    • It is immersed via the map

      geometry_6fd6169e555575e7dce50b2a2a8d3dafa3baf5a3.png

    • This immersion of geometry_e9b05f72af5d1684a8358a2358848e2f33c3a824.png in geometry_e4f375c26796781f71b7ae3026445db617a6e78b.png fails to be an embedding at the crossing point in the middle of the figure 8 (though the map itself is indeed injective)
    • Thus, geometry_e9b05f72af5d1684a8358a2358848e2f33c3a824.png is not homeomorphic to its image in the subspace / induced topology.

Riemannian manifold

A (smooth) Riemannian manifold or (smooth) Riemannian space geometry_c07ec01dcf20696d722a3604c2e9763e605ef7bd.png is a real smooth manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png equipped with an inner product geometry_f7da3924b368d24a145395ec3a47e4c4248609f4.png on the tangent space geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png at each point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png that varies smoothly from point to point in the sense that if geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and geometry_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png are vector fields on the space geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, then:

geometry_01488415910e7db67067401593fdd6146719a38d.png

is a smooth function .

The family geometry_f7da3924b368d24a145395ec3a47e4c4248609f4.png of inner products is called a Riemannian metric (tensor).

The Riemannian metric (tensor) makes it possible to define various geometric notions on Riemannian manifold, such as:

  • angles
  • lengths of curves
  • areas (or volumes)
  • curvature
  • gradients of functions and divergence of vector fields

Euclidean space is a subset of Riemannian manifold

Resolving some questions
  • Why do we need to map the point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to two vector-spaces geometry_88a185ba59d680dd9b8401e7aec99e98938de712.png before applying the metric geometry_f7da3924b368d24a145395ec3a47e4c4248609f4.png ? Because it's the tangent space geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png which is equipped with the metric geometry_f7da3924b368d24a145395ec3a47e4c4248609f4.png, not the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png itself, and since vector spaces are defined by a basis in geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png we need to map geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png into this space before applying geometry_f7da3924b368d24a145395ec3a47e4c4248609f4.png.
  • What do we really mean by the map geometry_771fc2a94d17fd39596f4f9bcc13b78c12d070d7.png being smooth ? This means that this maps varies smoothly wrt. the point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.
  • The Riemannian metric geometry_f7da3924b368d24a145395ec3a47e4c4248609f4.png is dependent on geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, why is that? Same reason as the first question: the inner product is equipped on the tangent space, not the manifold itself, and since we have a different tangent space at each point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, the inner product itself depends on the point chosen.

Differential manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

Submersion

A submersion is a differentiable map between differential manifolds whose differential is surjective everywhere.

Let geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png be differentiable manifolds and geometry_e872a467403103b86d1fbe9bafc9b501c3fe462a.png be a differentiable map between them. The map geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png is a submersion at the point geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png if its differential

geometry_d8175fd528a60b8416fde18544f5170a23f5552c.png

is a surjective linear map.

Homeomorphism

A homeomorphism or topological isomorphism is a continuous function between topological spaces that has a continuous inverse function.

Diffeomorphism

A diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are smooth.

Isometric

An isometry or isometric map is a distance-preserving transformation between metric-spaces, usually assumed to be bijective.

Tensor

In words

Let geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png be a vector space geometry_408f11eb148a975cb67aab291d62fab3bed0ad5e.png where geometry_3d136f0fc4860468633907421c098b9feb0eef24.png is some field, geometry_2d9eee9ed3f0530ef8fe709062f9dca828344e68.png is addition in the vector space and geometry_7cf66a4c3f602dfcf65c4a37701fa282ffb36c4f.png is scalar-multiplication.

Then a tensor geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is simply a linear map from some q-th Cartesian product of the dual space geometry_d10f7a265ad25e3977b2686a4f59771776876ad9.png and some p-th Cartesian product of the vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png to the reals geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png. In short:

geometry_8e437eaa1aef6d16ae5bf31fe42db29b1a07e4c9.png

where geometry_2237a11a01392d55e27ebd17c1024aa843291f1c.png denotes the (p, q) tensor-space on the vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, i.e. linear maps from Cartesian products of the vector space and it's dual space to a real number.

Maths

Tensors are geometric objects which describe linear relations between geometric vectors, scalars, and other tensors.

A tensor of type geometry_a3b6bdd6c50d8c2f4ee3e210c7ecc1524cd5c581.png is an assignment of a multidimensional array

geometry_e2a2267b13bf311f66438f58be0e27611458f1d0.png

to each basis geometry_01c94e9f9a2f29c9231f0dd832b60d52b028f2bb.png of an n-dimensional vector space such that, if we apply the change of basis

geometry_2330b5e6febfe79934e1aee1167cb49a42d9fefc.png

Then the multi-dimensional array obeys the transformation law

geometry_588aa80ada89cd071321aeb1d307b4a509d311f9.png

We say the order of a tensor is geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png if we require an n-dimensional array to describe the relation the tensor defines between the vector spaces.

Tensors are classified according to the number of contra-variant and co-variant indices, using the notation geometry_a3b6bdd6c50d8c2f4ee3e210c7ecc1524cd5c581.png, where

  • geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is # of contra-variant indices
  • geometry_f63749027365e29025a5cb867262c465d2de65bb.png is # of co-variant indices

Examples:

The tensor product takes two tensors, geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png and geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png, and produces a new tensor, geometry_9f38f2553140835ec25b60a8724e3e32b21f245a.png, whose order is the sum of the orders of the original tensors.

When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.

geometry_c098695a3a8c1750e13788afbf784253d608f751.png

which again produces a map that is linear in its arguments.

On the components, this corresponds to multply the components of the two input tensors pairwise, i.e.

geometry_67e01738af7aa7da844894db3e1ef4a81fbd9e8c.png

where

  • geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is of type geometry_866940d3c731e15432f152150b254af8a7a7db44.png
  • geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is of type geometry_805bacb6d81ec90254f1dac4a7390f37728d27d7.png

Then the tensor product geometry_9f38f2553140835ec25b60a8724e3e32b21f245a.png is of type geometry_af65c1c17366092b6287ffa35ea83c486bf157e5.png.

Examples

Linear maps - matrices

A linear map is represented as a matrix, and we say this is a tensor of order 2, since it requires 2-dimensional array to describe the relation.

Homomorphism

Algebra

A K-vector space geometry_d9a768b65fc8fb7c347047309fc29a92094b7797.png equipped by a product, i.e. a bilinear map

geometry_ab6f40f2a30e2dc5501d3d262c0b2571cc022b91.png

is called an algebra geometry_1f06f0a58777c81f5618917539c5105e893a5a56.png

Example: Algebra over differentiable functions

On some manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, we have the vector-space geometry_84d72864f9170a26b934dd3c7ef9c455c6465aed.png which is a geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png -vector space, and we define the product geometry_d6ada3514f11bc60253d88327733eddcc74bc866.png as

geometry_fccd7af1d6bc93fa3e61eba86e07492523798268.png

by the map, for some geometry_a7ab6b731be4a1793a432bb3d4c86d8d53b268f1.png,

geometry_d4fe3317b6a3253cb5b1fcc4e36fa15f2b96f912.png

where geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png and the product on the RHS is just the s-multiplication in geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png.

Derivation

A derivation geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a linear map

geometry_f4cc1a4242144ee01d5f3a23445b1ca823f4636a.png

for some algebras geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, which additionally satisfies the Leibniz rule, that is:

geometry_5037eb6cd6729972144b83f7615e8d35232aa04e.png

where geometry_0895d659200b4ac81090c18999d339b9d59577a4.png and geometry_235bd6cbb160ac49c9825bee90ed9b53ab479edc.png denotes the products in the algebras geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png respectively.

Example: derivative on algebra of continuous functions

We have the algebra geometry_9c009f1c8fc064d2e95fa5c3eaf6ff802c5e0247.png, for any geometry_81f7f869f08e2faebdacb41f455fde6c446b913c.png, we have

geometry_a5cb6be59cf41883bf7900a9b2e25c48d12268a3.png

which satisfies

geometry_abd2b0a8279bd2dde9f758046b49ed650be2773c.png

Example: Lie-algebra

Let geometry_7e756fa658b2eace58c40f5a0aa36908ec8c8b16.png for some K-vectorspace geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, then we define the map

geometry_a9b794b147768bb048b561e5efd86d76401c5b0e.png

defined by

geometry_9a673915ae4f847b0dc12f1591d997371b9b92aa.png

then geometry_4a3c8fb8e0b9fdf653383675668e8ae8ac8c2faf.png forms an algebra, which is an example of a Lie-algebra!

In fact, the definition of a Lie-algebra is because of some other property, but this is an example of a Lie-algebra.

Equations / Theorems

Frenet-Serret frame

The vector fields geometry_12a186ad7bc534c3e71a7cc30cc3bb3e595c44f5.png along a biregular curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png are an orthonormal basis for geometry_455d05a6d8b058d1856814e9128d0f920d211517.png for each geometry_aa1698cb8ee1665238ec3e91824191643c62ee93.png.

This is called the Frenet-Serret frame of geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png.

By definition of the unit tangent, geometry_283c03c42cdebb868a16df6b5aad62828a674adf.png. Differentiate this wrt. geometry_aa1698cb8ee1665238ec3e91824191643c62ee93.png to find geometry_255b08efba5a3316e6c179512d4eeea0292dd7a6.png. Thus, the principal normal satisfies geometry_0a91c0618b981934aeba617f44d85d950fcde843.png and geometry_ea1b27dbbe31b31831d49453e56f12bf6a76f5c0.png.

By definition of the binormal we also have geometry_92a1d194deb06491fb4ee166d57c2a943020260e.png and geometry_bd4dd69d1304428ae98330bb527ba2bb04ef9f8f.png. Hence, geometry_1b99a4118a311b02a95d9283b9765c084fbdbede.png form an orthonormal basis.

Structure equations

Let geometry_40fb973cd997849a029e392c385d32d4b8c40196.png be a unit-speed biregular curve in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png. The Frenet-Serret frame geometry_028cb2a6ff480e257df5a1821ceb9518f2421ff4.png along geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png satisfies:

geometry_26d5fed4446f40d5ad0b74cd724f378bed7f0f5e.png

These are called the structure equations for unit-speed space curve, or sometimes the "Frenet-Serret equations".

See p. 9 in the notes for a proof.

For a general parametrisation geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png of a biregular space curve the structure equations become

geometry_50148c77e6a38551426045852bd2254a62961a6b.png

where geometry_d198c0d06b0da5525f9a966222601066509ac01d.png is the speed of the curve.

Extras

A biregular curve geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png is a plane curve if and only if geometry_5f8108306255bd5ff1e9c3315799ffc497e4fdfe.png everywhere.

If geometry_da61cea9eb874581a1f4b5c8610337ebd4e767b9.png lies in a place, then geometry_fc9b3740d8cab303a8debd959c24a0b239392d4e.png and geometry_3f068851f9629ff3e371bc0b89748957264e96fb.png are tangent to the plane and so geometry_8d0b618205a8cc8d6e21fd38f550c9fd73d9998e.png must be a unit normal to this plane and hence constant.

The structure equations then imply geometry_5f8108306255bd5ff1e9c3315799ffc497e4fdfe.png.

The curvature and torsion of a biregular space curve in any parametrisation can be computed by

geometry_75bbaa79f200157ac326dcaeef68f1b2b6f834a3.png

Matrix formulation

The structure equations can also be expressed in matrix form:

geometry_0696d47cfb2a836ca1f7fcbdca156f90540c5397.png

By ODE theory, for given geometry_071ae907a116ba204861855b392b6ab769d9b38a.png and geometry_d4539e0902e9ed7ed5bc3c7f4f9c701b6114bed4.png and initial conditions

geometry_231a0a20dc4e561820f29132fcd254eb3e2e4690.png

there exists a unique solution

geometry_1607c9ef1f2786e8520d3fd986b2a8139c4f877b.png

to the ODE system, and hence it must conicide with the Frenet-Serret frame.

There is then a unique curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png satisfying

geometry_4326f9d78bad27eb8bc99659b2835e0920bd25ae.png

Equivalence problem

The equivalence problem is the problem of classifying all curves up to rigid motions.

Uniqueness of biregular curve

Let geometry_071ae907a116ba204861855b392b6ab769d9b38a.png and geometry_d4539e0902e9ed7ed5bc3c7f4f9c701b6114bed4.png be given, with geometry_071ae907a116ba204861855b392b6ab769d9b38a.png everywhere positive. Then there exists a unique unit-speed biregular curve geometry_009bb9c65df50f255d517020a4b755226c9fc06e.png with these as curvature and torsion such that geometry_bb1d7506c3c1268d194a35606c9dc9e59da1d216.png and geometry_ce81b575fea2556a54aa96148ff62d8e0dd1214c.png is any fixed oriented orthonormal basis in geometry_455d05a6d8b058d1856814e9128d0f920d211517.png.

Fundamental Theorem of Curves

If two biregular space curves have the same curvature geometry_b992266287b82994936475115099d107b16c51ca.png and torsion geometry_6eb95865ccf7f3fc048eb8bcc06c72901a0a7724.png then they differ at most by a Euclidean motion.

Tangent spaces

Orthogonality

geometry_77b7eb9548d2e7f9da07eb2706981cddd90f013f.png

Change of basis

Suppose we have two different bases for a space:

geometry_43de912447a8cd1a31260c216e08c8227cf48c4c.png

we have the following relationship

geometry_ebbdf42da56cdd720e92f6378d0530fd37b992a6.png

and for the dual-space

Implicit Function Theorem

Let geometry_58c02ddebefdb61ed707129a1061d669d4f2968b.png, where geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png is the base and geometry_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png is the "extra" dimension.

Let geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png be an open subset of geometry_6368e2fc21dd7cde355beeb3b2e21e4bfefb242b.png, and let geometry_761ca123def38ba014af79cd9c91e8cda1eda2e2.png denote standard coordinates in geometry_6368e2fc21dd7cde355beeb3b2e21e4bfefb242b.png.

Suppose geometry_366c766096aff2921420fe2b74651386ba136a14.png is smooth, geometry_6e01e4d29db37040937b354b9295fab423841795.png with geometry_11726898f555d18ee80836e938b072ffef7e1a68.png and geometry_e453b34317e1849a3fd0f20af5720f3e33a20ef0.png, and

geometry_18fc05e74be8ddd405efc742c7b5e057ae7bf295.png

If the geometry_40880b6ec9a59e60c6b691eab59baf5e8d76b3ce.png matrix

geometry_c54a39ecf71016e2e4c6ed7fe3677bcd48b78c27.png

is invertible, then there exists neighbourhoods geometry_e17e33b9a8959c31a9c6d8194ad7f548f5a8f7a7.png of geometry_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and geometry_de7cbd8ef8ad0943d51a187479192033ca004929.png of geometry_f8ba0bcc022c6d477cbfde325f315454468205bb.png and a smooth function

geometry_d2f526c63e7258dfafd4631b077af3f8cd6de4fb.png

such that:

geometry_36dcfd855a9b44a0c8b0917b73e2dc8d1f68d017.png

is the graph of geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png.

Or, equivalently, such that

geometry_cb5791ea842a4719860380808c472f38c3cf286c.png

I like to view it like this:

Suppose we have some geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png dimensional space, and we split it up into two subspaces of dimension geometry_6f51fce2570d7ad3b80f479fd51470ee5a78d81f.png such that

geometry_be1b38ec555514c1cd1a0ba75b0b90172614ef2c.png

Then, using Implicit Function Theorem, we can simply check the invertibility of the Jacobian of geometry_05594e8ce8f52728892c4c673d8bc28149b4b24d.png, as described in the theorem, to find out if there exists a function geometry_60a083de8a2af889292243a8cf450f17bc6d26c4.png where geometry_e17e33b9a8959c31a9c6d8194ad7f548f5a8f7a7.png and geometry_de7cbd8ef8ad0943d51a187479192033ca004929.png.

Where geometry_366c766096aff2921420fe2b74651386ba136a14.png, where geometry_a612c3cc50b065407489432f7f47311ee9fda09f.png, is a map "projecting" some neighbourhood / open set of geometry_6368e2fc21dd7cde355beeb3b2e21e4bfefb242b.png to geometry_967e180ce96954152b14ff0e07264fce177b3b29.png.

Example

Consider geometry_42818c8ce1e42b57391f6ca31faab3ca421e7d43.png and geometry_3fd9def7c793bd1606357c77c9449e235c8b2143.png. Ket

geometry_e6ed717d571486f1b0bbab63e94d9b91e63ecf2c.png

Then,

geometry_361f3993fe5ff354f7a1d125adcdf2988b4d67f0.png

Consider geometry_73a8ffee2b7e83ebfc9fd00be8cd3a33cac6882e.png and geometry_5d03a80607886cc76958690c2e1b9e4f09622360.png, and geometry_b50065f7b59d40eba0000eb2480dd89f5fee9345.png.

Thus,

geometry_2cfc70596b3854d284143860ca233049f23412e9.png

Thus, in the neighbourhood of geometry_8a89c460fad56a68a765f42cd59ad8c9016fa0b3.png in geometry_e4f375c26796781f71b7ae3026445db617a6e78b.png we can consider the level set geometry_960f55fae2df67a0c8ccefeb414bcee57ea19b4e.png and locally solve geometry_eb83d466c7d035356e9f39998f357cee73da1e26.png as a function of geometry_cfa691d59352dd23896c554a013c7254335640c4.png, i.e. there exists a function geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png such that geometry_6892ae6db3bbedf1c9728c627eeacd89bca4edc4.png.

Inverse Function Theorem

Let geometry_32c40fbd72518cf577d939959fa153205707ed70.png be a differentiable map, i.e. geometry_eb0200f5675385d11821070e9a7525c1d001f57b.png. If

geometry_e5a93a671013e5e945b20eb7e28787132c3d48e0.png

is a linear isomorphism at a point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, then there exists an open neighborhood geometry_5325f52c666f3f3d655c6e033df7b6983f652e5b.png such that

geometry_839cff7d182c28721fe72a9c150b8cb2c99c4652.png

is a diffeomorphism.

Note that this implies that geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png must have the same dimension at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.

If the above holds geometry_27d8a4057e99abe766632534767e53569717eb77.png, then geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is a local diffeomorphism.

Gauss-Bonnet

Let geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png be an oriented closed and bounded surface with no boundary. Then

geometry_4e23c0a3d496f157d6dd92c401438a50a6c9aab4.png

where geometry_ac0e5b46b9e13e7109a1940bcc4a0ee0e4004647.png is the Euler characteristic of the surface geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png, defined as

geometry_c0555b6a30b8886ef4da53c1ecc39de39ea3173a.png

where geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png denotes the vertices, geometry_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png the edges, and geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png the faces obtained by dissecting the surface geometry_017948b866be67b1a8e56a6b5f8848f823410c34.png into polygons (this turns out to be independent of choice of dissection).

Change of basis

Notation

  • Einstein summation notation
  • geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is a K-vector space, i.e. vector space over some field geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png
  • geometry_88eded808a21ab6cb7281eb507d556a108b63e30.png denotes the b-th basis-vector of some basis in geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png
  • geometry_17165a6a8c2a5c077edce38f179354db0b748076.png means isomorphic to
  • geometry_acced656eeb8aa11f6ce33bad648d78b6052a7a7.png denotes a tensor of order geometry_a3b6bdd6c50d8c2f4ee3e210c7ecc1524cd5c581.png

Stuff

Suppose we have two different bases in some K-vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png:

geometry_709ec026798c1f7ff3a27185dc4e35adbe01c8d6.png

Now, how does the this affect the 1-forms / covectors (contra-variant) change?

k-forms / covectors

Let geometry_e29af05574032ace665d996d46b3280fc49866ef.png be a covector:

geometry_85da6bdcdec45587d29afd87aab416be807e4db4.png

where geometry_b81499e04453389e140628b02298f95b9d3ac5ec.png denotes the m-th new basis!

geometry_1f7804ce6d40598adb522bf529237f6cfcb14a59.png

is true since geometry_e29af05574032ace665d996d46b3280fc49866ef.png is a linear map by definition.

vectors

geometry_a93060093653dd983fae5651e9e0e736e8d384d2.png

where the only thing which might seem a bit weird is the

geometry_0eed6975daeda9b15906143e0bad84b2c84424d1.png

which relies on

geometry_7c98d2b28a2a32ee6392c114c981ad957fdecf44.png

where

geometry_697766e76017fd9f1998cf4a6ddb23c08aa386d4.png

i.e. geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png is isomorphic to the the dual of the dual space, which is only true for a finite basis!

Apparently this can be shown using a "constructive proof", i.e. you build up the notation mentioned above and then show that it does indeed define an isomorphism between the vector-space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and the dual of the dual space.

Determinants

Notation

Stuff

Problem with matrix-representation

A matrix is a geometry_d3257a47827f15c036745363886df8d3e604d4ef.png tensor, and we can thus write it as

geometry_3a2ae25bf3d4f5de1ae040f88f06fd517b4d29cb.png

where geometry_507254e830909823544c4de69548fd2adeb3234d.png means that we can write out an exhaustive representation of all the geometry_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and geometry_f8ba0bcc022c6d477cbfde325f315454468205bb.png entries in such a way (in this case a matrix).

Also, it turns out that we can write a bilinear map as

geometry_274529076ce373d67886a1ddd27eb71800e0a070.png

See?! We can represent both as a matrix, but the way the change with the basis are completely different!

The usual matrix representation that we're used to (the one with the normal matrix-multiplication, etc.) is the geometry_172d213d790d0d2d4b0b1f02398fd51111354995.png tensor, and it's an endomorphism on geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, i.e. homomorphism which takes a vector to a vector.

Definition

Let geometry_9d0d324b066d6efcdcf2933af716111abea211d1.png. Then

geometry_46b1739a7e129ea247e2c005cf9a822646c0e294.png

for some volume form / top-form geometry_e29af05574032ace665d996d46b3280fc49866ef.png, and for some basis geometry_07c864825120fc50d2f7d9aff6b65e60be3922c9.png of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, i.e. it's completely independent from the choice of basis and top-form.

Due to the top-forms being equal up to some constant, we see that any constant in the above expression would cancel.

Topological Manifolds & Bundles

Tangent space and manifolds

Notation

  • geometry_4ade21757ee8fe8568d0be314491d1b70591f16c.png, that is a linear map from the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png, often called a chart map (as it is related to a chart geometry_a878f920ea298ab71f116d6960f416bd44aa38d0.png)
  • geometry_fbdc03bb4a5be7df8a46f84ef16c5802838f120e.png
  • geometry_8d2de7de783547a4db469f9aca6caf664e61631c.png
  • geometry_e79a0016f758fe16a2ad3a361d24a32f6a57dd56.png is a smooth curve, i.e. a smooth mapping taking in a single parameter and mapping it to a point on the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png
  • geometry_528c4d0a0311276d6929e87ccc43be670673eadf.png is the partial derivative, which at each point geometry_3d7acf4a9735d0ae577221d4fc837a246cd5e92b.png (since geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png) for some function geometry_b7f4da153b659373124de1abb0ae02c2ffcd49ba.png we take the partial derivative of geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png wrt. a-th entry of the Cartesian product geometry_750cf1cb750664ccd6b30e3257d45fcbc4cc0473.png (n times)

Stuff

We define a new symbol

geometry_f1c5659c50dca119d6d3cb6993f819edf25bb837.png

that is

geometry_a17f9db196ba7189815eb22dc2fed624ede01170.png

Why is this all necessary? It's pretty neat because we're first using the chart-map geometry_40fb973cd997849a029e392c385d32d4b8c40196.png to map the point geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png to Euclidean space.

Then, the composite function geometry_156229a7202f4e53701f96be5453158a1f560235.png

The tangent space geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png is an n-dimensional (real) vector space.

Addition structure: Consider two curves geometry_58408b77dac4de3232899b7905cb21260a3efa57.png in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png s.t.

geometry_86e2146269d5175a3b5162d70fa48b244cd77d37.png

with tangent vectors geometry_157dff168e895c5151648a60490f89c4dc313376.png and geometry_a1db8db88c20da5bff452620e19fc40fa624a332.png. We let

geometry_11a1faa904126790d9070f932d5c82632506dbf4.png

Need to show that geometry_91dc6504c3f20cea8cb22b05de65185de5f6342e.png curve geometry_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png, s.t.

geometry_5d02a89cc1304ca3f2ad136decf3adabf88c130e.png

Let geometry_9a33431a12f593f1dce98eea86510a08714e74d9.png be a chart, geometry_2eb23fb487bb1fe1a691ea5610b0682a5137bce2.png. Then we define geometry_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png by

geometry_bf0e36324370c9b4280895492cce662a893f61c5.png

so geometry_9601daae78faa41e0fa8a58042f64badfef7f9aa.png and

geometry_90b058221451fd47b1d71aa840837aea346e682a.png

where geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png are the components of the chart. Tangent vector geometry_ffc094702275ba71f91c573bf9490cc643285973.png to geometry_e2697d80120f3dc874ffb73b07cdc68259d6fd48.png at geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png:

geometry_bcfe3e3f8f28004ace3ed5373057ef85dcbc2ce4.png

where we have used the fact that

geometry_ab85524c7b583cfce4e5a5181b4c101803ad8bf3.png

N-dimensional follows from Theorem thm:tangent-vectors-form-a-basis: They form the basis

geometry_a6170600658a384e0ab78d336e0c77fd970af7c2.png

thus geometry_f90d98a5c7654f7ce9705e07806b140f0483d93a.png.

Construting a basis

From above, we can construct vectors

geometry_c99d6a2a7579c2da4bdf5e7884f61dc32216f2b2.png

which are the tangent vectors to the chart-induced curves geometry_a1df9805cab47e8e391595595c9f11d265f916d8.png.

Any geometry_81f7f869f08e2faebdacb41f455fde6c446b913c.png ca be written as

geometry_d778721023fb81cd1ad8fc765150c438506ed865.png

where we're using Einstein summation and geometry_7cf66a4c3f602dfcf65c4a37701fa282ffb36c4f.png refers to the s-multiplication of a in the vector space.

Further,

geometry_cbc6e355ab4b32c9883769942f6921c617c9fcc1.png

form a basis for geometry_b91a39240d7f4085e0f9f1fd5f11332c950a65fa.png.

geometry_fc4cf2e8c09bab304aad633de809621d5edf08da.png, geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a smooth curve through geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png

geometry_aea97679bb0593f34223d3e3dae1399160beaf2b.png

Then the map geometry_f7f8a2bfa9fa546c63b9197c490c46d071ae4002.png, which is the tangent vector of the curve geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png at point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png given by

geometry_dfab7eb98a530136a0ce27eabe11b6e30a29524d.png

Then by the chain rule, we have:

geometry_e423cc25d2019c59f1b38f71e066603b3c919b32.png

where we've used the fact that geometry_bce5c31a31b12d2c01f6787482334c3f81ce8fd5.png is just a real number, allowing us to move it to the front.

Hence, for any smooth curve geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png at some point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png we can "generate" the tangent of this curve geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png at point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png from the set

geometry_5b3f85227a53755d41ec8d3d31d422c0164af320.png

which we say to be a generating system of geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png.

Now, all we need to prove is that they are also linearly independent, that is

geometry_1c8fe9102a57a0a5879ea0636de829fb61e7bd01.png

This is just the definition of basis, that if a vector is zero in this basis, then either all the coefficients are zero or the vector itself is the zero-vector.

One really, really important thing to notice in this proof is the usage of

geometry_dfab7eb98a530136a0ce27eabe11b6e30a29524d.png

where we just "insert" the geometry_fc987b1f5ef46a9a3fcef6f95df615d860b33788.png since itself is just an identity operation, but which allows us to "work" in Euclidean space by mapping the point in the manifold to the Euclidean space, and then mapping it back to the manifold for geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png to finally act on it!

Push-forward and pullback

Let geometry_32c40fbd72518cf577d939959fa153205707ed70.png be a smooth map between smooth manifolds geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png.

Then the push-forward geometry_0a732d0dabac71257af68537632e005b1d489624.png at the point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png is the linear map

geometry_8a8dae51d77e9b43c169a8ac46f636379e7392b7.png

which defines the map

geometry_2a9dc5b42107a847c0d8fb8df4b7d11e4aa3b3a4.png

as

geometry_483003965124b6dda4a09870d60d25be3d0b33de.png

where:

  • geometry_5f942a94f81f873c28b7702351f3a4a6ae7647dc.png is a smooth function
  • geometry_81f7f869f08e2faebdacb41f455fde6c446b913c.png
  • geometry_3e32e2ebf918528d254984db961ec86f223676f6.png
  • Elements in geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png and geometry_18664241fca2396ad2cf895abe6cf760df9af2e1.png define maps of functions, hence we need to apply it to some function to define it's operation

A couple of remarks:

  • geometry_10b2c611fba6546e3a227128ef0e8e4ad9b02430.png defined as above, is the only linear map from geometry_12f8c0300fdd592772cca2d1e7c55beefcf0f55d.png to geometry_4b06ea43f4bb50fc1fe15bc83f52cff1b7db4d12.png one can actually define!
  • geometry_10b2c611fba6546e3a227128ef0e8e4ad9b02430.png is often referred to as the derivative of geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png at the point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png
  • The tangent vector geometry_0ad3aae3bbd5fb362e656ec7f8b11b8274a782c2.png of the curve geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png at the point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png in the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is pushed forward to the tangent vector of the curve geometry_1113aee4d2d96b35e732cd2269bb87590315cfa2.png at the point geometry_dee21f1860f432c0da5f9661b7559b68c99feb55.png in the manifold geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png; i.e. for a curve geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png we map the tangent vector at some point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to the tangent vector of the "new" curve geometry_1113aee4d2d96b35e732cd2269bb87590315cfa2.png in the target manifold at the resulting point geometry_dee21f1860f432c0da5f9661b7559b68c99feb55.png

Let geometry_32c40fbd72518cf577d939959fa153205707ed70.png be a smooth map between smooth manifolds geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png.

Then the pull-back geometry_50bc0cf9e45be601deec8d55e0e5e454cec8e09c.png of geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png at the point geometry_b424e7f6d02ff6a0436dff6a9b9b2c8fd61a8f11.png is the linear map

geometry_f333be5bada4ace2e9a820f59da6d2dc0cd68ac6.png

i.e. a linear map from the cotangent space at the target TO the cotagent space of the originating manifold at point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png!

We define the map

geometry_bb990fb41cd8c9292e067d77819d2b96877925c3.png

as, acting on geometry_81f7f869f08e2faebdacb41f455fde6c446b913c.png,

geometry_8aa59bf8b94b62adb57fefdfacaf473a79e128e7.png

which is linear since geometry_10b2c611fba6546e3a227128ef0e8e4ad9b02430.png and geometry_e29af05574032ace665d996d46b3280fc49866ef.png are linear, where geometry_10b2c611fba6546e3a227128ef0e8e4ad9b02430.png is the push-forward.

Comments on push-forward and exterior derivative

First I was introduced to the exterior derivative in the Geomtry course I was doing, and afterwards I was, through the lectures by Schuller, introduced to the concept of the push-forward, and the pull-back defined using the push-forward. Afterwards, in certain cotexts (e.g. in Geometry they defined the pull-back of a 2-form on geometry_06c52c91865c1b80c39679525c58e32834eaf132.png involving the exterior derivative), I kept thinking "Hmm, there seems to be some connection between the exterior derivative and the push-forward!

Then I read this stachexchange answer, where you'll find the following snippet:

Except in one special situation (described below), there is essentially no relationship between the exterior derivative of a differential form and the differential (or pushforward) of a smooth map between manifolds, other than the facts that they are both computed locally by taking derivatives and are both commonly denoted by the symbol geometry_f4e5566de1c75b7700b562b91a80d2308cc66d12.png.

And the special case he's referring to is; when the function is a smooth map geometry_fbdc03bb4a5be7df8a46f84ef16c5802838f120e.png, where the two are equivalent.

Immersion and embedding

Let geometry_6e3f4d384482c2d4ba369ff6211b8b60dc82a0cb.png be a smooth map on manifolds geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png to geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png.

We say geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is an immersion if and only if the derivative / push-forward geometry_10b2c611fba6546e3a227128ef0e8e4ad9b02430.png is injective for each point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, or equivalently geometry_c2a6f2dfe302b419125a7c3593ce1c3f59ba1221.png.

Remember the push-forward is a map from the tangent space of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png at point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png to the tangent space of geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png at geometry_dee21f1860f432c0da5f9661b7559b68c99feb55.png: geometry_2119e498acc2a22123767f1ac8fe91250c735281.png. We do NOT require the map geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png itself to be injective!

Let geometry_6e3f4d384482c2d4ba369ff6211b8b60dc82a0cb.png be a smooth map on manifolds geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png.

We say geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is an embedding of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png in geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png if and only if:

  1. geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is an immersion
  2. geometry_d5ff3d82e5b6190264e78264f52c596c9709eea9.png, where geometry_d89c5ed63b9b4346154dcdc1e980f0a8be50820a.png means a homeomorphism / topological isomorphism

Any smooth manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png can be:

  • embedded in geometry_dd8cfba1d06c203285f5960cada4b1fd0862670d.png
  • immersed in geometry_996ef06dc49b608fa271472d57c33f35b8e54873.png

Where geometry_23e7b3f838a0ddf35bb49ea3b136460606321796.png.

This is of course "worst-case scenarios", i.e. there exists manifolds which can be embedded / immersed in lower-dimensional manifolds than the rules mentioned here.

There exists even stronger / better lower bounds for a lot of target manifolds, which requires slightly more restrictions on the manifold.

Let geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png me differentiable manifolds. A function

geometry_f1c367daface0e0e4947a0395a5c9725279df447.png

is a local diffeomorphism, if, for each point geometry_2e159c5b3380548dcf6ab328cbfe3f5f22215d25.png, there exists an open set geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png such that geometry_11c99107ff0fedc65d9b30628f6d8cec4aace605.png, and the image geometry_1976f090dcc913914a17767ee18e66199f2622e8.png is open and

geometry_03608a7a181ab2143b3fc744138a01f05861f4ba.png

is a diffeomorphism.

A local diffeomorphism is then a special case of an immersion geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png from geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png to geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png , where the image geometry_1976f090dcc913914a17767ee18e66199f2622e8.png of geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png under geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png locally has the differentiable structure of a submanifold of geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png.

Example: 2D Klein bottle in 3D

The klein bottle is a 2D surface, as we can see below: klein_bottle.png

But due to the self-intersecting nature of the Klein bottle, it is not a manifold when it "sits" in geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png. Nonetheless, the mapping of the Klein bottle as shown in the picture, does in fact have a injective puh-forward! That is, we can injectively map each tangent vector at a point in such a manner that no two tangent vectors are mapped to the same tangent vector on the Klein bottle 2D surface in geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png.

Hence, the Klein bottle can be immersed in geometry_c6a2614c59fe1064e78014c5b3ae6360b65058e3.png but NOT embedded, as "predicted" by Whitney Theorem. And the same theorem tells us that we can in fact embed the Klein bottle in geometry_8c06529d7bf2008e2a7467207a3fff55abe292c9.png.

Tensor Fields and Modules

Notation

  • geometry_29dabe65c56073bc7d19675c2e230460c7e2b95d.png geometry_00b6a899c2814a48844cca77daa5543cc808b3a0.png where geometry_260fb19d62cd1add754a80212011f7883ed38dbe.png denotes the tangent bundle of the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.
  • geometry_e4ea1480f7b8c487ea65a66e57a574fc8c8a0307.png is taking the Cartesian product and equipping it with addition

Stuff

Let

A vector field is a smooth section geometry_de32b094129e4e03f562577e6b8b51603d8ca54e.png of geometry_260fb19d62cd1add754a80212011f7883ed38dbe.png, i.e.

  • geometry_8160dd2c98e4eff5a90cae4b2c5d578903dcab3c.png is smooth
  • geometry_1a8ce5256cda8973aaa2efc21409b1e5a71459b4.png is smooth
  • geometry_8142dc77548e67f88770b343909c04db22d6ea15.png

Informally, a vector field on a manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png can be defined as a function that inputs a point geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png and outputs an element of the tangent space geometry_bc21162c8303a508184c08031333210cfa24e8d5.png. Equivalently, a vector field is a section of the tangent bundle.

Module

We say geometry_b6ed2477b462f72603a429f4acfc32df32405e8a.png is a R-module, geometry_3d136f0fc4860468633907421c098b9feb0eef24.png being a ring, if

geometry_1955a43801f9015171a421e9a261a0de97e0b4b5.png

satisfying

geometry_ba367196ffd133919728e92184ca67f54a444453.png

Thus, we can view it as a "vector space" over a ring, but because it behaves wildly different from a vector space over a field, we give this space a special name: .

Important: geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png denotes a module here, NOT manifold as usual.

If geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a division ring, then geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png has a basis.

This is not a geometry_bd3cc82fdfe9254196ef619a200cd447f48fd25d.png but simply says that we guarantee the existence of a basis if geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a division ring.

First we require the Axiom of Choice, in the incarnation of Zorn's lemma, which is just the Axiom of Choice restated, given that we already have all the other axioms of Zermelo-Fraenkel set theory.

Zorn's Lemma: A partially ordered set geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png whose every totally ordered subset geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png has an upper bound in geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png contains a maximal element.

where:

partially ordered

Every module geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png over a divison ring has a basis

Notation
Theorem

Every module geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png over a division ring has a basis.

  1. Let geometry_502742c36950391175e8d93f60fb470e61022274.png be a generating system of geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, i.e.

    geometry_ac1da6e96e2581c80cd88c15bd2a6c7304c9f45c.png

    Observe that geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png, the generating system, always exists since we can simply have geometry_609fdd1149ffdd579ee9ed6ddb6edb9d6cf87f24.png

  2. Define a partially ordered set geometry_66ad96beb38e7773c60a5b64b264ae3072c7f2d6.png by

    geometry_c7432a160a4b17dee3c6e943c42e62bd03780965.png

    where geometry_f3efaf7ada91d90eb34ff42ad3b1d689e04b63b0.png denotes the powerset of geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png. We partially order it by inclusion:

    geometry_c040fe3f48008058c50d919a74446fecb03bc552.png

    i.e. if a set is a subset of another, then the it's smaller than the other subset.

  3. Let geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png be any totally ordered subset of geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png, then

    geometry_8d9f7d602a738ddf4588980811f0e7d2642025af.png

    and it is a lin. indep. subset of geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png. Thus, by Zorn's Lemma, geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png has a maximal element, one of which we call geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png. By construction, geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png is a maximal lin. indep. subset of geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png.

  4. Claim: geometry_d920935e62b946feffd15dc24bdcd3be9a78c9f8.png Proof: Let geometry_bc0f4df606dc91711d1c8a45e9c8915ba102415f.png. Since geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png is maximal lin. indep. subset, we have geometry_e28cce2c64338d9a43afac1b624e4281562b163b.png is linearly dependent. That is,

    geometry_2758863507ca196114abfbe0bfba3ab32a05905a.png

    and not all of geometry_4891051ca8737bcf6818e068192a42124d7c1e81.png, geometry_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png vanish, i.e. geometry_18e5a396975ded646551dead3146b93113210ab9.png. Now it is clear that geometry_360a2a8fa66955238dd4e460ad32f9a1dfd0b555.png, because

    geometry_8773c330682b1b793d1cdb38595545af48ddc288.png

    but this is a contradiction to geometry_1d5e6286accf9787576dea43292fa310b94cde18.png being linearly independent, as assumed previously. Hence we consider geometry_360a2a8fa66955238dd4e460ad32f9a1dfd0b555.png; then, since geometry_b6aebfbca83a3804326e206b0441e4315f9c6de3.png (remembering that geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a division ring)

    geometry_2b798bdb5f169d773c643d54e44ebbc7b8aca2bc.png

    Thus, if we multiply the equation above with the inverse of geometry_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png, we get

    geometry_7c3faf93d94e967b1a25b93ed62f55f7ca8c0398.png

    for the finite subsets geometry_4bb83bddea383e12e1bf04c1b698cf4bc8455872.png of B. Thus,

    geometry_c65ab1396d06add4882ac23e67ac08c152c8aa7a.png

    Hence, we have existence of a linear indepndent subset of geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png which also spans geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png.

As we see above, here we're making use of the fact that geometry_a9d4a02ab56ca60e657921091aa6c4c80c08e8f7.png is a division ring, when we're using the inverse of geometry_4c2e3ef0a630ca9217f04701920488b851892c87.png.

Observe that geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png is not a division ring, hence geometry_0f5d434871fe6e2cb9e27ca6fa367bd8d8b5c3de.png consider as a geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png module is not guaranteed to have a basis.

Definition of geometry_0f5d434871fe6e2cb9e27ca6fa367bd8d8b5c3de.png can be found here.

Examples

geometry_f89c92349319aa0b3df52f1e86b5d9fa1c767c5f.png module

One simply example of a module is

geometry_804746b0a296fcc11d4da12cabb384ca15573f3e.png

where:

  • geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a manifold
  • geometry_1a8ce5256cda8973aaa2efc21409b1e5a71459b4.png is the projection
  • geometry_0f5d434871fe6e2cb9e27ca6fa367bd8d8b5c3de.png denotes the set of all sections of geometry_260fb19d62cd1add754a80212011f7883ed38dbe.png, i.e. the total space of the bundle

which is a geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png module.

Terminology

A module over a ring is called free if it has a basis.

Examples:

geometry_0b30b0eb306a4c75a7f45980846642db659a6956.png

A module geometry_12b2a1240e0f4c77cdd07ea0cd6ddc2b4e98482a.png over a ring geometry_3d136f0fc4860468633907421c098b9feb0eef24.png is called projective if it is a directed summand of a free R-module geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png:

geometry_3e504fd7b6558fed4408d1aee66733d5c44b6445.png

where geometry_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is the R-module.

Remark: free geometry_85a48dfbdbafa5eb610a72628a4edeb4106be807.png projective

Theorems

geometry_a7aaedd52cb28154fdbb510111e9d9eaffcccd29.png

is a *finitely projective geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png module geometry_2d438540baab92f5646fb2fbd469e641c9194ebb.png.

From this have the following corollary:

geometry_e50d8abf72f7133928b10d96eca5ba6b39c1fc01.png

where geometry_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is the geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png module and geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png is a free module.

Thus, geometry_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png "quantifies" how much geometry_2d438540baab92f5646fb2fbd469e641c9194ebb.png fails to have a basis, since geometry_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is how much we have to add to geometry_2d438540baab92f5646fb2fbd469e641c9194ebb.png to make it free.

Let geometry_1bf2edcfec28caa18def50379d92f604d084d6e5.png be finitely generated projective modules over a commutative ring geometry_3d136f0fc4860468633907421c098b9feb0eef24.png.

Then,

geometry_1e0172bdf6e505e212ddeb6d416102e61d9cf02b.png

is again a finitely generated projective module.

This falls out of the commutativity of the ring geometry_3d136f0fc4860468633907421c098b9feb0eef24.png.

In particular:

geometry_829532cb34e2b600d4445bd4fda5393bb5a080a3.png

where the equality geometry_ab9438ee708461adefa543603801c274ee3dffb7.png can be shown (but we haven't done that here).

Finally, this gives us the "standard textbook definition" of a tensor-field:

A geometry_b322efb64a92d1334000ba99d97d7a0b6a44ce19.png tensor field geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png on a smooth manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png multilinear map

geometry_b6dd77f3cc0e4358fa76abad21481e39f2407527.png

We can then view

geometry_3e277b14631d48b5fa8399f6f74e448784c86242.png

as the space of all tensor-fields on geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png, which again, forms a module!

Hence, we when we talk about the mapping geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png being multilinear, of course it must be multilinear to the underlying ring-structure of the module, i.e. geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png multilinear!

Some textbooks give the above definition, and then note that geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png is not geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png linear, but geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png linear, and that this needs to be checked. But because we're aware of the commutative ring that "pops out" we know that of course it has to be multilinear wrt. the underlying ring-structure, which in this case geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png.

Grassman algebra and deRham cohomology

Notation

  • geometry_9095d957da7c7ce0318cf153c5b286f73c6b2b4a.png, i.e. maps in geometry_f89c92349319aa0b3df52f1e86b5d9fa1c767c5f.png s.t. composed with the projection from the total space to the base space of the tangent bundle form the identity. That is, geometry_de32b094129e4e03f562577e6b8b51603d8ca54e.png maps a point in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png to it's fibre in geometry_b91a39240d7f4085e0f9f1fd5f11332c950a65fa.png, known as sections
  • geometry_9e283f0a770977b9c5776238e85e4b1f2406cdd4.png denotes a permutation in what follows section, NOT a projection as seen earlier
  • geometry_ddf7ef8c1e82c2029c7ecf8062d572a8176b8d26.png denotest the set of permutations on geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png letters / digits
  • geometry_7c0fc9ef5775573914c9b11dc14cb1cf6255ba0b.png is the tensor product
  • geometry_c2192c2c6fb7f1f01ac368ecf92632005c2e0c41.png is called the anti-symmetric bracket notation, where geometry_fae8cf5a2f59dc3509ad87b158201ad79abd07bd.png denotes some "indexable object"
  • geometry_58173d06787b3b40b4ef5b126ba6e0046a4435ab.png is the same as anti-symmetric bracket notation, but dropping the geometry_56f7e5d823c069762dcb456cd9acc3dd7596ca5b.png, we call symmetric bracket notation
  • geometry_f83b6255ac2c6f03862c5edae9fbb450f09cc662.png

    where geometry_edf6fcd229c98784f98876546112b91104d73bff.png is the exact forms and geometry_e44b7b8e028d4222bfb5c27037c61b3d5c8bb825.png is the closed forms, where geometry_4d9f4fcc47839a0ceaea65d4a9c5bd633766731f.png denotes the previous and geometry_da69e7d6dd5a11a2c8d2617c0f4d3aacbf27e84a.png the next

Grassman Algebra

The set of all n-forms is denoted

geometry_969d15fadfb2b82a623a1bf9ee350da71e893b2e.png

which naturally is a geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png module since:

  • sum of two n-forms is a n-form
  • geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png multiple of some geometry_44b70af46d1656bbc572fdd3e0f72ba5ce06d8ee.png is again in geometry_13862c558d9c345e997847bdc242f5c6939153bb.png

We have a problem though: taking the tensor product of forms does not yield a form!

I.e. the space is not closed.

In what follows we're slowly building up towards a way of defining a product in such a way that we do indeed have the space of forms being closed under some additional operation (other than geometry_2d9eee9ed3f0530ef8fe709062f9dca828344e68.png and geometry_7cf66a4c3f602dfcf65c4a37701fa282ffb36c4f.png), which is called the Grassman algebra.

We define the wedge product as follows:

geometry_858051fa6c667e231fa5750589f0c32bf944dc3f.png

geometry_f77ca06899e31bc5e559aa9e25e3f1f66ee4247f.png

defined by

geometry_bb7cd115140cd28826f21f2b99302c1d6b6492b8.png

e.g.

geometry_90ca277bde4d3a8619011a91eed1fcdf87ff72d2.png

where the tensor product geometry_7c0fc9ef5775573914c9b11dc14cb1cf6255ba0b.png is just defined as

geometry_63ab243c053dd6b29fd08c1e1886b99b0f30d4fc.png

as usual.

Further, this allows us to construct the pull-back for some arbitrary geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png form:

Let geometry_7ae6a28f7998699517d6010960b741f955228c0a.png and geometry_a787b09fcedc51f145da8e5220fedaf3275d76d0.png be a smooth mapping between the manifolds.

This induces the pull-back:

geometry_7c059bc30ed935271cacb744375d24960f806df4.png

can be used to define

geometry_bdba15ab36133ff947b37c728e31f4d9b4ba430f.png

where geometry_4df3b2e6f4d016949cb6f13101592a05de0ce64a.png, which is the pull-back of the entire space rather than at a specific point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.

Then,

geometry_f285c4b26d10a30829d3b21db264abba3e64663a.png

where geometry_5014ac0478ec857c752d1ae9004b5b58446cc709.png is the push forward of a vector field.

The pull-back distributes over the wedge product:

geometry_bfe42af04bcc23fed08c1afdaf9edc90ff9f4f3b.png

The geometry_e6cbf658dc33c85508f2364f13d23dcb5225aa32.png module defined

geometry_00fff1a797e9c0dc5dcd69f872164a869fba4f5d.png

(where we've seen geometry_ca677064fa7d88596266cd14e422a0ddd2398d88.png and geometry_fa1ea6cd5d4f9bcfdf2466cbb107b0263c244dbc.png before!)

Then geometry_21f17b2ce0846d4cbc57b046b07daba16ec9bf0a.png defines the Grassman algebra / exterior algebra of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, with the geometry_879b987f908d17a36565fbdec6d1ee317b72da24.png being a bilinear map

geometry_2da06b6ee632376b49b77d82f4bef99453bf6f1d.png

is defined by linear contiuation of geometry_879b987f908d17a36565fbdec6d1ee317b72da24.png (wedge product for forms), which means that for example if we haven

geometry_70066b7ed3dc86de0a2be00ea1cce3eb200b1dcd.png

where geometry_e4877877380b2b258ab12259645336c67362e98f.png and geometry_e0a19a6ea00225f4b424863ee9068d9c4d64122c.png, and another geometry_962a43b7f6c845806819d2f756d0e291df5ea684.png, then

geometry_c3fa3c1b4bf47940f10ca79af64d0475e2d8f9b0.png

Now, as it turns out, we cannot define a differentiable structure on tensors on some manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png. But do not despair! We can in fact do this on anti-symmetric tensors, i.e. forms, we can indeed define a differentiable structure without any further restrictions on the manifold!

The exterior derivative operator

geometry_e6b795e56843502e8b884d14a742f289e35342a2.png

where

geometry_c1c5a21b92c647a38d2358e78cb50a3d6fcbaccd.png

i.e. since geometry_e29af05574032ace665d996d46b3280fc49866ef.png takes geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png entries we leave out the i-th entry, and geometry_70bb8f881a421d4305cb8401dfa1ad7e8a6307ce.png is the commutator.

Commutator of two vector fields geometry_6157262f7504734f472c73ef69b5993a4fd5e1ac.png is given by

geometry_851962b7772e35dc9bb68118fbded2206f8d8bf8.png

for geometry_41f8eb3d67bdf2b572518e662d7de34c5ce041b8.png.

Let geometry_7ae6a28f7998699517d6010960b741f955228c0a.png and geometry_c787f7b3b9b66ff31c6c0abd3ac046cb7280335c.png.

geometry_d690c7e5a1a6bee4ab116c67bbcf99330e417a56.png

If we have the smooth map geometry_a787b09fcedc51f145da8e5220fedaf3275d76d0.png, then

geometry_70297edbd6a2f1c9b0e425da2e3182469094f236.png

where we observe that the geometry_f4e5566de1c75b7700b562b91a80d2308cc66d12.png are "different":

  • LHS: geometry_6464e60fade03cdd9f3c1c26ffb3198288be7974.png
  • RHS: geometry_f45913df3d0dc7ff830a9b3e68146d63fb7eddfc.png

which is why we use the word "commute" rather than that they are "the same" (cuz they ain't)

Further, action of geometry_f4e5566de1c75b7700b562b91a80d2308cc66d12.png extends by linear continuation to geometry_373e9e0df8c153f4f5ebed85c8926c59b80df734.png:

geometry_c5a68ade28af8a132a61d6b36b5a3289a647d7c8.png

where geometry_373e9e0df8c153f4f5ebed85c8926c59b80df734.png denotes the Grassman algebra.

Physical examples

Maxwell electrodynamics

Let geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png be the field strength, i.e. the Lorentz force:

geometry_98569042b80dc3852f4b69facf066ebdf5c2bde9.png

then geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png is a two-form (since it maps both geometry_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png and geometry_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png to geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png), and we require

geometry_33106603e2edcb7192e33c79def643733ae21c03.png

which is called the homogenous Maxwell equations.

Since geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png is a two-form on the reals, we know from Poincaré lemma that if a n-form is closed on geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png and thus we must exact, thus

geometry_9ec3cecd1b26587696c2088bca664ab78b61c967.png

for some geometry_3985bda3fe8858515a726a83f1c3a30ba09f949d.png. geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is called the gauge potential.

de Rham Cohomology

The following theorem has already been stated before, but I'll restate it due to it's importance in this section:

geometry_6d66b79bfbd951f42f03763c474c9a8b481dd86f.png

where geometry_f4e5566de1c75b7700b562b91a80d2308cc66d12.png is the exterior derivative.

geometry_b4828c21fccbc02edfe8a0385f0765cb772a15c7.png

and

geometry_55d1b6f6b7dee4cc8300030a20f19fc34526dae1.png

in local coords:

geometry_85b904c59458fedabba29d9e757fd25c8d93de6e.png

(remember we're using Einstein summation), where we've used the fact that geometry_8c7062ce3482fe4c3c123ae568f207f7a3b076de.png, which gives us

geometry_fa9dc44b139cb062750b186e167c25c45dd9600d.png

where the last equality is due to Schwartz (geometry_079d2543b3122b60404d3e5fa02cb67cf90cb18c.png under certain conditions).

geometry_42777dcd4fbddf82000c9672c7a274847f00c282.png implies that there exists a sequence of maps such that

geometry_b4584962e973e67eb025b165b3940025b00283d0.png

We then observe that:

geometry_8df6fb9cb5de8195ae2e0a9f4fc94b2ad8b77033.png

where the above theorem tells us that:

geometry_a7f578c61b8e8fe895f35abf811b6efa7ad29176.png

Now, we introduce some terminology:

geometry_59f2d2a58b459f3718b1707b964670efc1eff5c4.png

We then say that geometry_7ae6a28f7998699517d6010960b741f955228c0a.png is called:

  • exact if geometry_2244fbe228c2d8fabecaed7056e68e51c4a2cfc6.png
  • closed if geometry_63e715e71fc897980555d8b5afa0e23a80f8f6d4.png

which is equivalent to the exact and closed definitions that we've seen before, since geometry_10e7397f2b4b9581eef3650ad7da9741324ab9ad.png for some geometry_64b9440c68d5efd9684d59904864bcfb917d3de7.png is exact, i.e. geometry_2244fbe228c2d8fabecaed7056e68e51c4a2cfc6.png. Observe that we here consider geometry_f4e5566de1c75b7700b562b91a80d2308cc66d12.png as a mapping geometry_58a65ada2efd03c31a9ac8d560651d7968f8adf2.png rather than geometry_6ebb86e4b3d0e3671318b30e97079d0a43f5e4f8.png, thus the different colored mappings are sort of the same but sort of the not :)

As we know from Poincaré lemma, there are cases where

geometry_d79d41689439eb6a302105ddfa2f5a8060fbf495.png

but then you might wonder, if it's not the case: how would one quantify the difference between geometry_e44b7b8e028d4222bfb5c27037c61b3d5c8bb825.png and geometry_edf6fcd229c98784f98876546112b91104d73bff.png?

The n-th deRham Comohomology group is the quotient vector space

geometry_407ff4293a6628c6587a2dc55c158793fc8f5bf2.png

where on geometry_e44b7b8e028d4222bfb5c27037c61b3d5c8bb825.png we have equivalence relation:

geometry_d481ee2a7208ee83978556b076549e84b0d0686b.png

and we write (this is just notation)

geometry_4dca47b9211f867324e582efa4fd0971216b5827.png

The idea of de Rham cohomology is to classify the different types of closed forms on a manifold.

One performs this classification by saying that two closed forms geometry_92383d8893aa1c1c00feeba43b31d250a2d36d32.png are cohomologous if they differ by an exact form, i.e. geometry_fda5f59b9f94562ff064781f27c4efcdc2af9b8e.png is a exact:

geometry_11197f5bb07efa77728bc68f545a7ad0a2d6122b.png

where geometry_ec55227b5b6588eab9d91c4b16fa67f563913ec0.png is the set of exact forms.

Thus, the definition of n-th de Rham cohomology group is the set of equivalence classes with the equiv. relation described above; that is, the set of closed forms in geometry_648631dbaa848d77a35ffa28e1898e79e02d94c7.png modulo the exact forms.

Further, framing it slightly different, it might become a bit more apparent what we're saying here.

Observe that

geometry_6064dd79c2ed01b137a5238c86fe549a9a1cfb26.png

Since geometry_7f66cfb57ff359773231ae1e7413fb6e42e39d8a.png, then clearly

geometry_7f89d050f2c2c37206974bdc88eab09ed78ce088.png

And further, it turns out that by partitioning all the closed forms by taking the "modulo" the exact forms, we get a set of unique and disjoint partitions (due to this being an equiv. relation).

That is,

geometry_ee7bb350a2a41fad1302a370f7cd936f725deab0.png

geometry_4c387f9faf467f6f22e90fb1b88c446816c6518e.png only depends on the global topology of the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png

This is quite a remarkable result, since all our "previous" work depends on the exterior derivative of the local structure, and then it turns out that geometry_4c387f9faf467f6f22e90fb1b88c446816c6518e.png only depends no the actual topology of the manifold! Woah dude!

We have the following example:

geometry_ad05a661138b20e0be5373fd2a53aa634ba06236.png

Summary

  • From geometry_42777dcd4fbddf82000c9672c7a274847f00c282.png we get the sequence with inclusions where always the images are included in the kernels of the next map.n
  • Then if we want to quantify how much the images diviate from the kernel, we can quantify by "modding out" the closed forms, geometry_edf6fcd229c98784f98876546112b91104d73bff.png, from the exact forms, geometry_e44b7b8e028d4222bfb5c27037c61b3d5c8bb825.png.
  • We then learn purely topological invariants, geometry_4c387f9faf467f6f22e90fb1b88c446816c6518e.png

Lie Theory

Notation

  • geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png often used as short-hand notation for the K-vectorspace geometry_65a851830e28eaa68b2d9131907bde8028f9bdfa.png when this vector space is further equipped with the Lie brackets geometry_db0604a541d4fb5722354cf5f9b81ac5416b76cf.png, i.e. when writing

    geometry_2544bf3bf705ef111ebd274d836029f107bfc7e1.png

    geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png refers the underlying K-vectorspace.

  • geometry_736b2940b85ee9b6dc127b5e075d2523a41d6a93.png is the set of left-invariant vector fields in the Lie group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png
  • geometry_d939f693c34af14e219a2ab903f58acd4df84b66.png refers to two vector spaces being isomorphic (which is not the same as, say, a group isomorphism)
  • geometry_db0604a541d4fb5722354cf5f9b81ac5416b76cf.png denotes the abstract Lie-brackets, i.e. any function which takes two arguments as satisfy the properties of a Lie bracket.
  • geometry_ca1fd9734446b1235e0941b5bcaff30302543c4e.png denotes the particular instance of a Lie bracket defined by

    geometry_62a3fbe70754cf61d68f813865055620e5e1c243.png

    known as the commutation relation

  • geometry_830e5657e29bb3f164284d41b3f2f8d2140ea40d.png refers to the 0th fundmantal group / path components
  • geometry_cee5a7fdeaf60f4f797c731de63a79e682b0a855.png refers to the 1st fundamental group
  • geometry_baaf7a72f57d34d1ee5b21e1bf6f226107979c92.png denotes the connected component of the identity
  • Submanifold refers to embedded submanifold
  • geometry_473e80d7c423d8bb37184de8b08a345f1997ea0f.png denotes the 2-torus
  • geometry_47ad38e3c2bceecf3262094aa0b5f1ad3c151ae0.png denotes the space of vector fields
  • geometry_55e86b933cadffea157ebf01f3f4adca52f54f15.png denotes that geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png acts on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png as a group

TODO Stuff

A Lie group geometry_a25a208ba09c77f50b364a437d6d12f554f8d011.png is

  • a group with group operation geometry_d6ada3514f11bc60253d88327733eddcc74bc866.png:

    geometry_024f09aaf8cc94576b0d108322317d45aec2f2d4.png

    where geometry_ae4f3efbeee8beb4a9dace660b19e971bcb1c339.png denotes that the group could be commutative, but is not necessarily so.

  • geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is a smooth manifold and the maps:

    geometry_adcb996a6e6c4f866ecac5359dd327621609f06f.png

    where geometry_9d2f223632ec8e57ab4fa01be3032ac1975272c4.png inherits smooth atlas from geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, thus geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a map between smooth manifolds.

    geometry_b57e574c84f082db0b76c8cd6de1e62cc24bde23.png

    are both smooth maps.

Let geometry_a25a208ba09c77f50b364a437d6d12f554f8d011.png be a Lie group.

Then for any geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png, there exists a map

geometry_1c57b4ebf801e9777ab55cde7fb432e91b886b1f.png

called the left translation wrt. geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png.

Each left translation of a Lie group geometry_d9caa147c2c578f84c27c7341d797fc5812dd300.png is an isomorphism but NOT a group isomorphism.

It is also a diffeomorphism on geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, by the definition of a Lie group.

Let geometry_05730b27b27c441938cef65c92a80addd7a92991.png and geometry_de6b911480f53fd52c1b79d0c0a029edf17ffa37.png be Lie groups

If geometry_07586759917e44e85bcf23a24fffb99e34f0ad89.png is a smooth / analytic map preserving group structure, i.e.

geometry_2b3515df784928dd6e13542e005c5f30f97c54aa.png

then geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png is a morphism of Lie groups.

  1. Lie groups do not have to be connected, neither simply-connected
  2. Discrete groups are Lie groups

Let

  • geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png be a Lie group
  • geometry_baaf7a72f57d34d1ee5b21e1bf6f226107979c92.png be the connected component (which always exists around identity) of geometry_1c143ae51231f110cb4bbb7d90bb1ffedd1c9192.png
  • Then geometry_baaf7a72f57d34d1ee5b21e1bf6f226107979c92.png is a normal subgroup in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png and geometry_2634620b6dfdbe4b71bed3a3d6d3871c6777f326.png is a Lie group itself.
  • geometry_5565210e649c6e2c959f5a2c424901cbcd70861d.png is a discrete group
  1. First we show that geometry_baaf7a72f57d34d1ee5b21e1bf6f226107979c92.png is indeed a Lie group. By definition of a Lie group, the inversion map

    geometry_7ac6bf1cbe3c1d2431a52cdbb58b2ab769670a72.png

    is continuous. The image of a connected topological space under a continous map is connected, hence geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png takes geometry_3113152577cea4efa87b8c9bf38db4d5b9c30bd8.png to a connected comp of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png containing the identity, since

    geometry_554f3280749493c57180b05beac071ccb3c4e6cf.png

    Similar argument for geometry_80ba04e53f9b1c9130fb1159d66f65973651f3ac.png. Hence geometry_baaf7a72f57d34d1ee5b21e1bf6f226107979c92.png is a Lie group. At the same time, conjugation

    geometry_a7f81969a277d7644b8fbe6ad3f0447be9e1cc2e.png

    is cont. in geometry_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png for all geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png. Thus geometry_62380477f98814a93e6085acf8e4b0b90182c076.png is a conn. comp. of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png which contains geometry_585b5774f4995982d133baed6ca9d784f121b44f.png since geometry_df71a8e646959edff42b22f39381c00b2502ed32.png.

    geometry_ec460383cde19f7a81311602aa945b70f34ced21.png

  2. Let geometry_85027c5a22fbe8e6d57e215df7811c14565bef81.png be the quotient map. geometry_f63749027365e29025a5cb867262c465d2de65bb.png is an open map (i.e. maps open to open) since geometry_284ca72b17b9eb391da392009e4bf4503a60b6e3.png is equipped with the quotient topology. This implies that for every geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png we have

    geometry_4066569eb475b2a650bebaa370c5da046f42feff.png

    i.e. it's open. This implies that every element of geometry_284ca72b17b9eb391da392009e4bf4503a60b6e3.png is an open subset, hence the union of all elements in geometry_284ca72b17b9eb391da392009e4bf4503a60b6e3.png cover geometry_284ca72b17b9eb391da392009e4bf4503a60b6e3.png and each of them open, i.e. we have an open covering in which every open subset contains exactly one element of geometry_284ca72b17b9eb391da392009e4bf4503a60b6e3.png (which is the definition of a discrete topological group).

Let

FINISH IT!

  1. Show that geometry_1711f7aa36f30608b649c812d06bec4b8385cdf3.png is a Lie group. Let geometry_9dc54cb072169a34d0110a86ac53e24ce75ba419.png be a connected manifolds, geometry_8b77b207c01cd271ddf687a3c8355493504ec78c.png, geometry_611d8e05de896324ce6b117fa5184c95c48ad821.png, and

    geometry_235f6a5bda79b4a6a2d43ca311a0b12110247f56.png

    be cont. geometry_14d0b31da84daf85138e123cdae8b21ea45a9433.png be universal covers, geometry_01398a3feb1148b4802119daf55c50b5ffe46db4.png and geometry_31bd4025aa7cca12c8a5ce9c9aa0ff6b708cba24.png with

    geometry_0c6e3dc6cf2864ecb8c3264ca0afee404d3bbb48.png

    Then geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png lifts to geometry_48b9b7a25bee4092d73e2e4d5c170adbcd70a991.png s.t.

    geometry_f3a57bf257bfa88644537d25d28dfc767907c762.png

    Choose geometry_665800d0a99614a438a89f255ef67a9a1b85dc07.png s.t. geometry_475d38921372b2496d172f3b3dba1961f1374681.png implies that geometry_f8025426fc65a7188bfc2ca8829c950628fe7a5c.png lifts in a unique way to geometry_e16f0da851090057d86b9c8e56821abd61084184.png taking geometry_fa8797be9dd4c9251da8e95334be17b532c3bec6.png. Same trick works for geometry_8b4d428e5c2d9d8b54dcbf413c177b6900cebfbf.png.

  2. geometry_f56d9e857313129777efeff7d6b2781eaaf1d4a8.png is discrete and central

Lie subgroups

A closed Lie subgroup geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png of a Lie group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is a (embedded) submanifold which is also a subgroup.

A Lie subgroup geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png of a Lie group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is an immersed (as opposed to embedded) submanifold which is also a subgroup.

  1. Any closed Lie subgroup is closed (as a submanifold)
  2. Any subgroup of a Lie group which is a closed subset is a closed Lie group.
  1. geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png connected Lie group, geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png neighborhood of geometry_37608566d766623099382589c8db47d09a460da0.png, then geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png generates geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png.
  2. geometry_07586759917e44e85bcf23a24fffb99e34f0ad89.png is a morphism of Lie groups, and geometry_de6b911480f53fd52c1b79d0c0a029edf17ffa37.png is connected. If geometry_6d2fc397164e1c987493b7ff111d8e6ae93317e4.png is surjective, then geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png is surjective.
  1. geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png subgroup generated by geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, then geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png is open in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png because geometry_6f3a2793b8e95df5116e8693eea353a54435c122.png, we have geometry_3ef55d4953115788f08573ea06853d621773fabc.png is open neighborhood of geometry_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png. Then
  2. Inverse function theorem says that geometry_1b77ed6cedb617cc944a7e6059abe9421d80cf0a.png is surjective onto some neighborhood geometry_2fb32fbc679c6b85e94fe78561166111f062d6f1.png, Since an image of a group morphism is a subgroup, and geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png generates geometry_de6b911480f53fd52c1b79d0c0a029edf17ffa37.png, geometry_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is surjective.
Example

geometry_b3cd726fd2c2372524130d3d7c10c7f39e5f4184.png and geometry_47ff01058b13e9967ab864bb2287ed4d839ab85c.png with

geometry_b8821e94ca831b5b0a099cc6369ea3415afe013d.png

Then it is well-known (apparently) that the image of this map is everywhere dense in geometry_08360ce5700ce47dc9d71cb7b70758a33d530677.png, and is often called the irrational or dense winding of geometry_08360ce5700ce47dc9d71cb7b70758a33d530677.png, and the map is open "one way" but the "other way".

This is an example of a Lie subgroup which is NOT a closed Lie subgroup. The image of the map geometry_d67c6bbcc046be9229dd099cfe9ad5d04cee36fb.png is a Lie subgroup which is not closed. It can be shown that if a Lie subgroup is closed in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, then it is automatically a closed Lie subgroup. We do not get a proof of that though, apparently.

Factor groups

  • As in for discrete groups, given a closed Lie subgroup geometry_c71f1b861cc4b4fe9b9da6ebbf7a74d09cf1315c.png, we can define notion of cosets and define geometry_ff11e83f2881f66557ac80c217281a1260d5ae8d.png as the set of equivalence classes.
  • Following theorem shows that the coset space is actually a manifold

Let

Then geometry_ff11e83f2881f66557ac80c217281a1260d5ae8d.png is a submanifold of geometry_0e244f5e69d40d6e023f814480619ea2606b9768.png and there exists a fibre bundle with geometry_95cf89358263c54e996bac87f1b2e022181ab545.png, where geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is the canonical map, with geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png as it's fibre. The tangent space is given by

geometry_e59ad2021babd88e5830f0f0dee019a00767dcdf.png

Further, if geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png is a normal closed Lie subgroup then geometry_ff11e83f2881f66557ac80c217281a1260d5ae8d.png has a canonical structure of a Lie group (i.e. transition maps are smooth and the smooth structure does not depend on the choice of geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png and geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png (see proof).

Let

  • geometry_1d1fcaf56430b502ef4e12e62e65b2f7925c1c53.png be the canonical map
  • geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png and geometry_23209de7f06fb0fb215c2d80e54bb681263c6c7a.png

Then geometry_3e752f066bd3a0a506d12b6f6ce4a61930f27020.png is a (embedded) submanifold in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png as it's an image of geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png under diffeomorphism geometry_64a2090cc26ce246144685999518b65667972e3b.png. Choose a submanifold geometry_7c02e0335a8812884ebabcb2a56c0422508bf25b.png such that geometry_65ae4213776231a8c9db11172d24115b9b1d4bea.png and geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is traversal to the manifold geometry_3e752f066bd3a0a506d12b6f6ce4a61930f27020.png, i.e.

geometry_f36efc7e9fb12358443880a63a6e2e012d930a0c.png

which implies that geometry_f141c024d17cf1b336508b7a0779af1fc1cf012d.png.

Let geometry_ab4862d234eeebd11968ded3b86bcb74e00618a2.png be a sufficently small neighborhood of geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png. Then the set

geometry_a162085e9578b9df2c11838e97a892fa8273a3ea.png

is open in geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png. This follows from the IFT applied to the map geometry_9a4930177f9d8a187a9e255705f696a5be17b7ff.png.

Consider geometry_59aa82b75065da8d91dec9270a159563223a6d12.png. Since geometry_9db9083698c277164e56a5db0649caaba76912cc.png is open, geometry_793079f554edc8e95ab2c3be17ad4e2e038e64b5.png is an open neighborhood of geometry_0f33899f006e3c6682b51b4aa998ae32b1df14c4.png in geometry_ff11e83f2881f66557ac80c217281a1260d5ae8d.png and the map geometry_0eecd726203b1091daf0d51adc39c5ac1ee40e6c.png is a homeomorphism. This gives a local chart for geometry_ff11e83f2881f66557ac80c217281a1260d5ae8d.png by geometry_6b11cdfbd45e8af6a69111335893cf699b7dbe3c.png, where geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png denotes a chart map for geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png. At the same time this shows that geometry_0ad04b9db679b2571235d0da9f6230f8abb72533.png is a fibre bundle with fibre geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png.

GET CONFIRMATION ABOUT THIS. With the the atlas geometry_e6cd80329be296e3cc8305cb4c6955b444234246.png we see that the transition maps are smooth by the smoothness of geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png and geometry_495537152c5f02a795eef2b0a349c2ce4f4d30b2.png. Further, observe that choosing any other geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png and geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png does not alter the proof, since geometry_f141c024d17cf1b336508b7a0779af1fc1cf012d.png still holds, and therefore ???

The above argument also shows that the push-forward of geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png, i.e. geometry_9cec43a5a09e309f3c5c43e7a0a85fff94044871.png has the kernel

geometry_d57523fe85296426184b6f9a994a3b675dde5bf5.png

In particular, geometry_2cff500c928153f776864344b6404abbc47a4871.png gives the isomorphism (since geometry_44b3c384281b78c53ec6679e899a7ce08bdc9de3.png is an isomorphism)

geometry_85790829908f6be535fd75854b7730df4b8f1d76.png

as wanted.

REMINDER: If geometry_95cf89358263c54e996bac87f1b2e022181ab545.png is a fibre bundle with fibre geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png, then there exists a long exact sequence of homotopy groups

geometry_038eac0503bfa9cfe974de7597a7680a01b96b4c.png

Exact means that geometry_9041e6ceb31f3e11fd6219515c3e89d737a244c0.png with geometry_90568fc1f664aa6a120629c09213e8d53d5f4bf6.png.

Let geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png be a closed Lie subgroup of a Lie group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png.

  1. geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png connected geometry_85a48dfbdbafa5eb610a72628a4edeb4106be807.png geometry_86f439d143f987483d910e42937693bca9a44c4b.png where geometry_e9f1264986817367da70a31df4165a59a7ac75f7.png. In paricular, if geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png and geometry_ff11e83f2881f66557ac80c217281a1260d5ae8d.png are both connected, then so is geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png.
  2. geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png connected, geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png connected geometry_85a48dfbdbafa5eb610a72628a4edeb4106be807.png

    geometry_b84228362fef1f2f4782c2c5cc5fd9c4ba5da589.png

Push-forward on fields

What does this mean? It means that on a Lie group we can in fact construct a diffeomorphism using the left translations, and thus a push-forward from geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png to geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, i.e. we can map vector fields to vector fields in the group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png!

We can push forward on vector field geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png on geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png to another vector field, defined

geometry_67b1332d479b1ee4f6c17f6cdceb72b5772353ef.png

where geometry_b08776de587c81c874b913120d81b1a8ef55ab97.png, geometry_11fb7406358d92c731aaa23eb21520c28a593a44.png.

Let geometry_a25a208ba09c77f50b364a437d6d12f554f8d011.png be a Lie group, and geometry_7a84c9a383f9772338016d101ccc096be06af784.png a vector field on geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, then geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is called left invariant vector field, if for any geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png

geometry_4ad7c3c708114187b7498299447dc8771e60c3e2.png

Alternatively, one can write this as

geometry_d515a237bf98c5fe8ea995d5f62d7e52a22c8de0.png

where we write the map pointwise on the vector field.

Alternatively, again, geometry_a22307456a4d8b3317cd8e22e89eb9cd030a5efb.png and geometry_09870da307dfe45805ebc5a359fedb4596ad8ebb.png

geometry_9754c6451f5a0e7b9bc8d03d3a19f1465e928e57.png

The set of left-invariant vector fields of a Lie group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png can be denoted

geometry_58f186e4d6b4e082040bcba389b4a1f7188c614f.png

where geometry_3d4e3bfc07425ad6f96f873213f771b617992998.png is a geometry_f67ff8a47407eebbc544e3c7cba2e7ac458d3e75.png module.

We observe that the following is true:

geometry_2090b2fc6d02f8d1f9b1e443236610f97f8b9b55.png

which implies that geometry_736b2940b85ee9b6dc127b5e075d2523a41d6a93.png is a geometry_f67ff8a47407eebbc544e3c7cba2e7ac458d3e75.png module.

Lie Algebra

An abstract Lie algebra geometry_85e24fe5495c188f404fdd5dab62293112f24ea9.png is a K-vectorspace geometry_65a851830e28eaa68b2d9131907bde8028f9bdfa.png equipped with an abstract lie bracket geometry_db0604a541d4fb5722354cf5f9b81ac5416b76cf.png that satisfies:

  • bilinear (in geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png): geometry_f79e6c556ad4ef566da8892a698289d3b1d1aeb5.png
  • anti-symmetric: geometry_ff41091e9f0bd144e141ce966bab7cbc06dc2dd6.png
  • Jacobi identity:

    geometry_f7792737df9f03c7a38dcb503bda825d34b5c091.png

One might wonder why we bother with these weird brackets, or Lie algebras at all; as we'll see, there is a correspondance between Lie groups, which are geometrical objects, and these Lie algebras, which are linear objects.

geometry_ecd25a19f5ef073d39e33451738a488d45bc665c.png

We start with the following corollary

geometry_8d060d9f51e22f38ead61f84569551c4fbfb28a1.png

We need to construct a linear isomorphism

geometry_ba509e213fe9447efdfdb7173460017a06e5f187.png

where

geometry_844ad35f63cca997ce359138b263c492b0d9e215.png

where geometry_3fc9faea0561fe1587069b91ab7a0b166e8817e0.png denotes at the point geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png.

That is, we push forward the vector-field at the point geometry_1a4a18e739bc08439a04efa4dc6013a508ff1163.png for every point geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png, thus creating a vector at every point.

  1. We now prove that it's left-invariant vector field:

    geometry_2c43eed8e0ef1bcd463f631215d5fb6820d0b81b.png

  2. It's clearly linear, since geometry_08eab138c7a3f944965544fc699869dcde94f6dc.png and the push-forward on a vector-field at the point geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png, geometry_68ad91e91ab3e71dca1cbfbbc5ff4e371fbc43a0.png, is by definition linear.
  3. geometry_d8797d487554fc78e460f5843167fcb01b53e76b.png is injective:

    geometry_e6caa47d70386c6dee127d808bc56da74e4bc93b.png

    which can be seen from

    geometry_e9ae9346f22a4edeba15f4524b9ac981a58db57f.png

  4. geometry_d8797d487554fc78e460f5843167fcb01b53e76b.png is surjective: Let geometry_adc9c0dde1f411cb3f78fc1daef9d88160f6d7f1.png, i.e. geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is a left-invariant vector field. Then we let geometry_807502a8f63359155f272c34501a373045b0989b.png be some vector field geometry_e46729bc781c25bbc7120ee2892cc1c0215af7da.png associated with geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png, defined by

    geometry_1ba79df3bf83f36691b1769313f8b98bba1bd542.png

    Consider:

    geometry_ef7d6623c2e120d34f29e940a0ad6af4ca7877b4.png

    which implies geometry_3becfbd89f6a27c04c38a80fc59ef5d3adf6bf57.png

Hence, as claimed,

geometry_3566e5bf8fe4804c101d4de82e254a97dc35695e.png

Which means that as a vector space we can work with geometry_ede6de58b50a8ca1fec46e614cdbb5c6d0344c41.png to prove properties of the vector-field of left-invariant vector field!

Only problem is that we do not have the same algebra, i.e.

geometry_6d4e4ae652f602d618ed2ba46c88fc0ea95f1e2b.png

and we would really like for the following to be the case

geometry_11d7f0f255d2ff527de2c2bc8f645ba81e9b50f7.png

that is, we want some bilinear map geometry_99b796f37f888e5d4ab75887564db3e6f6c52267.png s.t.

geometry_8b92e8318447aae76e829851c84dcd6ed19a3fcf.png

Thus, we simply define the commutation brackets on geometry_ede6de58b50a8ca1fec46e614cdbb5c6d0344c41.png, geometry_db0604a541d4fb5722354cf5f9b81ac5416b76cf.png such that

geometry_f62e92d9cb619213754d2e86a4d44f564543acbb.png

as desired we get

geometry_11d7f0f255d2ff527de2c2bc8f645ba81e9b50f7.png

Example of Lie Algebra

Let geometry_9e2b340804b0f651110a9c5926d6b93522f0530b.png is a geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png vectorspace. Then

geometry_ae59ab9d833596224d697dca25035989a303fb1b.png

is an infinite-dimensional (abstract) Lie algebra.

Examples of Lie groups

Unit circle

geometry_0d2f7b1ced897a343316dab9c44de6ef4a326d5c.png

where we let the group operation geometry_9bf95a23c3847a29f0a60b149cdcdf23a2c15e5d.png, i.e. multiplication in geometry_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png.

Whenever we multiply a two complex numbers which are both unit length, then we still end up on the unit-circle.

General linear group

geometry_50aa72b0a05935d0c955951a508e61378379bd9d.png

equipped with the geometry_3d4fd6de0d71bddfdc5feee738a7a6b8796b7657.png operation, i.e. composition. Due to the nature of linear maps, this group is clearly satisfies geometry_b8a0304801e5f4df1235f109cd3b1805d5a8e3c1.png (but not geometry_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png), hence it's a Lie group.

Why is GL a manifold?

geometry_e5cb24a50057011a6b5c2f1d385a865b6bea9389.png can be represented in geometry_eb726a79566e5df5933f26d0063111e385571270.png (as matrices), and due to the geometry_c9ccc7957380daa6c3d3fec59b21d58247878619.png the set is also open.

Thus we have an open set of on geometry_eb726a79566e5df5933f26d0063111e385571270.png which we can represent as

Relativistic Spin Group

In the definition of the relativistic spin groups we make use of the very useful method for constructing a topology over some set by inheriting a topology from some other space.

  1. Define topology on the "components" of the larger set
  2. Take product topology
  3. Take induced subset-topology
Proof / derivation

To define the relativistic spin group geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png we start with the set

geometry_3729d14c44207cce668da2bce42f77629de473b1.png

  • As a group

    We make geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png into a group geometry_06020d2b8f8a74de6f88f366e9472f86ac264f53.png:

    geometry_250592418812fcc719547d6b529a4ea5cffce358.png

    i.e. matrix multiplication, which we know is ANI (but not commutative):

    • Associative
    • Exists neutral element
    • Invertible (since we recognize geometry_8fc58a5b3ab17794e2051b7bffe4d77f0329eda4.png)
  • As a topological space

    From this group, we can create a topological space geometry_464a59a262bd8c72e8e30e686b240c73d78e51d0.png:

    1. Define topology geometry_83a38ff9c86942cdb8639d06e3a94f37bd73205c.png on geometry_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png by virtue of defining "open balls":

      geometry_2de812e1a30731a4c726c141f4762913ee2ab78a.png

      which is the same as we do for the standard topology in geometry_abf17ab8d81a8a9c9a3b8d37b4e26b6e2983cdc7.png.

    2. Take the product topology:

      geometry_278818feed3a1fef6ef6b32dad5d2c4c409f19e7.png

    3. Equip geometry_ea1eb0735c016a6c8bb87115f3a1902c5d0545df.png with the induced subset topology of the product topology over geometry_669880a2895bdc620623dbc0affa66723b0b1bbd.png, i.e.

      geometry_0da03777d6cf0abf57c65405bcb4cbe04e849766.png

    Verify that we have geometry_464a59a262bd8c72e8e30e686b240c73d78e51d0.png, with geometry_3dd4548540bf408f0290b00437fe6082a46e5fee.png as given above, is a topological manifold. We do this by explicitly constructing the charts which together fully covers geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png, i.e. defines an atlas of geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png:

    1. First chart geometry_1f083b63e361efe45649c11b0e75777aaffc2a2b.png:

      geometry_ba527097daf2d958c8d49b51785a510d52bd7c41.png

      and the map

      geometry_e9c50b06f33694b22d7248b1a734f02e814335b1.png

      which is continuous and invertible, with the inverse:

      geometry_577e8ebc3a3812d98e3e0165985bd3ac3680ae5d.png

      hence geometry_7ad7017d275cd9ba530592dc752ef3a9681cbd6c.png is a homeomorphism, and thus geometry_1f083b63e361efe45649c11b0e75777aaffc2a2b.png is a coordinate chart of geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png.

    2. Second chart geometry_170ca3290727e21109e94001a087259322bc4e67.png:

      geometry_5470de6bd6e8528e710c278cde3a793359360e32.png

      and the map

      geometry_db057719749bec0a568789d38e341bb14b9cee6b.png

      which is continuous and invertible, with the inverse:

      geometry_7be367f77913b3f623f0c28a76077655697cfcd9.png

    3. Third chart geometry_bd1532aa036e2e0c410b1a468a39fedb1f04ec11.png:

      geometry_06e2c4817668c61139713fb81476de8df8c831bb.png

      and the map

      geometry_78841d841e93c490ac92e9fb34944b505d604c31.png

      which is continuous and invertible, with the inverse:

      geometry_08e248f759c1622d8338dabb47adbbd30ad11b5b.png

    Then we have an atlast in geometry_07259204849853d9217b0c293f3c17b126a967cc.png, since these cover all of geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png, since the only case we're missing is when all geometry_e83602c8f3ab227852fe3e609743ad1f2d2b48bd.png, which does not have determinant geometry_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png, therefore is not in geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png. Hence, geometry_464a59a262bd8c72e8e30e686b240c73d78e51d0.png is a complex topological manifold, with

    geometry_d558192c0f1ee669b884faeb4c59d45fc40daad0.png

  • As a differentiable manifold

    Now we need to check if geometry_ebeef6bb2eb15d7d3f4eb80b00428249b0a38826.png is differentiable manifold (specifically, a geometry_e30b2d6b59f4a90334ec1ac444c506bc886a6b06.png); that is, we need the transition maps to be "geometry_6ef08c22258d59b778d6063e82f7118de45597be.png compatible", where ? specifies the order of differentiability.

    One can show that the atlast geometry_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png defined above is differentiable to arbitrary degree. We therefore let geometry_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png be the maximal atlast with differentiability to arbitrary degree, containing the atlast we constructed above. This is just to ensure that in the case we late realize we need some other chart with these properties, then we don't have to redefine our atlas to also contain this new chart. By using the maximum atlas, we're implicitly including all these possible charts, which is convenient.

    One can show that geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png above defines open subsets, by observing that the subset where geometry_5aee29f7f90a15773f4be86b227da01a9139f0a5.png is closed, hence the complement (geometry_360a2a8fa66955238dd4e460ad32f9a1dfd0b555.png) is open.

  • As a Lie group

    As seen above, we have the group geometry_22791ad8f8d41ece690f2827131c70f72dac556e.png, where geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png is a geometry_e30b2d6b59f4a90334ec1ac444c506bc886a6b06.png manifold to arbitrary degree, with the maximum atlas geometry_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png containing geometry_bc81f637deea7e4e14560e98d14265e2b150c32e.png as defined previously.

    To prove that this is indeed a Lie group, we need to show that both the following maps:

    geometry_4176c202021c796eb2cf78b3027c4d72e4bfc0db.png

    and

    geometry_b08b8a0b8536a01db8fa16ebe87c72e7dc89accc.png

    are both smooth.

    Differentiability is a rather strong notion in the complex case, and so one needs to be careful in checking this. Nonetheless, we can check this to the arbitrary degree.

    relativistic-spin-group-smoothness-of-inverse-map.png

    We observe that the Fig. fig:commutation-diagram-inverse-relativistic-spin-group-charts is the case, since the inverse map restricted to geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, where geometry_360a2a8fa66955238dd4e460ad32f9a1dfd0b555.png, is then mapped to geometry_85a0420edfa3fd96112b828a2434e84b94bd9c25.png, i.e. the image is geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png.

    We cannot talk about differentiablility on the manifold itself, hence we say geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png is differentiable if and only if the map geometry_2a31ddd5362d2c669305c5809b545ed9bf364559.png is differentiable (since we already know geometry_7a84c9a383f9772338016d101ccc096be06af784.png and geometry_eb83d466c7d035356e9f39998f357cee73da1e26.png are differentiable). We observe that

    geometry_62020d58281a61c0b47edd9c4bf9ffe637e229ad.png

    which is most certainly a differentiable map. We've used the fact that all these matrices have geometry_85c2bbe9a428d64625c98ba8c26d4595ef8b6c72.png in the inverse above.

    Performing the same verification for the other charts, we find the same behavior. Hence, we say that geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png, on the manifold-level, is differentiable.

    For geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png we can simply let the product-space geometry_5738f5cf5bbae2e4579640e25b3452bec30208ac.png inherit the smooth atlast geometry_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png on geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png, hence geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png is also smooth.

    That is, the composition map geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png and the inverse map geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png are both smooth for the group geometry_22791ad8f8d41ece690f2827131c70f72dac556e.png, hence geometry_22791ad8f8d41ece690f2827131c70f72dac556e.png is a complex 3-dimensional Lie group!

  • TODO As a Lie algebra

    In this section our aim is to construct the Lie algebra geometry_54017e335048e49427fb923e63567b1781c0606f.png of the Lie group geometry_22791ad8f8d41ece690f2827131c70f72dac556e.png.

    We will use the standard notation of

    geometry_40d3c02ede98ccd547f6c6dd49d5bbd76af5216a.png

    Recall

    geometry_bed813cac688a88def50739b065e0de48ccd3509.png

    i.e. it's the set of left-invariant vector fields.

    Further, recall

    geometry_0da94fcb786d9bfd27e9a1dbdad31b8408d41e1e.png

    where

    geometry_cc0ba628a9f33a4f7e918e93a63ff50365afe1a6.png

    Now, we need to equip geometry_b3bdff46df4eab74584b1609a0c40fdb71fd4ecc.png with the Lie brackets:

    geometry_70229582a812a72dc6da148b9610435f4e76cd96.png

    where

    geometry_12ba2cf2eeafb0772bea54440db4840ee3bd0115.png

    To explicitly write out this geometry_db0604a541d4fb5722354cf5f9b81ac5416b76cf.png, we use the chart geometry_1f083b63e361efe45649c11b0e75777aaffc2a2b.png since geometry_2801fe6656f35823e782b5fad51a2cf55d251d1c.png. For any geometry_3045665c28ce248ec0a0d8672a843bbbcb0a0dd3.png, we have

    geometry_8b77a348af36602c2d7ceb50782ed387335d5056.png

geometry_96fbc2166f7b357767e85e35a5c9db014f371222.png

geometry_c75b2ebfc7a78cbcebdaaed67ee7d809cc58bf92.png

Observe that if we write

geometry_8330aef40e1de783d05e7e66f02dad7ad13a7d73.png

then we observe that geometry_96fbc2166f7b357767e85e35a5c9db014f371222.png is diffeomorphic to geometry_ad57d72263bb98f1765a127956aa38fce5f3d9fc.png.

Classification of Lie Algebras

Notation

Stuff

Every finite-dim. complex Lie algebra geometry_e24cd4ef8d335253b731fb840aff27765fd056e0.png can be decomposed as

geometry_c4b1cfd005289290498897bf91135dd1e97767b8.png

where:

  1. geometry_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png is a Lie sub-algebra of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png which is solvable, i.e.:

    geometry_4d5f71d7bea6b081b727bef750d005a537853366.png

  2. geometry_10c3b343fa99072b92224200bd129a6be268b329.png are simple Lie algeabras, i.e.
    • geometry_7a98ccdca6eb82221fc52a067f9e6cf66e6bc97c.png is non-abelian
    • geometry_5598d6f0cbd57f0181244b466afdebfce3e787b6.png contains no non-trivial ideals, where:
      • An ideal means some sub-vector space geometry_4871ecbe4fc3c023086e03887cedb000e835e6ef.png s.t. geometry_c1d37e24fb736493c0dbc2f94ec158b50b8e363a.png, i.e. if you bracket the sub vector spaces from the outside, then you're still in the ideal geometry_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png.
      • geometry_45daf3c13bdff98253e0199590771b1c1a70e34e.png is clearly ideal since geometry_6dbe51bb2d69eb9e95bcd994e43223d21b04bacb.png
  3. The direct sum geometry_9773abd4b3e2bdbcb7488e51c7c7ec1f33435fd8.png between Lie algebras is defined as:

    geometry_2f73f84223694e0a3522cad9fe38ca61e6691e95.png

  4. semi-direct sum geometry_f9122ed559533934d9ab91a36009949e1b052bb8.png

    geometry_525f634d1dd454b5a05a8602963abb99267ebe9f.png

Which saying that we can always decompose a complex Lie algeabra in a solvable geometry_3d136f0fc4860468633907421c098b9feb0eef24.png, semi-direct sum, and a direct sum between simple Lie algebras.

Every Lie algebra can be decomposed into a semi-direct sum geometry_f9122ed559533934d9ab91a36009949e1b052bb8.png between a solvable Lie algeabra geometry_3d136f0fc4860468633907421c098b9feb0eef24.png and a direct sum geometry_9773abd4b3e2bdbcb7488e51c7c7ec1f33435fd8.png between simple Lie algebras

A Lie algeabra that has no solvable part, i.e. geometry_6607bec1e7eefa03c95ecb0b7887abac40237578.png is called semi-simple.

It turns out it's quite hard to classify the solvable Lie algebras geometry_3d136f0fc4860468633907421c098b9feb0eef24.png, and simpler to classify the semi-simple Lie algeabras. Thus we put our focus towards classifying the semi-simple Lie algebras and then using these as building blocks to classify the full Lie algebra of interest.

geometry_a083e411a63efafc1a24fa9b31ea568e3c2ced99.png is a complex Lie algebra and geometry_fd20f1dff526e84e16876ebc835b7d2690483d7a.png, then define

geometry_43735abb70ba44b93a74f71bd0a02f00c981e659.png

is the adjoint map wrt. geometry_fd20f1dff526e84e16876ebc835b7d2690483d7a.png.

The bililinear map

geometry_3ddfbacd9b85500c80e5ab9fd5428ecd16c4fee2.png

is called the killing "form" (it's a symmetric map, so not the kind of form you're used to).

And we make the following remarks about the Killing form:

  • geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is finite-dim. thus the geometry_ddef676c364a38258561ff3a3f2c185c00a4284e.png is cyclic, hence geometry_fa29a1829913253c500c296953e3ab7094c32ba4.png
  • geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png semi-simple (i.e. no solvable part) if and only if geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png is non-degenerate:

    geometry_6969a5d903e77890725db1c547d2aa95543a1a37.png

Now, how would we then compute these simple forms?

Now consider, for actual calculations, components of geometry_5fb57b658bdca591159fc1a7c7575988aef7949e.png and geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png wrt. a basis:

geometry_5a6f3d0a0e560e1494e80b643ecb751869035d51.png

Then

geometry_b349eb4954fc1ee1bf4afa3a7bd7f627413e5c14.png

where geometry_9120ed1e3dd420d6f8183239d9be0f961b0dd4c2.png are just coefficients of expanding the commutation in the space of the complex numbers, which we clearly can do since geometry_27defa7afc1d68486df0a4263a76e7f3d47a10a6.png. These coefficients are called the structure constants of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png wrt. chosen basis.

The killing form in components is:

geometry_ae08147eed798463a7fd54f7199f3c091082c759.png

Where the last bit is taking the geometry_ddef676c364a38258561ff3a3f2c185c00a4284e.png. Thus, each component of the killing form is given by

geometry_fcc3586847711a15b058dc8254b674a0373f5cfe.png

We then empasize that geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is semi-simple if and only if geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png is a psuedo-inner product (i.e. inner product but instead of being positive-definite, we only require it to be non-degenerate).

One can check that geometry_5fb57b658bdca591159fc1a7c7575988aef7949e.png is anti-symmetric wrt. killing form geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png a (for a simple Lie algeabra, which implies semi-simple).

A Cartau subalgebra geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png of a Lie algebra geometry_85a52284b971cf3b87ff28884b40247b1eeaa024.png is:

  • geometry_52f90a032b410d50e75ea30109f53bc468c573f9.png as a vector subspace
  • maximal subalgeabra of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png and that there exists a basis:

    geometry_ae044a61367e33622dabe07d03294fb622b74c04.png

    of geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png that can be extended to a basis of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png

    geometry_23fe973a630527178911087863aedc85b016c224.png

    such that the extension vectors geometry_b6fac1e10a5fd1f8a01f6f4d35d28b9f4c179b92.png are eigenvectors for any geometry_5fb57b658bdca591159fc1a7c7575988aef7949e.png where geometry_656379462a0acd1da6acb6b8e46afd7b70d41c84.png :

    geometry_5c217b979e0319ded2ac5f127344c2e42156b528.png

    where the eigenvalue geometry_bacf59cae18c391aadad97a3db6304a4b4e5feaf.png depends on geometry_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png since with any other geometry_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png we have a different map geometry_5fb57b658bdca591159fc1a7c7575988aef7949e.png

Now, one might wonder, does such a Cartau subalgebra exists?!

  1. Any finite-dimensional Lie algebra posses a Cartau subalgebra
  2. If geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is simple Lie algebra then geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png is abelian, i.e. geometry_8127b33646b9b72c5492d52e9175668b238e29c2.png

geometry_a8ab04bbe6df59a70645a09cf1b62dd7adf7b880.png

is linear in geometry_656379462a0acd1da6acb6b8e46afd7b70d41c84.png. Thus,

geometry_d74699868790d4a05ace031ab42832d37ff94b37.png

I.e. we can either view geometry_eb7337a8de0374b86293aaca301fb32bea358cec.png as a linear map, OR as a specific value geometry_58134d362705b1b227c78b4d7359e381afb96065.png. This is simply equivalent of saying that

geometry_cae4e4612734ff8710a99336e077ff538ddfefa2.png

The geometry_c110da095adec83946b87c385983cc0e2ad124a5.png are the roots of the Lie algebra, and we call

geometry_f04baf3e6bab42a4d2bc9bb9c864204ef2a3b231.png

the root set.

Since geometry_5fb57b658bdca591159fc1a7c7575988aef7949e.png is anti-symmetric wrt. killing form, then

geometry_84408d0de20a4b857c424f8fc730d4758eed863e.png

Also, geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png are not linearly independent.

A set of fundamental roots geometry_1d2ba920e1ecccf5b8f8792e005dfccbcc63b211.png such that

  1. geometry_0f866b70a0684772c90606caebe35825a6234426.png linearly independent, geometry_496bb037fb8568dc5a664eaf8e58024dedc69ae3.png
  2. Then geometry_1c9f9114969faa41b877fc263e44d5a336fdcf69.png such that

    geometry_853bb92273a870a8fca25cb73a485666c1fedb12.png

Observe that the geometry_224ae917d74dc2133d4403064c971bf562d4db50.png makes it so that we're basically choosing either to take all the positive geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png or all the negative geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, since they are linearly independent, AND this is different from saying that geometry_ab6d779abe94aff64fe6a9c38800418f147bd448.png!!! Since we could then have some negative and some positive. We still need to be able to produce geometry_802dfc6aed8b35d3a752ad3d22d5a3d729a82f9f.png from geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, therefore we need the geometry_224ae917d74dc2133d4403064c971bf562d4db50.png.

And as it turns out, such a geometry_1d2ba920e1ecccf5b8f8792e005dfccbcc63b211.png can always be found!

The fundamental roots of geometry_05594e8ce8f52728892c4c673d8bc28149b4b24d.png span the Cartau subalgebra geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png

geometry_8e002c119abd3cf520bc7ebc6541532107c4e15b.png

buuut note that geometry_0f866b70a0684772c90606caebe35825a6234426.png is not unique (which is apparent by the fact that we can choose geometry_a71cfad80661849a64d33f8ea6e49a6d80c2e01a.png for the expressing the roots, see definition of fundamental roots).

If we let geometry_583d3c17d3bca0f725012f93df66b6a73e97ad87.png, then we have

geometry_5e075209307a4fd4b0d6ab0a1d3ba272d5cfbcab.png

We define the dual of the Killing form as geometry_8884d5ceb3487b5e5beb3a45635688bfc4c4e47a.png defined by

geometry_f4c98716f218c30fb253a5b06cf6fbbbfcf6c449.png

where we define

geometry_82c622e192153e9149fabc94b1ffd7751bd99767.png

where geometry_937a9fafe319f7169579836a8bf41e7c2c648398.png exists if geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is a semi-simple Lie algebra (and thus of course if it's a simple Lie algebra).

If we restrict the dual of the Killing form geometry_2590543aacd81d19801870f2325d74015cc6d828.png to geometry_0e61a384479ef1dd97338610de0b2f31c52436df.png (as opposed to geometry_b2893e8b28f70632066ff612c7276d72a6642456.png), that is,

geometry_472e363d7185e0a94ad2c99b0e6f335c4555fe21.png

and

geometry_96111510296d5608ff6cf8f8d47ef41314cb737a.png

with equality if and only if geometry_3b011c80014b4e5941b355d7e72165d65ae431a0.png.

Then, on geometry_e9529260f43527c4bdce7ef5cced2467d2eadde4.png we can calculate lengths and angles.

In particular, one can calculate lengths and angles of the fundamental roots of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png (all roots are spanned by fundamental roots => can calculate such on all roots).

Now, we wonder, can we recover precisely the set geometry_05594e8ce8f52728892c4c673d8bc28149b4b24d.png from the set geometry_0f866b70a0684772c90606caebe35825a6234426.png?

For any geometry_7d6466cdcef247727d7dbeacf384799ffd85966e.png define

geometry_ca34b36e5e6b6633e29ab50c402499364bd9cd3e.png

which is:

  • linear in geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png
  • non-linear in geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png

and such geometry_1d2fca5782bde2c2fe67efc7bfe702ec2a9cf2b7.png is called a Weyl transformation, and

geometry_97d9a7c8b3d1f9216872dd5aed8fe1f5a08dd91b.png

called Weyl group with the group operation being the composition of maps.

  1. The Weyl group is generated by the fundamental roots in geometry_0f866b70a0684772c90606caebe35825a6234426.png:

    geometry_6f804bfb44c1051e5919c2628a93b77df2451c08.png

  2. Every root geometry_7d6466cdcef247727d7dbeacf384799ffd85966e.png can be produced from a fundamental root geometry_5c4b3ea7610e31a18d0b523517b5ef81ecab652c.png by action of the Weyl group geometry_f53cf2c3985c80987bd9707fd5a0b28594db886f.png:

    geometry_2a3550b21b192a9c81a55e4e618f4c5c67608f0c.png

  3. The Weyl group merely permutates the roots:

    geometry_2f190dc60007caa77671ccd164600cf928f75bac.png

Thus, if we know the fundamental roots geometry_0f866b70a0684772c90606caebe35825a6234426.png, we can, by 1., find the entire Weyl group, and thus, by 2., we can find all the roots geometry_05594e8ce8f52728892c4c673d8bc28149b4b24d.png!

Conclusion

Consider: for any fundamental roots geometry_08285fdef2a85963ad83eee9581b462ce2eb0a5e.png, by definition of geometry_c21bca363cfae57ad9d216e17ec63e7f1ff1340b.png we have

geometry_299312ad405a2b0275a1f5eaecadff44cb938827.png

And

geometry_0f811df88de6083447ad3db913faaedd29456db1.png

Which means that both terms on LHS of geometry_563b8074859cdfd19bc14435719116349fb89d69.png must be the same sign, and further because it's an element in geometry_05594e8ce8f52728892c4c673d8bc28149b4b24d.png, we know the coefficient must be in the integers (for geometry_1a3020ccbbd7970168c3dc344384360d791dfc41.png):

geometry_0f9ed020bd60eabc21c44cb95a6e2493d214e0c1.png

where, for geometry_1a3020ccbbd7970168c3dc344384360d791dfc41.png, we have $ - Cij ∈ $.

Observe, geometry_4764f1db94b346022f9a0fb75ab6b6bdbe7e268e.png is not symmetric.

We call the matrix defined by geometry_4764f1db94b346022f9a0fb75ab6b6bdbe7e268e.png the Cartau matrix, and observe that geometry_c5b1828dae2a8b87bf6ca8e87d7eedbc4d062181.png while every other entry is some non-positive number!

No we define the bond number:

geometry_fc5a134522e9097be3df92f18409265d5ac59326.png

which implies

geometry_144878a1065726cdb8a13337d3fbdc4261969f44.png

where geometry_4764f1db94b346022f9a0fb75ab6b6bdbe7e268e.png and geometry_2f5e819b1a32b139ce0513279d3eaa9cfeabeaef.png are non-positive numbers, hence:

geometry_b1e3c0ee6debd4eacbfd4845bde1cae2875b1321.png

Therefore:

geometry_4764f1db94b346022f9a0fb75ab6b6bdbe7e268e.png geometry_2f5e819b1a32b139ce0513279d3eaa9cfeabeaef.png geometry_a104980ad1f1599b67e2712db7c6577fcda5c4fa.png
0 0 0
-1 -1 1
-1 -2 2
-2 -1 2
-1 -3 3
-3 -1 3

Which further implies that

geometry_981886b14beef1ba34ba838849475625fb1b72ea.png

Dynkin diagrams

We draw these diagrams as follows:

  1. for every fundamental root draw circle:

    fundamental_roots.png

  2. if two circles represent geometry_08285fdef2a85963ad83eee9581b462ce2eb0a5e.png, draw geometry_a104980ad1f1599b67e2712db7c6577fcda5c4fa.png lines between them:

    fundamental_roots_connected.png

  3. if there are 2 or 3 lines between two roots, use the geometry_b1987047c45ca71bc29776798bd2287cf170efd5.png sign on the lines between to indicate which is the greatest root

Any fininte-dimensional simple geometry_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png - Lie algebra can be reconstructed from the set geometry_0f866b70a0684772c90606caebe35825a6234426.png of fundamental roots and the latter only comes in the following forms:

dynkin-diagrams.png

Taken from [[https:/commons.wikimedia.org/wiki/]] /

Representation Theory of Lie groups and Lie algebras

TODO Representation of Lie Algebras

Let geometry_1270398a6fa3833437db0bca322f80cb9c943748.png be a Lie algebra.

Then a representation geometry_b7c65dd8ef6a2f3c96ab9371776fdbf97113f841.png of this Lie algebra is:

geometry_2cb4f6462986f19668668609a4b341bb8aacbe77.png

s.t.

geometry_81f936288c05cbf6f375ae1eb9d9beaac9751c5f.png

where the vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png (a finite-dimensional vector space) is called the representation space.

An example of a representation is geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png acts on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, then

geometry_02db7a0700ae7afc7bedf8b50875e758a9e4ecff.png

is a representation of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png via

geometry_dc14bc4308ec0e1b2073254fa73eec1dc4e83d5d.png

In general, geometry_68e47a162f70e725e0060d4711d1b8b654e89ef4.png is aan equiv. calss of curves where geometry_6bc2bbac2d1d622542e9e1c84ee2d9b4f0eee3c9.png if geometry_6b98d09c686bde4a8c7583098c8ef4cf03f0b1e7.png and geometry_d565de5b9c2eeff2a84e3e16e78ab5b315be8c03.png.

geometry_9dc54cb072169a34d0110a86ac53e24ce75ba419.png are manifolds, geometry_f8748e779d9738342ec918f186adcc635a9f6461.png with geometry_bd2dba7a33e54ec6398aa6cf82c1157cdce7ff9a.png. Then geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png takes curves through geometry_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png to curves through geometry_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, and

geometry_42daac42a45506fd18f1ab9672da6921244bd853.png

And geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png takes equiv. calsses to equiv. classes (CHECK THIS). Thus differential of geometry_078b85cd3478400338e3a1ee425c2a468644be7e.png, geometry_6da1486ef9c839f587f534edfe18ab223760d9ee.png.

A representation geometry_9e3e50ffe78099cc34efc83ac9d7b790995451c4.png is called reducible if there exists a vector subspace geometry_47bc9b6c03e21bf4d816e3a22279ee2c5b73b7f0.png s.t.

geometry_17f66c188546d1443b0364eb95d104758c4f5201.png

in other words, the representation map geometry_b7c65dd8ef6a2f3c96ab9371776fdbf97113f841.png restricts to

geometry_194fc113d73a86d61b873bbafd4133796bf13bb9.png

Otherwise, geometry_e2c0afeefc3cbd4af58cf9843eaba920ac0fcc76.png is called irreducible.

Adjoint representation

Let geometry_f855a33d0e80ae0f04f0f00c8c03e39ee83657d7.png be a Lie algebra over a field geometry_1641d18cc980f8db14cdff95d7417a8526eef446.png.

Given an element geometry_7a84c9a383f9772338016d101ccc096be06af784.png of a Lie algebra geometry_f855a33d0e80ae0f04f0f00c8c03e39ee83657d7.png, one defines the adjoint action of geometry_7a84c9a383f9772338016d101ccc096be06af784.png on geometry_f855a33d0e80ae0f04f0f00c8c03e39ee83657d7.png is given by the adjoint map.

Then there is a linear mapping

geometry_89e6e4331bb2c8d962be0f5a1e208137f3625f7b.png

Within geometry_ba8b744df896df090dea9dfce64b788a96f52529.png, the Lie bracket is, by definition, given by the commutator of the two operators:

geometry_e656c51ca7c1366f5b25be8838f7023e8c7a6653.png

Using the above definition of the Lie bracket, the Jacobi idenity

geometry_c8e8a60dc5f330a21db8296d4cfcc1173a8bb838.png

takes the form

geometry_772b9f71db11a4f4de33501c33469d408babc242.png

where geometry_7a84c9a383f9772338016d101ccc096be06af784.png, geometry_eb83d466c7d035356e9f39998f357cee73da1e26.png, and geometry_fa87ab9a5006bb5bd8b6c54e2ce82b0507c166d1.png are arbitrary elements of geometry_f855a33d0e80ae0f04f0f00c8c03e39ee83657d7.png.

This last identity says that geometry_4884ba6067a996f9a8540ce1ce686949eae2ae5e.png is a Lie algebra homomorphism; i.e. a linear mapping that takes brackets to brackets. Hence geometry_4884ba6067a996f9a8540ce1ce686949eae2ae5e.png is a representation of a Lie algebra and is called the adjoint representation of the algebra geometry_f855a33d0e80ae0f04f0f00c8c03e39ee83657d7.png.

Casamir operator

Let geometry_fed46254b4e995ec5dcf79282a056f485b41e979.png be the representation of complex Lie algebra geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png.

We define the ρ-Killing form on geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png as

geometry_ad78a8e4a193a666de64528819d153254ebb38f7.png

/Note that this is not the same as the "standard" Killing form, where we only consider geometry_de742a1aff4c6a43688ab0ebabbc5b6f93479db4.png.

Let geometry_fed46254b4e995ec5dcf79282a056f485b41e979.png be a faithful representation of a complex semi-simple Lie algebra geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png.

Then geometry_0220356aac9c63f0021fea7a293d9da79234aab1.png is non-degenerate.

Hence geometry_0220356aac9c63f0021fea7a293d9da79234aab1.png induces an isomorphism geometry_40032c93b6a35b3cba9baa93a8b5c0dbe35751a6.png by

geometry_97e17f732369400332831c587d20303286fbbf7c.png

Recall that if geometry_e3a629798bca1d4a38626259caf847495942a87c.png is a basis of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png, then the dual basis geometry_1e9bbe6de363a1948a0f8a0c9bedf61b62fd086b.png of geometry_4aeb11395e34c30abc5b48916262687cadb9ad7f.png which is defined by

geometry_50e3b3f0c43b8a9ea5c5a1ac29e9ffe5856d2914.png

By using the isomorphism induced by geometry_0220356aac9c63f0021fea7a293d9da79234aab1.png (when geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is a faithful representation, by proposition:ρ-killing-form-induces-isomorphism-for-complex-semi-simple-algebras), we can find some geometry_1d365f7478ad8e7abc6c33bea02a5a9cbcd6ab16.png such that we have

geometry_9b2a065d8d812c375a6b4c576048688404aee4eb.png

or equivalently,

geometry_f6dff849aa38201f514c173162d3df3c5fa3dc4a.png

We thus have,

geometry_6a8789f5cebc338211f3f68852122ec65f4626a7.png

This seems awfully much like Reproducing Kernel Hilbert Spaces (RKHSs), doesn't it?

At least geometry_0220356aac9c63f0021fea7a293d9da79234aab1.png seems to define a kernel of the space?

Let geometry_d5ca39203e91aefc22be9a6729419ca62c5a96c1.png and geometry_e9688ced43dabb1ad2de30a87f1cd97de9467db1.png be defined as above. Then

geometry_6047d9d05d2da3bd8c2e4684966928e299a7a41b.png

where geometry_ca25c295924cae6810e29d9ad3f67ae5d868d05b.png are the structure constatns wrt. geometry_d5ca39203e91aefc22be9a6729419ca62c5a96c1.png.

Let geometry_fed46254b4e995ec5dcf79282a056f485b41e979.png be a faithful representation of a complex (compact) Lie algebra geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png and let geometry_e3a629798bca1d4a38626259caf847495942a87c.png be a basis of geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png.

The Casimir operator associated to the representation geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is the endomorphisms geometry_babb8f8743a360e01244eac2fe7bb223ea1a5038.png

geometry_17640cc0fbe8bfb79275bf8ab3527694214edf87.png

Let geometry_1f2f854c3c84790eed7e114e7ad5dffd5eb47d30.png be the Casimir operator of a representation geometry_fed46254b4e995ec5dcf79282a056f485b41e979.png.

Then

geometry_c8be2a8f79e71a2e37fbd473c41de2a959b960e2.png

that is, geometry_1f2f854c3c84790eed7e114e7ad5dffd5eb47d30.png commutes with every endormorphism in geometry_a4133720d545ed0660efe0ca9a104b05ae0612dd.png.

If geometry_fed46254b4e995ec5dcf79282a056f485b41e979.png is irreducible, then any operator geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png which commutes with every endomorphism in geometry_a4133720d545ed0660efe0ca9a104b05ae0612dd.png, i.e.

geometry_77773d5eae4db018ec145fddf01d6b50a035c157.png

has the form

geometry_3fb3e277b3374be2e94edc690d8ef8cc47bc87ce.png

for some constant geometry_a91c88fe5119b0911945505c21c5a1d1f990eec2.png (or geometry_492d525117d0dcc93d066c8759f46b98cf9980ca.png, if geometry_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png is a real Lie algebra).

The Casimir operator of geometry_fed46254b4e995ec5dcf79282a056f485b41e979.png is

geometry_2c9c0672e8252bdcb800450ab6379537f7eda350.png

where

geometry_f33abfdd2bc47de93cb284a09592c5e6a7dfe053.png

The first part follows from Schurs lemma and Thm. thm:casimir-operator-representation-of-lie-algebra.

Representation of Lie groups

A representation of a Lie group geometry_d5ac7c7e4f128a60ddc4f3de9e5f1a4172dacd14.png is a Lie group homomorphism

geometry_0a18771f70d4848608cde1d2c399929965b4df82.png

for some finite-dimensional vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png.

Recall that geometry_1cf0c8b4bc9da2c0ad1e4673cdb7ba813ad2b0b2.png is a Lie group homomorphism if it is smooth and

geometry_7198bcba73dc86bc4d944e6cf4651696c74fbdb9.png

Let geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png be a Lie group. For each geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png, we define the Adjoint map:

geometry_b8abfd80d58f6413ba4b309a26fbfe43c7305ba4.png

Notice the capital "A" here to distinguish from the adjoint map of a Lie algebra.

Since geometry_c8c9182a6b7d28f63c29adab5f0f25df6ee11bf8.png is a composition of a Lie group, then it's a smooth map. Further,

geometry_53c843e302f3abf49530c9fb96612de2e7144b2f.png

Thus, geometry_d724762bf70fd4b8a2f002dc9735c635642e6adc.png.

Reconstruction of Lie group from it's Lie algebra

Notation

  • geometry_5f737627dfab02da643f3a3d20d77f347e59be56.png denotes the geometry_6b3c4f1c0d7ff56462979c13b1d0afbf02c35e0a.png map restricted to the vector space geometry_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png

Stuff

geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a smooth manifold and geometry_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png be is a smooth vector field on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

Then a smooth curve

geometry_711e582de4807ec5a0f70558a9297563ffbf7b0e.png

is called an integral curve if

geometry_f5865caf539100fc07970b83e90eebf113fdcbb9.png

There is a unique integral curve of geometry_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png through each point of the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

This follows from the existence and uniqueness of solutions to ordinary differential equations.

An integral curve geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png of a vector field geometry_93fb8ee5d35a52b19e02d5f83883e7eff903843c.png is called complete if its domain can be extended to geometry_1cf39e2cf7ecd4c843aebac443a9b00088b0c122.png.

On a compact manifold, every vector field is complete.

Every left-invariant vector field on a Lie group geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is complete.

Let geometry_ac0e2df69c08e54685eb918fa44186ac11b32c73.png and define the thus unqiuely determined left-invariant vector field geometry_04487c873eb3ade9b169b58021342cf212326678.png:

geometry_3c5f92b48310adda9e71fd67d4b421758fa1b12e.png

Then let geometry_a423ecc218b2fcf9ceab0bd0f345d6fa744ee824.png be the integral curve of geometry_04487c873eb3ade9b169b58021342cf212326678.png (which we can due to this theorem) through the point

geometry_f34dab8a6a175ae533bc879c83c917ae9f04fb8d.png

This defines the so-called exponential map

geometry_196978caf29399b61dc3f8dcf1cac738e1402285.png

It might be a bit clearer if one writes

geometry_1a4b28f2a183016d4d1918c89e5fe6360371b9f2.png

Some places you might see people talking about using infinitesimally small generators, say geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png. Then they will write something like "we can generate the group from the generator geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png by expanding about the identity:

geometry_c4bb2b001a34d518011185ed4cda269a62ad7657.png

where geometry_1d2732c3617f8bec9310d1577ab032fa7243a44c.png is our Lie group element generated from the Lie group element geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png!

Now, there are several things to notice here:

  • Factor of geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png; this is just convention, and might be convenient in some cases
  • geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png is just what we refer to as geometry_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png
  • This is really not a rigorous way to do things…

Finally, and this is the interesting bit IMO, one thing that this notation might make a bit clearer than our definition of geometry_6b3c4f1c0d7ff56462979c13b1d0afbf02c35e0a.png is:

  • We don't need an expression for geometry_36b331ea3405932ca775d487bd9eb45cdc5125b9.png; geometry_36b331ea3405932ca775d487bd9eb45cdc5125b9.png is a smooth curve, and so we can
    1. Taylor expand geometry_36b331ea3405932ca775d487bd9eb45cdc5125b9.png to obtain new Lie group element
    2. geometry_6b3c4f1c0d7ff56462979c13b1d0afbf02c35e0a.png new Lie group element
    3. Goto 1. and repeat until generated entire group
  • Matching with the expression above, only considering first-order expansion, we see that

    geometry_99b2bbbc0de4c899bfc9ffb0b250fa70c7c99d30.png

    where

    geometry_3268c5b5fcedcc3588242fcd96601513cf4857c9.png

  • Neat, innit?!
  1. geometry_6b3c4f1c0d7ff56462979c13b1d0afbf02c35e0a.png is a local diffeomorphism:

    geometry_e36ec8f7eeee1e0b8a24dd8139d24e684fae57b1.png

    1. This restricted map is bijective
    2. geometry_9c0d031cf8fdc5f4e1c0c4a5bf58f4e1f5a62c19.png and geometry_4872bc17b46e5eacb9f230f4d1bad909a6dc54b8.png are smooth
  2. If geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is compact then geometry_6b3c4f1c0d7ff56462979c13b1d0afbf02c35e0a.png is surjective: geometry_4bf78a226e3f3228cf1468c013ae3a7f6f9cef21.png. This is super-nice, as it means we can recover the entire Lie group from the Lie algebra!!! (due to this theorem) However:
    • geometry_ede6de58b50a8ca1fec46e614cdbb5c6d0344c41.png is non-compact
    • geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is compact
    • Hence geometry_6b3c4f1c0d7ff56462979c13b1d0afbf02c35e0a.png cannot be injective!
  3. If geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is non-compact: it may be not surjective, may be injective and may be bijective. Just saying it can be whatever if geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is non-compact.

The ordinary exponential function is a special case of the exponential map when geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers).

The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.

Lie group action, on a manifold

Notation

  • Unless otherwise specified, assume maps in this section to be continuous
  • geometry_c7565a6bb8a3dbf3f5f316636699172f028e2115.png is sometimes used to the denote a specific equivalence class related to the group-element geometry_224ae917d74dc2133d4403064c971bf562d4db50.png
  • geometry_23e7b3f838a0ddf35bb49ea3b136460606321796.png
  • geometry_df045b146261b7b1682cd92a58c9ac8408a2b321.png denotes the orbit of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png of the element geometry_8b77b207c01cd271ddf687a3c8355493504ec78c.png, i.e. geometry_76c6274c6fbb6bfcf68cd9b4c710045dbf413639.png
  • geometry_290d2927bc2fd2247c49e3af10118a976fca1646.png denotes the stabilizer of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png of the element geometry_8b77b207c01cd271ddf687a3c8355493504ec78c.png, i.e. geometry_ac9c72980c3e4c5caffd5b4f23a9c6318fbcac47.png

Preparation

Let geometry_a25a208ba09c77f50b364a437d6d12f554f8d011.png be a Lie group, and geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png be a smooth manifold.

Then a smooth map

geometry_a33f7fb0068b633fec81f0084e2b93a3dc473794.png

satisfying

  1. geometry_79db7eb0d3fb818836aa182ea9685be439b28f52.png
  2. geometry_749f3879a3420157e2ab159d2e264c1d4d2c0ea5.png

is called a left G-action on the manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

Similarily, we define a right action:

geometry_7b296cf2e3b0a365ecb6a31ff4215a57dec31c47.png

s.t.

  1. geometry_d4fc8905a632a0ee09385238d4f14b273504d206.png
  2. geometry_6637e565e614ebca897a6e98b065dd16f953ce95.png

Observe that we can define the right action geometry_74c7ecfc8e882d81bda5d9e226928520e5b47cc1.png using the left action geometry_9abf02c71d4c61b73cfb7beaf72fd1d4f035d114.png:

geometry_4b4948e1acfd3120207a4fcaa5a48494631205a0.png

geometry_4ca1a8ffb86b2c4233554963c47fccf576fb7aba.png

and

geometry_29d9fad627a7d2c9a547aae418a62e95f9ba8c9c.png

can be understood as a right action of geometry_fc1c688ee252236ba50f96afc05cef07e1a158cb.png on the basis geometry_34dd97d52e71f76366362427597380b22c23536a.png and a left action of geometry_fc1c688ee252236ba50f96afc05cef07e1a158cb.png on the components geometry_c01d3af05b51390c64763afafa94c52754232c08.png.

Let:

  • two Lie group geometry_a25a208ba09c77f50b364a437d6d12f554f8d011.png and geometry_aea1ceff4ae7a4dfc4c7f1978b8a4b2122b67bce.png and a Lie group homomorphism geometry_029461635b8978d3db618e5415f62ad55260d65f.png.
  • geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png be two smooth manifolds
  • two left actions

    geometry_296b770724b573e5cec86f1d117fcce2a0d03400.png

  • geometry_e872a467403103b86d1fbe9bafc9b501c3fe462a.png be a smooth map

Then geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png is called geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png equivariant if the following diagram commutes:

equivariant-lie-group-action.png

where geometry_49daba0dbda25af7a540f055f6df8a3d71a61be1.png is a function where geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png takes the first entry and maps to geometry_14a40f189f2ce698341c03cd5c099a337431c49f.png, and geometry_93369077affb352dbdb97e8b3182fd50784f2b14.png takes the second entry and maps to geometry_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png.

Let geometry_55cb0b0da368f4b43107feaab43b80cea372058b.png be a left action.

  1. For any geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png we define it's orbit under the action as the set

    geometry_b9cfe1c73f4e7125c94f7424a4f7bb13def345c6.png

  2. Let

    geometry_e99931e55a98aaa163fe4f38a4f307b23e2f4f4a.png

    which defines a equivalence relation, thus defining a partition of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png called orbit space

    geometry_134eb22d2be9717042e67519d1b4df5e1772a1a6.png

  3. For any geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png we define the stabilizer

    geometry_e4914b37528a3b27a2bbacc0554d10ced540373a.png

An action geometry_d59b024f1409d6812befa2ea3592a61a64f18c45.png is called free if and only if

geometry_57639c0485b72170d7cd84098512a76eb933821a.png

where geometry_1a4a18e739bc08439a04efa4dc6013a508ff1163.png is the identity of the group.

Examples of Lie group actions

  1. geometry_07ef3d3a36da6703c5a6293bae91b3ba332be558.png acts on geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png and is a Lie group
  2. geometry_7613df2709e5256630dae6652e5f9f19ee3b5e24.png acts on geometry_5718791916ddba2787815c28e6f9830bd7ccd16f.png since

    geometry_efd96db25ed114c09bd5afc13029cf7f2e6cac96.png

    and thus geometry_2fcbaf86660a326a0b2673a1967427ff0a702e31.png we have

    geometry_137078d50a3dbcbc5192a2388ff8175abe506777.png

    • Of course, geometry_06c1f35912be254f9b69db0146b6b26b04ae67e4.png also acts on geometry_d2b47b30887d606006181335bb3584202a43fe5d.png
  3. geometry_a221ec3b3542d8d6d92fea94db673f6a7686301b.png acts on geometry_de1c14469b1325043c6dd05b7cf9674654cdec97.png

G-homogenous space

Let geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png act on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, then geometry_275c44858a555098cf1c1422d335832159f38e80.png

  1. geometry_290d2927bc2fd2247c49e3af10118a976fca1646.png is a closed Lie subgroup of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png
  2. geometry_b391a1596e7e171e87d736d3b214f5a811b63ce5.png is an injective immersion geometry_2b57a7701a71f8f98d8749a89620ab58d78f03ec.png

In particular, geometry_7210671021a1df41cec7afba7dd8d61e503bd704.png is an immersed submanifold in geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png and if it is a submanifold, then geometry_b391a1596e7e171e87d736d3b214f5a811b63ce5.png is a diffeomorphism geometry_631053c95fadcc7ba803bdc6615fc12f16c5ac10.png.

A G-homogenous space is a manifold geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png with transitive action of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, i.e. geometry_5e84f0239009fdfb42c1d14a2af922250b3b3675.png

geometry_0652de7d64ed142d46543dcc728f5937fa12d02b.png

If geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a G-homogenous space, then geometry_275c44858a555098cf1c1422d335832159f38e80.png there is a fibre bundle geometry_4112dcfb273f0a5238fa250245686960d43acb63.png with fibre geometry_290d2927bc2fd2247c49e3af10118a976fca1646.png, where

geometry_7b69382381f2ba44a08d8c2dd87f216b0b325dfc.png

The proof follows from the fact that we have a diffeomorphism between geometry_ed068cf21b1e90925546607a28e72d65e71c49b0.png and geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png following thm:stabilizer-is-closed-lie-group-and-action-diffeomorphism-to-manifold , and we already know that geometry_9c7612637beb487675e7579be877755ec6c0807b.png is a fibre bundle with fibre geometry_290d2927bc2fd2247c49e3af10118a976fca1646.png from earlier.

Example of application of G-homogenous space

geometry_368a97d629e521a8d76420a7d6d0a450f3d7cfd1.png

geometry_45fba7f3fdd874b009104e0725a476fb76586eaa.png acts on geometry_cccf893444e5a93f5a65afa849ff21bb8e40d400.png with

geometry_4108d51b16854af0b5ada6bdabfee47976cd155e.png

So geometry_9d7f0ada8211a889c17b98582e41890443c1c236.png has geometry_7a84c9a383f9772338016d101ccc096be06af784.png on the first column, and then zeros on the rest of the first row, and then the rest of the matrix is geometry_e37e871bafc0b5a95c3f91178e951b1adefe1b84.png. MAKE THE MATRIX.

We get the fibre fundle geometry_7d342ad64dc2a7a9746505d523f831abca45cbba.png with fibre geometry_e37e871bafc0b5a95c3f91178e951b1adefe1b84.png. We then have the exact sequence

geometry_480c3f9076e0bd68d47edb83d2855aa79b96e607.png

If geometry_d84750a4583c79e1eb011cfb435835dbe0e26465.png and geometry_5da838648421faf72cedf7b0ca76dc6f1506c9df.png then

geometry_3f6d1397012a59213265091512b149e41fc88c1f.png

If geometry_340ac06ee792f85023d2206be205eb51f78efe3e.png, we have geometry_5b92547d9d2b02712331937b64cb5c36a32f1de3.png. This implies

geometry_0a915556c55cc67c99ccf6056f99b7df14f622c8.png

geometry_6175b4d22638184b9cf4a7e1accee432807928fe.png is a point, hence connceted and simply connected, hence so is geometry_45fba7f3fdd874b009104e0725a476fb76586eaa.png for all geometry_ddadf38e9b6f2913a5b64670ddde97d2cddaf0d0.png!

Principal fibre bundles

A bundle geometry_d343a1098f9e6fec7a05744fe96592b1bc7cdfef.png is called a principal G-bundle if

  1. geometry_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a right G-space, i.e. equipped with a right G-action
  2. geometry_7b47e169825307cd2afea6a3acc9a9e40edc32e0.png is free
  3. geometry_d343a1098f9e6fec7a05744fe96592b1bc7cdfef.png and geometry_d2c12e804db7840d4e3d288a7bf3c91d907a96f8.png are isomorphic as bundles, where
    • geometry_99fa11f46828009dee295bab1caf49a55552e97d.png, takes a point geometry_224ae917d74dc2133d4403064c971bf562d4db50.png to it's orbit / equivalence class
    • Since geometry_7b47e169825307cd2afea6a3acc9a9e40edc32e0.png is free:

      geometry_c2d506214fd0de04b90cbbd88c279809c99518d6.png

Suppose we have two principal G-bundles:

Then a principal bundle morphism or map needs to satisfy the following commutation relations:

principal-bundle-morphism.png

and a further restriction is also that there exists some Lie group homomorphism geometry_c8eb011ee3ca155709cf64432e7a0f42dacfaaae.png, i.e. geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png has to be a smooth map satisfying:

geometry_71c5b4d83f55610b3ec2fb110ba1ae88215f61f8.png

A principal bundle map is a diffeomorphism.

A principal G-bundle geometry_11fd7db38ad4ba05c23709c2da5670e88876d4a2.png under the action geometry_7b47e169825307cd2afea6a3acc9a9e40edc32e0.png by geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is called trivial if it is diffeomorphic to the principle G-bundle geometry_70fd6901ec02130d0c305df8307c1300a4b1a81e.png equipped with

geometry_3175b6ccf0c02c1f466e2e35cd3cb00dbf7589a2.png

and

geometry_e9dcae5c8f752b256016661399193a3bde80b2cf.png

That is, geometry_11fd7db38ad4ba05c23709c2da5670e88876d4a2.png is trivial if and only if it's diffeomorphic to the bundle where the total space is the mfd. geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png with geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png attached as the fibre to each point, or equivalently (due to this theorem) if there exists a principal bundle map between these bundles.

trivial-principal-bundle.png

A principal G-bundle geometry_11fd7db38ad4ba05c23709c2da5670e88876d4a2.png is trivial if and only if

geometry_f2430b383181c69247d3f7bfa6087167fa26578f.png

i.e. there exists a smooth global section from geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png to geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png.

Let geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png be a manifold. First observe that

geometry_717a7a0fd00605f26364de7cc26d2751a40c56bb.png

i.e. the bundle is simply the set of all possible bases of geometry_fea0c700da319ef1432b9b27352c42cd98440610.png.

The frame bundle is then

geometry_6b454e2100ca9c02bc2cdfc03524a00e36aef580.png

where geometry_83082a71f5c860362618d550f2729a1476f784e7.png denotes the unique union.

Equip geometry_c0a6a0a82fb77cf3b6cfcdd4c8ec5ac7bd2b8f5d.png with a smooth atlas inherited from geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

Further we define the projection geometry_f4dc0ccf03e2f3105ed93dcbfafa3d48e16662b0.png by

geometry_62d55fa615415b03b3daccff433fab5df3907b65.png

which implies that geometry_b56fbaab45b6a52934cc7ab7e8d2be8321cc105b.png is a smooth bundle.

Now, to get a principal bundle, we need to establish a right action geometry_7770ed45e6e8c3511bc362a8b159778bf9868540.png geometry_7b47e169825307cd2afea6a3acc9a9e40edc32e0.png on geometry_c0a6a0a82fb77cf3b6cfcdd4c8ec5ac7bd2b8f5d.png, which we define to be

geometry_e2ea78abf48aec870c550d0dc20ae9379f186bf5.png

which is just change of basis, and geometry_7b47e169825307cd2afea6a3acc9a9e40edc32e0.png is therefore free.

Checking that this in fact a principal geometry_7770ed45e6e8c3511bc362a8b159778bf9868540.png bundle by verifying that this bundle is in fact isomorphic to geometry_34699223dd2798e7e43774ce5fc7df5e9a194b4b.png.

Observe that the Frame bundle allows us to represent a choice of basis by a choice of section geometry_de32b094129e4e03f562577e6b8b51603d8ca54e.png in each neighborhood geometry_c0602cbb7fa0fbddcb3ef769ca192057e559855a.png, with geometry_41bde08662019dbc12e10445ed3a7f19cef5a551.png.

That is, any geometry_1959bc1263a144f7b5cd243841092f3d6b47790f.png is a choice of basis for the tangent space at geometry_41bde08662019dbc12e10445ed3a7f19cef5a551.png. This is then equipped with the general linear group geometry_7770ed45e6e8c3511bc362a8b159778bf9868540.png, i.e invertible transformations, which just happens to be used for to construct change of bases!!!

Ya'll see what this Frame bundle is all about now?

Associated bundles

Given a G-principal bundle geometry_11fd7db38ad4ba05c23709c2da5670e88876d4a2.png (where the total space geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png is equipped with geometry_e5f070f1d5413347a07b4a1c283dbd7b7b9ca7f0.png for geometry_0e3abaf7346b8b13bdf539f63ca9c75094dd786b.png) and a smooth manifold geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png on which we can have a left G-action:

geometry_fb93eb7ad7a9951851f84564a52c309d1f1a73cd.png

we define the associated bundle

geometry_cc36d9e50078c55daf0679090eaa321c93e36d0c.png

by:

  1. let geometry_74f2f7d339204421a7a83ff157bdfa12ac4357da.png be the equivalence relation:

    geometry_4bcf10992370f666d37a2812b1a40a3e8e995aff.png

    Thus, consider the quotient space:

    geometry_660b20c2e9bc381b1bb55ff44b8ee2bcc03e1537.png

    In other words, the elements of geometry_c56ab93e1fbbdccbf869ccb0d9b09876bbec27a8.png are the equivalence classes geometry_42b3ad6dc784f20502b8f070041249c4fd3aa55a.png (short-hand sometimes used geometry_9e595b79aec35024e72817682eb3bee06ef207eb.png) where geometry_18337a0035635b91cb717630ba58f94cc640692f.png, geometry_4100ea22c75ff38f4db49d8987335d2f16041673.png.

  2. Define geometry_9fc8ddcdbb5fcf84a6b06905086a86dea5e41143.png by

    geometry_d124bd8f682ebdd1931dbf5873dd4390544a2c47.png

    which is well-defined since

    geometry_3ad1ad6827ac528ce0590645ccd43888a3400dc0.png

This defines a fibre bundle with typical fibre geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png:

geometry_cc36d9e50078c55daf0679090eaa321c93e36d0c.png

Example of associated bundle: tangent bundle

That is, if we change the frame by the right-action (which represents a change of basis), then we must change the components of the tangent space (if we let geometry_ae9511ebea37d9eb8b10327effb374163f382251.png be the tangent space).

Then geometry_77cf6187b0f5f47f1bac0327baf418e9bd23bb2d.png is the associated bundle of the frame bundle.

Example: tensor associated bundle
  • geometry_21f7a6a7daa0f5ab7b32209ac8d9e5e2f26dcce9.png
  • geometry_a056f8785c35ea60fc66b9c732b6aea35b333127.png
  • geometry_6089b9a32b32f43ff58c8fcaab539ceb8db466cc.png

With the left action of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png on geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png:

geometry_be9b2fe6847828784ec7763e9a387d352e7ce2ac.png

Which defines the tensor bundle wrt. some frame bundle geometry_41dfd6ccdcd3bf88a716d7dcaa273fbad03f599c.png.

So what we observe here is that "changes of basis" in the frame bundle, which is the principal bundle for the associated bundles tangent bundle and tensor bundle, corresponds to the changes to the tangent and tensors as we are familiar with!

That is, upon having the group geometry_a056f8785c35ea60fc66b9c732b6aea35b333127.png act on the frame bundle, thus changing bases, we also have this same group act on the associated bundles!

Example of associated bundle: tensor densitites
  • geometry_21f7a6a7daa0f5ab7b32209ac8d9e5e2f26dcce9.png
  • geometry_a056f8785c35ea60fc66b9c732b6aea35b333127.png
  • geometry_6089b9a32b32f43ff58c8fcaab539ceb8db466cc.png

But now, left action of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png on geometry_b2db59aeb94cbc1050079892ff07b21b493513b7.png:

geometry_23960017843182449ddad09c408f6e7a2276ad1f.png

for some geometry_f18a12de272c4edd68750fb00de9e8136a8f4fac.png.

Then geometry_3c54e698747806ea01f8e931cf69740d08fbf1d4.png is called the (p, q)-tensor geometry_e29af05574032ace665d996d46b3280fc49866ef.png density bundle over geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

Observe that this is the same as the tensor bundle, but with the factor of geometry_ce5477fdbca6178a22943491d73be8d0b0d55818.png in front, thus if we had instead used geometry_06d17e590299a54c429fe669120f92ee298965d3.png, i.e. orthogonal group, then geometry_b86eef3308c3ead3e3d4c426cdb4ac0485a39898.png, hence it would be exactly the same as the tensor bundle!

Associated bundle map

An associated bundle map between two associated bundles geometry_ebc064faad688ccd05b8e166bf78dcefe7871e0b.png (sharing the same fibre, but being associated to arbitrarily different respective G-principal bundle geometry_9d6f453b829a28eb3a8a4c3be17f438a317f55b0.png)

associated-bundle-maps.png

is a bundle map geometry_eab632c31d958f6077c96f412dc97564cb1a3710.png (structure-preserving map of bundles) which can be costructed from a principal bundle map between the underlying principal bundles,

principal-bundle-map-in-associated-bundle-map-definition.png

where

geometry_7b3a888b3a4bce09717bd6c85a309158c14966bb.png

as

geometry_2a02cf03a52b7ac7354ec417a1d5add961ddbd19.png

Restricted Associative bundles

Let

If there exists a bundle morphism (NOT principal) such that

principal-bundle-map-in-restricted-association-bundle-map-definition.png

with:

  • geometry_8029800121063b340fc866cea266e45eccbd390c.png
  • geometry_1a4f497bca2beee6aaf8b5db6db05b6e06ebd47c.png

Then

  • geometry_b4d9fd581bbca34c7ac3af242c8396681a5fab69.png is called a G-extension of the H-principal bundle geometry_b731a9422a070ab0a12a8a107fffcf1448b6abbd.png
  • geometry_b731a9422a070ab0a12a8a107fffcf1448b6abbd.png is called an H-restriction of the /G-principal bundle geometry_b4d9fd581bbca34c7ac3af242c8396681a5fab69.png

i.e. if one is an extension of the other, then the other is a restriction of the one.

Connections

Let geometry_b4d9fd581bbca34c7ac3af242c8396681a5fab69.png be a principal G-bundle.

Then each geometry_ac0e2df69c08e54685eb918fa44186ac11b32c73.png induces a vector field on geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png

geometry_c13d7929086ded6b655735eee22f41753d359c20.png

It is useful to define the map

geometry_9172a449042fb08b9d63d33e891e9141fa6f5321.png

which can be shown to be a Lie algebra homomorphism

geometry_75895701feebee85625d51f3379c3987d5ce14c3.png

where:

  • geometry_db0604a541d4fb5722354cf5f9b81ac5416b76cf.png is the Lie bracket on geometry_ede6de58b50a8ca1fec46e614cdbb5c6d0344c41.png
  • geometry_ca1fd9734446b1235e0941b5bcaff30302543c4e.png is the commutation bracket on geometry_c66661834c85dea1ecb895df1f20ab422a0fe7de.png

Let geometry_18337a0035635b91cb717630ba58f94cc640692f.png. Then

geometry_4f16b8b60ca9dbe0a018b65d6a020804d1578c28.png

where:

geometry_2f7528c8e1a838e1085b68a15bd396d3886f83f8.png is called the vertical subspace at point geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.

Idea of a connection is to make a choice of how to "connect" the individual points of "neighboring" fibres in a principal bundle.

A connection on a principal G-bundle geometry_b4d9fd581bbca34c7ac3af242c8396681a5fab69.png is an assignment where every for geometry_18337a0035635b91cb717630ba58f94cc640692f.png a vector subspace geometry_4955b52578eb11e72c5539e3e6174ae87c3ced42.png of geometry_6daa33699e6065139c559f02d8995d195d3371a6.png is chosen such that

  1. geometry_ca602b7ee62fc82cf196e4a56e4e5516694289af.png where geometry_e4d5268724839498a9402b989e6add38ac08edff.png is the vertical subspace
  2. The push-forward by the right-action of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png satisfy:

    geometry_d8950053f913931bee9ab0f5b19ff3d3e20c3965.png

  3. The unique decomposition:

    geometry_b075b25ba0280aa3208ad9202aba9ebd17bef899.png

    leads, for every smooth vector field geometry_ce7a0794ee750a650308034ab863869f4d2d06a1.png, to two smooth vector fields geometry_3f92a8630b267ff0e4455260170dfb07c5db2f12.png and geometry_d70ea0f5ca6116f44c75931b1ddf3c699a7981dd.png.

some principal G-bundle geometry_b4d9fd581bbca34c7ac3af242c8396681a5fab69.png

The choice of horizontal subspace geometry_4955b52578eb11e72c5539e3e6174ae87c3ced42.png at each geometry_18337a0035635b91cb717630ba58f94cc640692f.png, which is required to provide a connection, is conveniently "encoded" in the thus induced Lie-algebra-valued one-form geometry_e29af05574032ace665d996d46b3280fc49866ef.png, which is defined as follows:

geometry_d5f4b1c280c7970edfa2ccde899bf8315ad143de.png

where we need to remember that geometry_8daf6b09479fdc52a6dad5657e2cd8c48571657d.png depends on the choice of the horizontal subspace geometry_4955b52578eb11e72c5539e3e6174ae87c3ced42.png!

Recall that geometry_b9e7b05a3c53282ccd19c3bd3eb0a59867ccbd40.png is the map defined in the beginning of this section:

geometry_f273311f29898c6e4d18e67a4f7330e4399dd216.png

where geometry_e29af05574032ace665d996d46b3280fc49866ef.png is called the connection 1-form wrt. the connection.

That is, the connection is a choice of horizontal subspaces geometry_4955b52578eb11e72c5539e3e6174ae87c3ced42.png of the fibre of the principal bundle, and once we have such a space, we can define this connection 1-form.

Therefore one might wonder "can we go the other way around?":

Yes, we can!

geometry_99ae6e7abc6f0797be55cf457db3ca464659d97c.png

A connection 1-form geometry_e29af05574032ace665d996d46b3280fc49866ef.png wrt. a given connection has the properties

  1. geometry_f344daedff19607b32a817e641b1bb6b1bac7133.png
  2. Pull-back

    geometry_aa7bd79855db3c1d15d1a30308f7c463e5f9b29b.png

    where we recall

    geometry_b4fa558bdb18a5e6ca903999e796f2e15a2da67f.png

  3. Smooth, since

    geometry_611ab84c0cd57afaf400c17691ddd9bc86f71361.png

    where geometry_937a9fafe319f7169579836a8bf41e7c2c648398.png is smooth since the exponential map is smooth.

Different approach to connections

This approach is the one used by Schuller in the International Winter School on Gravity and Light 2015.

A connection (or covariant derivative) geometry_a8364d4d7ee67f4e517ea28d99131f12497f0435.png on a smooth manifold geometry_5bcd7abae18b07cb9a663b6d81e33b3de91ad823.png is a map which takes a pair of consisting of a vector field geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and a (p, q)-tensor field geometry_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png and sends this to a (p, q)-tensor field geometry_f2e8dc2b7d6859e92e189b724b20d4269c47cc8f.png, satisfying the following properties:

  1. geometry_58cfd17708c72da4084a0c90a761db54d5a88a99.png
  2. geometry_9e79348f2deaa0e4436bbcbd65722a9118705bcf.png for geometry_052f229d7e2fe6593c756addfac06d6335269a98.png
  3. The Leibnitz rule for a (1, 1)-tensor:

    geometry_b5d32003855ba573fecbed699fb722d189cb94e4.png

    and the generalization to a (p, q)-tensor follows from including further terms corresponding to geometry_3cc2a4fd41d60cd5da2707a686a1aee807241c08.png of the arguments. It's worth noting that this is actually the definition obtained from

    geometry_cd2105e7a0b5754219770ec4e4232ecf1121cca3.png

    which is the more familiar form of the Leibnitz rule.

  4. geometry_a26cfd61171418ae69f438d3334d59f6421248c4.png

    geometry_9dce031d31b971d792850850e5da38651f066468.png

Consider vector fields geometry_6157262f7504734f472c73ef69b5993a4fd5e1ac.png. Then

geometry_fa2ddf6d574daa8fc868deeee1c45b59901bd196.png

  1. A vector field geometry_aa07b3a8458adb2855b54064282b1f340d44fbd4.png on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is said to be parallely transported along a smooth curve geometry_e79a0016f758fe16a2ad3a361d24a32f6a57dd56.png if

    geometry_edbce31b08acd4922aa1e7714e4cd5d2443c7fc3.png

    i.e.

    geometry_1efa70d323c86c75db4acf1f3ab77e151d673420.png

  2. A weaker notion is

    geometry_fe00e2b6c0e47f19febd21185aa2ed6bca172680.png

    for some geometry_88a7652d7454f07f9cd6522d2226b2d806740288.png, i.e. it's "parallel".

Local representations of connections the base manifold: "Yang-Mills" fields

In practice, e.g. for computational purposes, one whishes to restrict attention to some geometry_c0602cbb7fa0fbddcb3ef769ca192057e559855a.png:

  • Choose local section geometry_4ef5e25cd1cdd3f05e0b59b4de5b5ffc96c6dc37.png, thus geometry_8142dc77548e67f88770b343909c04db22d6ea15.png

Such a local section induces:

  1. "Yang-Mills" field, geometry_35792cfc7db813dd5d385e7cd3e119e3c9575f58.png:

    geometry_6f12457c2b55aa8ec6639a676909cfab57530f4b.png

    i.e. a "local" version of the connection 1-form on the principle fiber geometry_e29af05574032ace665d996d46b3280fc49866ef.png, defined through the pull-back of the chosen section geometry_de32b094129e4e03f562577e6b8b51603d8ca54e.png.

  2. Local trivialization , geometry_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png, of the principal bundle

    geometry_cd2e14b02139a2868f9e2cffc11ea2f1d7d82132.png

    Then we can define the local representation of geometry_e29af05574032ace665d996d46b3280fc49866ef.png

    geometry_3251ec9224641053c63d54f94b76cd675218d348.png

local-trivialization-yang-mills.png

Suppose we have chosen a local section geometry_4ef5e25cd1cdd3f05e0b59b4de5b5ffc96c6dc37.png.

The Yang-Mills field geometry_35792cfc7db813dd5d385e7cd3e119e3c9575f58.png, i.e. the connection 1-form restricted to the subspace geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, is then defined

geometry_6f12457c2b55aa8ec6639a676909cfab57530f4b.png

Thus, this is a "Lie algebra"-valued 1-form.

Choosing the local trivialization geometry_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png:

geometry_cd2e14b02139a2868f9e2cffc11ea2f1d7d82132.png

Then we can define the local representation of the global connection geometry_e29af05574032ace665d996d46b3280fc49866ef.png:

geometry_1859e437428c95cc7ccdbf4be0dcbf07a6800587.png

given by

geometry_e65bdb4fec08fbbebd9a4f52dd8782a9e1196786.png

where

local-trivialization-yang-mills.png

geometry_8206592d89c41a2657b4a1284831438b5e175d57.png

where we have

  • geometry_0a57c2732390206dc3ac720bc28acf87718f5ac7.png
Gauge map
  • Can we use the "local understading" / restriction to geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, to construct a global connection 1-form?
  • Can do so by defining the Yang-Mills field for different, overlapping subspaces of the base-manifold!
    • Need to be able to map between the intersection of these subspaces; introduce gauge map

Suppose we have two subspaces of the base manifold geometry_5a74a9dff975c112c49d7101c701565bc19e4e3a.png, such that geometry_cf275653c9f72b1e3a67c85b783e56832dd60287.png for which we also have chosen two corresponding sections geometry_be5aabe25859a08874decdbd56472b20e3110cec.png, such that

gauge-map-definition.png

We then define the gauge map as

geometry_43ab05c8d84758e29e560a8e84c70897aaaee074.png

where geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png is the underlying Lie group (on geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png), defined on the unique geometry_eefff5e1f8e1c8315b01f9916e5b8bdba301964b.png for all geometry_3aedc8fb72e946a684a9507f8a68970d7128defa.png:

geometry_8220a944ee2ca2282b88fc93b62bd14031c8f1f0.png

where geometry_567ecd7330dc92f9f658a7ecce6418a8dff97340.png is the Maurer-Cartan form.

From the definition of gauge map, get

geometry_79c1a4037ad31f5b5a35df1ee211cb994b4344ea.png

where:

  • geometry_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png denotes a point in the base-manifold
  • geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png denotes the the component (which geometry_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is NOT, hence the comma)

This theorem gives us the relationship between the Yang-Mill fields on the two different subspaces geometry_bd05792a975c871fe2d42cdbf309e1f07ae301cc.png of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png!

Example: Frame bundle

Recall that in the case of the Frame budle we have

geometry_f6c18f2e9be1f82c109290248ed82c7e9d981895.png

Then a particular choice of section geometry_792d1b660c210d9a0ff63ba4728b5ecdf877daa0.png for some chart geometry_19b0619723cdbe5c8c849bfeb710a7cc40026c31.png is equivalent of a specific choice of coordinates:

geometry_a7d200625c111a45f2b9d66f66a2b567428009c4.png

Let's first consider this as an instance of a Yang-Mills field:

geometry_ff5ab6649310af78a8467cea8461b8041b20a569.png

is then a Lie-algebra valued one-form on geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, with components

geometry_490412ce25fd444f255055714781ac1d88430578.png

where

  • geometry_3ba2462ecbfb3800b17a7a00e0c2c6dc15925c6c.png comes from being a one-form on geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png, hence geometry_c0fb46b67fdfc1039ddcfbd1c29a947362b9e76b.png components
  • geometry_382578e5248464fd6f2744548b8d92d7b28bd8c4.png from geometry_df4ce82e3947d2f1b52d371b16d779310c1c9323.png

We can obtain the gauge map for the Frame bundle. For this we first need to compute the Maurer-Cartan form geometry_567ecd7330dc92f9f658a7ecce6418a8dff97340.png in geometry_fe4484a86fe83a6253a4b20d3b6b25abbce78e21.png. We do this as follows:

  1. Choose coords an open set geometry_d0b8377ddf8daa05ddb8c928d4e495c79feeb471.png on geometry_7c7a794acef9eb38149a1f97fc567595d1ade7e1.png containing geometry_c59f0f9102710067b0f921ef16ecac7ab4ae446d.png:

    geometry_84199155c750549ccad15232d75abcdc0e8dc656.png

    where

    geometry_847af6c64286592539275ab6a79c71a9e4af63fe.png

    which are the "matrix entries".

  2. We then consider

    geometry_a493502556b87c86cc8dae4ad17b0bf7d60dfc3c.png

    since

    • geometry_a990dd58841582271199c0f9294b4d98e671a402.png is the unique integral curev of geometry_c079c0f3d0494a78f741aa735ca2dad53c3c8e08.png
    • by def. of vector-field acting on a map

This can then be written

geometry_e2c0f288bb5a90a24297f80dbe2c236790f83cd0.png

Hence,

geometry_01423c433964c62a5954446f96713e5aca711e03.png

Since

geometry_0132f794f59e6c2c1d7c32068db421a040a44471.png

we, in this case, have

geometry_b14d7104ff87ab1a6c6e3c0b421294a046c86d45.png

Then,

geometry_026254f69fb46592d4eeed2e7b6acae3f6be4295.png

as we wanted!

Recalling the definition of the gauge map, we now want to compute

geometry_72e805ebaf5082218622340baa50af38a564e09b.png

where geometry_acb0106122b7f90d9bd5639367a141a7e53d8327.png is the index of the components. To summarize, we're interested in

geometry_512c59b5d23669f95d9baa6ce63fd186d0fc2532.png

Let geometry_b32f254956e62c3386d6a9b1ef0febd02df5b724.png, then

geometry_763c3e9530bd64401afafc48097029db0f0959f8.png

Hence, considering the components of this

geometry_bc6f7090b9f51da20f6ab8772d57af00d4bdbb5b.png

where geometry_209b471bcc504502ec4e9d29977369c97cfc6dde.png denotes the corresponding matrix.

Now, we need to compute the second term:

geometry_4f6a4dee685e76f72922e684ef709c5dce596911.png

Observe that

geometry_bafc132b837d58d1065e54f008c53adc037e42d6.png

Thus,

geometry_7d0b193a7591c891cd96e704a80f6594a102f69c.png

The above can be seen from:

geometry_3cb7a87d7868b567e713f159bfed692ee69d1a80.png

Finally, this gives us the transition between the two Yang-Mills fields

geometry_669d1f45fc2b5549b1e18d357e1f3a21a6b2d07c.png

That is, we got it yo!

Parallel Transport

This is the proper way of introducing the concept of Parallel transport.

The idea behind Parallel Transport is to make a choice of a curve geometry_be2c50d326827aaf7f95c8ecf698320958a7b947.png from the curve in the manifold geometry_5fdff8e8e2a5e5053aa859caa7ed56d7ca27cd38.png, with

geometry_7f266d4140d68b46772299e3163a735cffef6ad3.png

Since we have connections between the fibres, one might think of constructing "curves" in the total space geometry_a9090c77ce9916955c745920bf8f134a0932d59d.png by connecting a chosen point geometry_3d6610a5110694791ecc072c341a86815eaff5f8.png for geometry_5a788aab9fab590cb30298059c1a15a73e43056a.png to some other chosen point geometry_f26d3973e6e7937c781b0d9607f7789710d6b2da.png for geometry_67d13962c24a45cbd9099bc47a5d8ecfd50eeadc.png, with geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png and geometry_41fe0d1ee1215ccc423346f2efcb3a70c41da4a0.png being "near" each other.

Then the unique curve

geometry_7b2595587dc5080fab2b83529fd8ac62ec52465f.png

through a point geometry_7715b36932a471d23990cb4281dd90f133d3561b.png which satisfies:

  1. geometry_60998cfd13a9a8da96238bf03c0c51c2cc764113.png
  2. geometry_7957eb2da1e45ea11f763452cf117d56267fbdd2.png, for all geometry_44a15174f3675c7b929a408a60552c136c1681ef.png
  3. geometry_5ed7571f73266f4129c7002a7f042042bcd137ae.png, for all geometry_44a15174f3675c7b929a408a60552c136c1681ef.png

is called the lift of geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png through geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png.

My initial thoughts for how to approach this would be to make a choice of section at each geometry_41b998a8533c617ee9f08ac05dd78fdf6b97fcf2.png, such that

geometry_b7760297b594753070f35d6fe3b8ff8b8b2697d6.png

where geometry_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png would be a choice for every geometry_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png. Just by looking at this it becomes apparent that this is not a very good way of going about this, since we would have to choose these sections for each geometry_c0602cbb7fa0fbddcb3ef769ca192057e559855a.png such that it's all well-defined. But this seems like a very "intricate" way of going about this.

The idea is to take some such curve as a "starting curve", denoted geometry_10218405c0b0ae92ec0be9ef0feb5695550d3f17.png, and then construct every other geometry_81b130cfdad7ff5015328a4f8669f35096cc25d6.png from this geometry_18f46e29ae5df6d58883632ac9a73c6a3ea7e102.png by acting on it at each point by elements of geometry_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, or rather, choosing a curve geometry_7b294c1c8d18213c84f381e98d8982d7c22d3bf2.png in the Lie-group, such that

geometry_370d4cd5086e1f58930e457577328dadd449c95c.png

i.e. we can generate any lift of geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png simply by composing the arbitrary curve geometry_18f46e29ae5df6d58883632ac9a73c6a3ea7e102.png with some curve in the Lie-group.

It turns out that the choice of geometry_7b294c1c8d18213c84f381e98d8982d7c22d3bf2.png is the solution to an ODE with the initial condition

geometry_b534f8c48cf852385c092c25cb49c4f54aef7b69.png

where geometry_c028dc5b2403a4f7c248b27ae3e2432cd6240a53.png is the unique group element such that

geometry_260323da4f9d71a710407aaa9194b8e2d35007a8.png

The ODE for geometry_7b294c1c8d18213c84f381e98d8982d7c22d3bf2.png is

geometry_8da208e636f745b12b4a3e489296aaf6bc15ceb2.png

with initial condition

geometry_b534f8c48cf852385c092c25cb49c4f54aef7b69.png

such that

geometry_260323da4f9d71a710407aaa9194b8e2d35007a8.png

Worth noting that this is a first-order ODE.

Spinors on curves spaces

Spin group is a double cover of the special orthogonal group

geometry_198d21459d40a3b28e6731d94dfd0f1cdcd94f51.png

where geometry_aeaa3b69b1e5879a358172c76546d6c6d881a1d4.png (i.e. inner product), i.e. we can construct a Lie group homomorphism

geometry_d81c55da362e4879b115a77608dd38cb07a09634.png

with

geometry_b67e7fd13e8225602edb826c23d715422612bf4d.png

In other words, the map geometry_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png is 2-to-1, where the "kernel geometry_f56d9e857313129777efeff7d6b2781eaaf1d4a8.png is the definition of spins".

Seems awfully similar to a bundle dunnit? With the addition that

geometry_8b6f6f0b8cb0e461a92a4b667a498e3edee16745.png

Complex dynamics

This section mainly started out as notes from the very interesting article by Danny Stoll called "A Brief Introduction to Complex Dynamics" (http://math.uchicago.edu/~may/REUDOCS/Stoll.pdf).

Notation

  • geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png denotes a Riemann surface

Definitions

A Riemann surface is a complex one-dimensional manifold.

That is, a topological space geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is a Riemann surface if for any point geometry_55817b477cbc7a068b3c934a0c4478369d0c512a.png, there is a neighborhood geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png of geometry_7225b076f6e6326f1636b11d1aad8de58bcc4761.png and a local uniformizing parameter (i.e. complex charts)

geometry_6efab5b9e930e84f13f5d49610a6474be655ece9.png

mapping geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png homeomorphically into an open subset of the complex plane.

Moreover, for any two such charts geometry_a4b480008193c50ca4186489bfe7949886b0772d.png and geometry_b1ee6d8c2b2fa6a86a03e018548d7d1cf69501d0.png such that

geometry_90f333810f6754d7d4e646f3f65f17b8056cf86a.png

we require the transition map geometry_d8de7d61c5b284ff6f7cd2882d15c0ef049a6b0f.png to be holomorphic on geometry_941b09d80be35a226414d4f22e383331dc57782c.png.

Let geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png be a simply connected Riemann surface. Then geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png is conformally isomorphic to either

  1. the complex plane geometry_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png
  2. the open disk geometry_1aeac941721258347fe2c5ff93495a1ab5ac4be1.png consisting of all geometry_cdc66794ed24fa48c75f9cf0311242af235a9ba3.png with absolute value geometry_5996d90dfc00a72362be13bee9945ab12ccfecea.png, or
  3. the Riemann sphere geometry_107ebeffb72b19129cf19677f681ded194a0c066.png consisting of geometry_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png together with the point geometry_f6f02b10995dc1353548c5ff27838dfc549df734.png with the transition map geometry_f37eeb4a1b0d244166db0cd286d736bf22660a17.png in a neighborhood of the point at infinity.

Normal cover

A deck transformation is a transformation geometry_b432e2e5aebc10d2968bd9e6e6c1cc6cefbd46d7.png such that geometry_db717f4255e8f917a0a9c0ab475008c4b95678f8.png.

A covering space is said to be normal if

geometry_58a06da8ee98f2780337eea8e0cbc700159e70c8.png

and there exists a unique deck transformation geometry_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png such that geometry_2ece6544172b180586a663903b3ada1c263ddbd6.png.

The group geometry_0885bdeebee973b8c0e45802dfd02d4b2a7d56be.png of deck transformations for the universal covering is known as the fundamental group of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png. Thus,

geometry_fd9a1c9452d692bdab012aaab65dd2fbddff310d.png

And we're just going to state a theorem here:

Let geometry_92757546978c1aad3468f8a69e8d334af2bfadac.png be a family of maps from a Riemann surface geometry_e1606e7346bacb31c433e48ab9a27150bcbdc1f8.png to the thrice-punctured Riemann sphere geometry_5e33ebb5d6e11c58e42e7b2c801820b90cf0ca8d.png. Then geometry_92757546978c1aad3468f8a69e8d334af2bfadac.png is normal.

Partitions of Unity

Suppose geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a topological space, and let geometry_27849543c12a2346422ca53df2b28d88e1dda5c4.png be an arbitrary open cover of geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png.

A partition of unity subordinate to geometry_8c8d4992aa385c22d79f66db8edced75a661b728.png is a family geometry_8e7cc6e83138e4ca3f1620f0b2b88061aadc90d3.png of continuous functions geometry_84420ee5478932e06093da5d5f420a05c154c44f.png with the following properties:

  1. geometry_a5687f292e2853e4efa4d5717961bf70b868ac7b.png for all geometry_13ac77bee96f71b2538ceab06ff6bc6a9621063d.png and all geometry_2e159c5b3380548dcf6ab328cbfe3f5f22215d25.png
  2. geometry_1658f356482c05e5a62c5f48eda775bb885eb742.png for each geometry_13ac77bee96f71b2538ceab06ff6bc6a9621063d.png
  3. The family of supports geometry_612e0d5ae34c868dc690b17e56412281ebd842bb.png is locally finite, i.e. every point has a neighborhood that intersects geometry_30f7b7e26fd8414e78085cb99e84cd476c3390c7.png for only finitely many values of geometry_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png:

    geometry_4199313a7a92fafa31be3804a1a223ba67b07c63.png

    for some geometry_40b36ed008e06781e4bc8d73db929d9a0c8b3b33.png.

  4. Sums to one:

    geometry_29619e7f4f4239792634578e4f9a859e95b6a21b.png

    Since on each open geometry_a52e9f1d3f8d4dc20bb0f4f3be4f6501ae831537.png there is only finitely many non-zero geometry_99f6313989d94edba80f6dc62143a62f83dc4efe.png, the issue of convergence of the above sum is gone, as it will only contain finitely many non-zero terms.

If geometry_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is a smooth manifold, a smooth partition of unity is one for which every function geometry_99f6313989d94edba80f6dc62143a62f83dc4efe.png is smooth.

For a proof, see p. 40 in lee2003smooth.

Integrations on chains

Notation

k-cubes

Let geometry_c5f6ed1f06af26900b01b5f8fad816b815350a42.png be open.

A singular k-cube in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png is a continuous map geometry_dab31135200208c7ff2ccaf6af0ea74ef6f7c5b1.png.

  • A singular 0-cube in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png is, in effect, just a point of geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png
  • A singular 1-cube in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png is parametrized cuvre in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png

The standard (singular) k-cube is the inclusion map geometry_0530487829f67b04eff1088999e788e5bdb47e04.png of the standard unit cube.

A (singular) k-chain in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png is a formal finite sum of singular k-cubes in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png with integer coefficients, e.g.

geometry_d441929c2ef653bd1772da4c362e7f1f7798fd68.png

k-chains are then equipped with addition and scalar multiplication by integers.

Boundary of chain

Consider inclusion mapping to the standard k-cube geometry_0530487829f67b04eff1088999e788e5bdb47e04.png.

For each geometry_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png with geometry_ed3cb1bec694673ed79f858a89960da0f64e5ef5.png we define two singular (k - 1)-cubes:

geometry_0b7bd20c32bc4e782cfccd5246f958580889dbf6.png

and

geometry_5f9bc0c34bcdb0fe9cb2e59bae8d98ac0307436a.png

We refer to these as the geometry_225913d78e352f98f6ddcbf57e6dd10878c4e0fd.png and geometry_d4b87fbb34580f0ce405692239a7848d7d398be8.png of geometry_5574010febeca82ffdf93c34903a33001d76ed98.png, respectively.

Then we define

geometry_d449cfae4a6012605f19278d6b63df2955deb695.png

Now, if geometry_dab31135200208c7ff2ccaf6af0ea74ef6f7c5b1.png is a singular k-cube in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, we define its geometry_aa3d03a07a2cfb5dfbef2eed1bcc8f8b86a274a3.png by

geometry_47e254937167a7372618de06aba917dfe93cc308.png

and then define

geometry_d00b1482c32e8879410d0da98888fc4883842ece.png

We extend the definition of boundary to k-chains by linearity:

geometry_795de1d8584655fe6b7c105ba7556115e018130d.png

If geometry_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png is a k-chain in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, then geometry_1b81c43da6ec4f050639ac9c633118afc4a5adbc.png, i.e. geometry_fd5d8b0b2bd85964986a7c7acd7db71e677340a5.png

Woah! Seems to corresponding heavily with geometry_42777dcd4fbddf82000c9672c7a274847f00c282.png as stated thm:exterior-derivative-squred-is-zero.

Integration

Now let geometry_cd36a7ad197e683413e83154e86581ceaa018865.png, i.e. geometry_e29af05574032ace665d996d46b3280fc49866ef.png is a differential k-form on the unit k-cube geometry_b9dfe52d0bf1addda4cdff8b01e5f6efe962f468.png in geometry_967e180ce96954152b14ff0e07264fce177b3b29.png. Then

geometry_60bdd4e75c763ed06dffa67770f1807cb1479750.png

In which case we define the integral

geometry_5037d24957794651b365e7f84980e98e8af57254.png

If instead geometry_3c0da6a1ed34edbc60e1ccbf66e81298e19cb34e.png for open geometry_c5f6ed1f06af26900b01b5f8fad816b815350a42.png, and geometry_dab31135200208c7ff2ccaf6af0ea74ef6f7c5b1.png is a singular k-cube in geometry_da9cb51849b13210fa778a80b9907f86fe90c379.png, we define

geometry_582506581d935aa5172fbe6d2aed513327bc45d6.png

i.e. integration of a k-form over a singular k-cube is defined by pulling the k-form back to the unit k-cube in geometry_967e180ce96954152b14ff0e07264fce177b3b29.png and then doing ordinary integration.

By linearity, for a singular k-chain geometry_77099dec2360c53d4d237b46f95779df1f379b49.png,

geometry_1ea68483a2c6039656272d7ed41763cc4efc653f.png

Let

Then

geometry_31741cc0261786a8b58797980e39c75e2cbddfc4.png

Q & A

DONE Tangent spaces and their basis

  • Note taken on [2017-10-10 Tue 05:03]
    See my note at the end of the question

When we're talking about a tangent space , we say the space is defined by the basis of all the differential operators , i.e.

geometry_6f55ec41dd4ed362981e6efa671e3e413f3fcb48.png

Now, these geometry_bb6c54e095d0cc098c522518b126cfc6ec38d72f.png, are they arbitrary? Or are they dependent on the fact that we can project them onto Euclidean space in some way, i.e. reparametrise some arbitrary basis geometry_b58948878c945ef6b16cee1c64ec17bdea377649.png to geometry_d630033d37300697172fc9e8502d8b188469a9af.png.

I believe so. In fact, the notation we're using here is that geometry_c6229db7b8ec8292287aca9c874abf03cc6409d4.png, hence we're already assuming the domain to actually be in geometry_004097ff73cb85a0f596c8a3b60218ece0e16be1.png, i.e. there always exists a reparametrisation to "normal" Euclidean coordinates.

DONE Is the tangent space at each point an intrinsic property or does it depend on the embedding space?

Bibliography