Editing Lecture 2 - Thursday Week 1







== Lecture 2 - 15 Jan 2026 ==

Review from diff geo:

* Tangent space <math display="inline">T_{m}M</math>
* Tangent bundle <math display="inline">TM</math> with map <math display="inline">\pi:TM \rightarrow M</math>
* Vector field on <math display="inline">M</math> is:
** derivation of <math display="inline">C^{\infty}(M)</math>
** map <math display="inline">X:M \rightarrow TM</math> such that <math display="inline">\pi \circ X = \text{id}_{M}</math>.

We can do same construction after cooking up new vector spaces from the tangent space <math display="inline">T_{m}M</math>.

e.g. <math display="inline">T_{m}M \rightarrow {T^{*}}_{m}M = \left( T_{m}M \right)^{*}</math> is the dual space of <math display="inline">T_{m}M</math>, called '''cotangent space'''. Then we have <math display="inline">T^{*}M = \cup_{m}T_{m}^{*}M</math>, also a manifold, the '''cotangent bundle'''.

A map <math display="inline">\alpha:M \rightarrow {T_{m}^{*}M}</math> such that <math display="inline">\pi^{*} \circ \alpha = \text{id}_{M}</math> (i.e. a section of <math display="inline">\pi^{*}</math>) is a '''one-form'''.

note <math display="inline">\Lambda</math> is '''exterior power'''. <math display="inline">\Lambda^{k}{T_{m}^{*}M} \rightarrow {T^{*}M}</math> …

More generally for <math display="inline">{T_{m}M}\text{tensor}\ldots\text{tensor}{T_{m}M}\text{tensor}{T_{m}^{*}M}\text{tensor}\ldots\text{tensor}{T_{m}^{*}M}</math> we get bundle <math display="inline">{TM}\text{tensor}\ldots\text{tensor}{TM}\text{tensor}{T^{*}M}\text{tensor}\ldots\text{tensor}{T^{*}M}</math>

in particular

* <math display="inline">\Lambda:\Omega^{k}(M) \times \Omega^{l}(M) \rightarrow \Omega^{k + l}(M)</math> where <math display="inline">(\alpha,\beta)| \rightarrow \alpha\Lambda\beta</math>



* <math display="inline">X</math> vector field, <math display="inline">\alpha \in \Omega^{k}(M)</math>, then <math display="inline">\iota_{X}:{\Omega^{k}(M)} \rightarrow \Omega^{k - 1}(M)</math>, where <math display="inline">\alpha| \rightarrow \iota_{X}\alpha</math>, which has equation

<math display="block">\left( \iota_{X}\alpha \right)\left( X_{2},\ldots,X_{k} \right) = \alpha(X,X_{2},\ldots,K_{k}).</math> So we plug in <math display="inline">X</math> to first arg of <math display="inline">\alpha</math>.

=== Lie bracket ===

For <math display="inline">X,Y</math> in <math display="inline">\Gamma({TM})</math> where <math display="inline">\Gamma</math> is the space of sections of …

<math display="inline">\rightarrow \lbrack X,Y\rbrack \in \Gamma(TM)</math> defined by: for all <math display="block">f \in C^{\infty(M)}</math>, <math display="block">\lbrack X,Y\rbrack(f) = X\left( Y(f) \right) - Y\left( X(f) \right).</math>

Note that the above is a derivation, whereas <math display="inline">X\left( Y(f) \right)</math> on it’s own is not.

Lie bracket has properties of

* symmetry
* bilinearity
* jacobi identity

=== Flow Map ===

Given <math display="inline">X \in \Gamma({TM}),m \in M</math>, we get curve <math display="inline">c:(a,b) \rightarrow M</math> such that <math display="inline">c(0) = m</math> and <math display="inline">c\prime(t) = X_{c}(t)</math>. that is to say flow curves exist at each point.

If <math display="inline">M</math> is compact then all such curves are defined on <math display="inline">(a,b) = ( - \infty,\infty) = {\mathbb{R}}</math>. These curves combine to give a group action of <math display="inline">({\mathbb{R}}, + )</math>, <math display="inline">\Phi^{\times}:{\mathbb{R}} \times M \rightarrow M</math> with formula <math display="inline">\Phi^{\times (t,\Phi^{\times ()})} = \ldots</math>

i.e. we have a group morphism <math display="inline">({\mathbb{R}}, + ) \rightarrow \text{ Diffeo}(M)</math>.

care is needed if <math display="inline">M</math> not compact, flow map may not be defined for all of <math display="inline">\mathbb{R}</math>. but given <math display="inline">X \in \Gamma({TM})</math> we still have local flow map defined on open containing <math display="inline">\left\{ 0 \right\} \times M</math>, i.e. <math display="inline">\Phi^{\times}:D \subset {\mathbb{R}} \times M \rightarrow M</math>.

=== Lie Bracket in terms of flows ===

<math display="inline">\lbrack X,Y\rbrack = \frac{d^{2}}{{dt}^{2}}\left| \_(t = 0)\left( {\Phi^{Y}}_{- t} \circ {\Phi^{X}}_{- t} \circ {\Phi^{Y}}_{t} \circ {\Phi^{X}}_{t} \right) = \frac{d}{dt} \right|_{t = 0}(\left( {\Phi^{Y}}_{\sqrt{- t}} \circ {\Phi^{X}}_{\sqrt{- t}} \circ {\Phi^{Y}}_{\sqrt{t}} \circ {\Phi^{X}}_{\sqrt{t}} \right)</math>.

The lie bracket measures how far the vector fields <math display="inline">X,Y</math> are from commuting with each other.

=== Lie Derivative ===

Let <math display="inline">X \in \Gamma({TM})</math> we can take the '''Lie derivative''' of

* vector fields
* differential forms
* tensor fieds

and <math display="block">{LX}(Y) = \lbrack X,Y\rbrack</math> for vector fields. For differential forms, we have cartans formula to tell us <math display="inline">{LX}(\alpha) = \left( d \circ \iota_{X} + \iota \circ d \right)(\alpha)</math>. The lie derivative satisfies product rule <math display="inline">{LX}(A \otimes B) = {LX}(A) \otimes B + A \rightarrow {LX}(B)</math>.

apart from functions: <math display="inline">{LX}(f) = X(f) = df(X) = \iota_{X}(df)</math>, evaluating <math display="inline">{{LX}(A)}_{m}</math> depends on more than just <math display="inline">X_{m}</math>. i.e. the lie dervative is <math display="inline">\mathbb{R}</math>-linear but not <math display="inline">C^{\infty}(M)</math>-linear. So <math display="block">L_{aX + bY} = aL_{X} + bL_{Y}\text{     }\forall a,b \in {\mathbb{R}}</math> but <math display="block">L_{fX} \neq fL_{X}</math>

we will be using a riemannian metric to give a connection that satisfies the above, which the lie derivative doesnt.