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Lecture 2 - Thursday Week 1

From Our Interactions in AGT




Lecture 2 - 15 Jan 2026

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Review from diff geo:

  • Tangent space [math]\displaystyle{ T_{m}M }[/math]
  • Tangent bundle [math]\displaystyle{ TM }[/math] with map [math]\displaystyle{ \pi:TM \rightarrow M }[/math]
  • Vector field on [math]\displaystyle{ M }[/math] is:
    • derivation of [math]\displaystyle{ C^{\infty}(M) }[/math]
    • map [math]\displaystyle{ X:M \rightarrow TM }[/math] such that [math]\displaystyle{ \pi \circ X = \text{id}_{M} }[/math].

We can do same construction after cooking up new vector spaces from the tangent space [math]\displaystyle{ T_{m}M }[/math].

e.g. [math]\displaystyle{ T_{m}M \rightarrow {T^{*}}_{m}M = \left( T_{m}M \right)^{*} }[/math] is the dual space of [math]\displaystyle{ T_{m}M }[/math], called cotangent space. Then we have [math]\displaystyle{ T^{*}M = \cup_{m}T_{m}^{*}M }[/math], also a manifold, the cotangent bundle.

A map [math]\displaystyle{ \alpha:M \rightarrow {T_{m}^{*}M} }[/math] such that [math]\displaystyle{ \pi^{*} \circ \alpha = \text{id}_{M} }[/math] (i.e. a section of [math]\displaystyle{ \pi^{*} }[/math]) is a one-form.

note [math]\displaystyle{ \Lambda }[/math] is exterior power. [math]\displaystyle{ \Lambda^{k}{T_{m}^{*}M} \rightarrow {T^{*}M} }[/math] …

More generally for [math]\displaystyle{ {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]\displaystyle{ {TM}\text{tensor}\ldots\text{tensor}{TM}\text{tensor}{T^{*}M}\text{tensor}\ldots\text{tensor}{T^{*}M} }[/math]

in particular

  • [math]\displaystyle{ \Lambda:\Omega^{k}(M) \times \Omega^{l}(M) \rightarrow \Omega^{k + l}(M) }[/math] where [math]\displaystyle{ (\alpha,\beta)| \rightarrow \alpha\Lambda\beta }[/math]


  • [math]\displaystyle{ X }[/math] vector field, [math]\displaystyle{ \alpha \in \Omega^{k}(M) }[/math], then [math]\displaystyle{ \iota_{X}:{\Omega^{k}(M)} \rightarrow \Omega^{k - 1}(M) }[/math], where [math]\displaystyle{ \alpha| \rightarrow \iota_{X}\alpha }[/math], which has equation

[math]\displaystyle{ \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]\displaystyle{ X }[/math] to first arg of [math]\displaystyle{ \alpha }[/math].

Lie bracket

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For [math]\displaystyle{ X,Y }[/math] in [math]\displaystyle{ \Gamma({TM}) }[/math] where [math]\displaystyle{ \Gamma }[/math] is the space of sections of …

[math]\displaystyle{ \rightarrow \lbrack X,Y\rbrack \in \Gamma(TM) }[/math] defined by: for all [math]\displaystyle{ f \in C^{\infty(M)} }[/math], [math]\displaystyle{ \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]\displaystyle{ X\left( Y(f) \right) }[/math] on it’s own is not.

Lie bracket has properties of

  • symmetry
  • bilinearity
  • jacobi identity

Flow Map

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

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

i.e. we have a group morphism [math]\displaystyle{ ({\mathbb{R}}, + ) \rightarrow \text{ Diffeo}(M) }[/math].

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

Lie Bracket in terms of flows

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[math]\displaystyle{ \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]\displaystyle{ X,Y }[/math] are from commuting with each other.

Lie Derivative

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Let [math]\displaystyle{ X \in \Gamma({TM}) }[/math] we can take the Lie derivative of

  • vector fields
  • differential forms
  • tensor fieds

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

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

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