0708-1300/Class notes for Tuesday, October 2

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Class Notes

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

General class comments

1) The class photo is up, please add yourself

2) A questionnaire was passed out in class

3) Homework one is due on thursday

First Hour

Today's Theme: Locally a function looks like its differential

Pushforward/Pullback

Let $\theta :X\rightarrow Y$ be a smooth map.

We consider various objects, defined with respect to X or Y, and see in which direction it makes sense to consider corresponding objects on the other space. In general $\theta _{*}$ will denote the push forward, and $\theta ^{*}$ will denote the pullback.

1) points pushforward $x\mapsto \theta _{*}(x):=\theta (x)$

2) Paths $\gamma :R\rightarrow X$, ie a bunch of points, pushforward, $\gamma \rightarrow \theta _{*}(\gamma ):=\theta \circ \gamma$

3) Sets $B\subset Y$ pullback via $B\rightarrow \theta ^{*}(B):=\theta ^{-1}(B)$ Note that if one tried to pushforward sets A in X, the set operations compliment and intersection would not commute appropriately with the map $\theta$

4) A measures $\mu$ pushforward via $\mu \rightarrow (\theta _{*}\mu )(B):=\mu (\theta ^{*}B)$

5)In some sense, we consider functions, "dual" to points and thus should go in the opposite direction of points, namely $\theta ^{*}f=f\circ \theta$

6) Tangent vectors, defined in the sense of equivalence classes of paths, [$\gamma$] pushforward as we would expect since each path pushes forward. $[\gamma ]\rightarrow \theta _{*}[\gamma ]:=[\theta _{*}\gamma ]=[\theta \circ \gamma ]$

CHECK: This definition is well defined, that is, independent of the representative choice of $\gamma$

7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, pushforward via $D\rightarrow (\theta _{*}D)(f):=D(\theta ^{*}f)$

CHECK: This definition satisfies linearity and Liebnitz property.

Theorem 1

The two definitions for the pushforward of a tangent vector coincide.

Proof:

Given a $\gamma$ we can construct $\theta _{*}\gamma$ as above. However from both $\gamma$ and $\theta _{*}\gamma$ we can also construct $D_{\gamma }f$ and $D_{\theta _{*}\gamma }f$ because we have previously shown our two definitions for the tangent vector are equivalent. We can then pushforward $D_{\gamma }f$ to get $\theta _{*}D_{\gamma }f$. The theorem is reduced to the claim that:

$\theta _{*}D_{\gamma }f=D_{\theta _{*}\gamma }f$

for functions $f:Y\rightarrow R$

Now, $D_{\theta _{*}\gamma }f={\frac {d}{dt}}f\circ (\theta _{*}\gamma )|_{t=0}={\frac {d}{dt}}f\circ (\theta \circ \gamma )|_{t=0}={\frac {d}{dt}}(f\circ \theta \gamma |_{t=0}=D_{\gamma }(f\circ \theta )=\theta _{*}D_{\gamma }f$

Q.E.D

Functorality

let $\theta :X\rightarrow Y,\lambda :Y\rightarrow Z$

Consider some "object" s defined with respect to X and some "object u" defined with respect to Z. Something has the property of functorality if

$\lambda _{*}(\theta _{*}s)=(\lambda \circ \theta )_{*}s$

and

$\theta ^{*}(\lambda ^{*}u)=(\lambda \circ \theta )^{*}u$

Claim: All the classes we considered previously have the functorality property; in particular, the pushforward of tangent vectors does.

Let us consider $\theta _{*}$ on $T_{p}M$ given a $\theta :M\rightarrow N$

We can arrange for charts $\varphi$ on a subset of M into $R^{m}$ (with coordinates denoted $(x_{1},...,x_{m})$)and $\psi$ on a subset of N into $R^{n}$ (with coordinates denoted $(y_{1},...,y_{n})$)such that $\varphi (p)=0$ and $\psi (\theta (p)=0$

Define $\theta ^{o}=\psi \circ \theta \circ \varphi ^{-1}=(\theta _{1}(x_{1},...,x_{m}),...,\theta _{n}(x_{1},...,x_{m}))$

Now, for a $D\in T_{p}M$ we can write $D=\sum _{i}a_{i=1}^{m}{\frac {\partial }{\partial x_{i}}}$

So, $(\theta _{*}D)(f)=\sum _{i=1}^{m}a_{i}{\frac {\partial }{\partial x_{i}}}(f\circ \varphi )=\sum _{i=1}^{m}a_{i}\sum _{j=1}^{n}{\frac {\partial f}{\partial y_{j}}}{\frac {\partial \theta _{j}}{\partial x_{i}}}=$ $={\begin{bmatrix}{\frac {\partial f}{\partial y_{1}}}&...&{\frac {\partial f}{\partial y_{n}}}\\\end{bmatrix}}{\begin{bmatrix}{\frac {\partial \theta _{1}}{\partial x_{1}}}&...&{\frac {\partial \theta _{1}}{\partial x_{m}}}\\...&&...\\{\frac {\partial \theta _{n}}{\partial x_{1}}}&...&{\frac {\partial \theta _{n}}{\partial x_{m}}}\\\end{bmatrix}}{\begin{bmatrix}a_{1}\\...\\a_{m}\\\end{bmatrix}}$

Now, we want to write $\theta _{*}D=\sum b_{j}{\frac {\partial }{\partial y_{j}}}$

and so, $b_{k}=(\theta _{*}D)y_{k}={\begin{bmatrix}0&...,i,...&0\\\end{bmatrix}}{\begin{bmatrix}{\frac {\partial \theta _{1}}{\partial x_{1}}}&...&{\frac {\partial \theta _{1}}{\partial x_{m}}}\\...&&...\\{\frac {\partial \theta _{n}}{\partial x_{1}}}&...&{\frac {\partial \theta _{n}}{\partial x_{m}}}\\\end{bmatrix}}{\begin{bmatrix}a_{1}\\...\\a_{m}\\\end{bmatrix}}$

where the i is at the kth location.

So, $\theta _{*}=d\theta _{p}$, i.e., $\theta _{*}$ is the differential of $\theta$ at p

We can check the functorality, $(\lambda \circ \theta )_{*}=\lambda _{*}\circ \theta _{*}$, then $d(\lambda \circ \theta )=d\lambda \circ d\theta$ This is just the chain rule.