# Difference between revisions of "0708-1300/Class notes for Tuesday, October 2"

Announcements go here

## 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.

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 $T_pM$ 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 $D\in T_pM$ 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.