# Thursday, October 14


:::{.remark}
Some background from smooth manifolds: a map $f:M\to N$ of manifolds is **smooth** if for any smooth charts $(U, \phi_U)$ and $(V, \psi_V)$ on $M, N$ respectively, the transition map $\psi_V \circ f \circ \phi_U \inv\RR^m\to\RR^n$ is smooth.
:::


:::{.exercise title="?"}
\envlist

1. Any smooth map $f:M\to N$ induces a bundle map $df: \T M\to f^* \T N$.
2. There is a canonical isomorphism $\T \RR^n \to \RR^n\times \RR^n$.
3. Let $(U, \phi_U)$ be a chart on $M$, then show that there are trivializing charts for $\T M$:
\[
d\phi_U: \ro{\T M}{U} \to \ro{ \phi_U^* }{ \T \RR^n }{U} \cong U \times \RR^n
.\]
In particular, the following diagram commutes:

\begin{tikzcd}
	{\ro{\T M}{U}} && {U\times \RR^n} \\
	\\
	U
	\arrow["\pi", from=1-1, to=3-1]
	\arrow["{d\phi_U}", from=1-1, to=1-3]
	\arrow[from=1-3, to=3-1]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHJve1xcVCBNfXtVfSJdLFswLDIsIlUiXSxbMiwwLCJVXFx0aW1lcyBcXFJSXm4iXSxbMCwxLCJcXHBpIl0sWzAsMiwiZFxccGhpX1UiXSxbMiwxXV0=)

:::


:::{.exercise title="?"}
Show that there is a canonical isomorphism 
\[
T_p M \iso \Der(M, \RR) \leq C^\infty(M, \RR)
,\]
where a derivation at $p$ is a smooth functional $v: C^\infty(M, \RR)\to \RR$ such that $v(fg) = v(f) g(p) + f(p) v(g)$.
:::


:::{.remark}
$N \subseteq M$ is a **smooth submanifold** if for any $p\in N$ there exists a smooth chat $(U, \varphi)$ on $M$ such that $p\in U$ and $\phi(U \intersect N) \subseteq \RR^k \times\ts{0} \subseteq \RR^n$.
:::


:::{.exercise title="?"}
Show that this is equivalent to $N = f(\tilde N)$ where $\tilde N$ is some smooth manifold and $f:\tilde N\to M$ is a smooth embedding and $d_p f: \T_p \tilde N\injects F_{f(p)} M$ is injective for all $p\in \tilde N$.
:::


:::{.definition title="?"}
Given a Riemannian metric on $\T M$ and a smooth submanifold $N \subseteq M$, let $\nu N$ denote $(\T N)^\perp \subseteq \ro{ \T M}{N}$ for the orthogonal complement of $\T N$.
This is a vector bundle $\nu N \to N$.
:::


:::{.exercise title="?"}
Show that up to a canonical isomorphism, this is independent of the choice of Riemannian metric.
:::


:::{.definition title="?"}
Given a curve $\gamma: I\to M$, then $\gamma'(t) \in \T_{\gamma(t)} M$ is the following derivation:
\[
\gamma'(t) . f \da \lim_{h\to 0} { f(\gamma(t+h)) - f(\gamma(t)) \over h}
.\]
A **vector field** on $M$ is a section of $\T M$, and a vector field along $\gamma$ is a section of $\gamma^* \T M$:

\begin{tikzpicture}
\fontsize{43pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/CharacteristicClasses/sections/figures}{2021-10-14_13-15.pdf_tex} };
\end{tikzpicture}

:::


:::{.example title="?"}
$\gamma'(t)$ is a vector field along $\gamma$.
:::


:::{.remark}
A **Lie group** is a group $G$ with the structure of a smooth manifold where multiplication and inversion are smooth self-diffeomorphisms.
:::


:::{.definition title="?"}
Given a Lie group $G$ and smooth principal $G\dash$bundle $P \mapsvia{\pi} M^n$, a **connection** on $P$ is a choice of subspaces $\xi_p \subseteq T_p P$ for all $p\in P$ such that $d\pi: \xi_p \to \T_{\pi(p)} M$ is an isomorphisms for all $p$, where $\xi_p$ is the horizontal subspace:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/CharacteristicClasses/sections/figures}{2021-10-14_13-25.pdf_tex} };
\end{tikzpicture}

:::


:::{.remark}
Given a connection, curves $\gamma$ in $M$ can be lifted to curves $\tilde \gamma$ in $P$ in such a way that tangents of $\tilde \gamma$ are projected to tangents of $\gamma$:


:::


:::{.definition title="?"}
Given a connection on $P$ and a smooth path $\gamma: I\to M$, a horizontal lift $\tilde \gamma$ of $\gamma$ is a path $\tilde \gamma: I\to P$ such that $\tilde \gamma'(t) \in \xi_{\tilde \gamma(t)}$ and $\pi \circ \tilde \gamma = \gamma$.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/CharacteristicClasses/sections/figures}{2021-10-14_13-32.pdf_tex} };
\end{tikzpicture}

:::


:::{.lemma title="?"}
Given a smooth path \( \gamma \) and $\tilde \gamma(0)$, there is a unique horizontal lift $\tilde \gamma$ starting at $\tilde \gamma(0)$.
:::


:::{.remark}
Consider $\Frame(\T M)$, then recall that $\TM = \Frame(\T M) \mix{\GL_n(\RR)} \RR^n$ by the mixing construction.
Given a connection on $\Frame(\T M)$, we can parallel transport vectors in $\T M$ along curves.
This comes from taking $[F_0, v_0]$ and evolving $F_0$ along $F_t \da \tilde \gamma(t)$, choosing pairs $[F_t, v_0]$ for all $t$, and ending at $[F_t, v_0]$.
:::


:::{.exercise title="?"}
Show that parallel transport yields a well-defined map $\T_{\gamma(0)}M \to \T_{\gamma(1))}M$.
:::


:::{.theorem title="?"}
Given a Riemannian metric on $\T M$, there is a canonical connection $\nabla$, the **Levi-Cevita** connection, which is torsionfree (i.e. $\nabla_X Y - \nabla_Y X = [X, Y]$) for $X, Y$ vector fields and $\nabla_X Y$ denotes parallel transporting $Y$ along $X$.
:::


:::{.definition title="?"}
A curve $\gamma$ is a **geodesic** if $\gamma'(t)$ is parallel.
:::


:::{.theorem title="?"}
For any $v\in \TM$, there is a unique geodesic $\gamma_v$ with $\gamma_v'(0) = v$.
:::


:::{.definition title="Exponential map"}
There is a map
\[
\exp: T_p M &\to M \\
v &\mapsto \gamma_v(1)
.\]
This is well-defined for $v$ of small norm, and for all $v$ if $M$ is closed.
:::



:::{.theorem title="?"}
If $N \leq M$ is a submanifold, then $\exp$ defines a diffeomorphism from a neighborhood of the zero section in $\nu N$ to a neighborhood of $N$.

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Fall/CharacteristicClasses/sections/figures}{2021-10-14_14-04.pdf_tex} };
\end{tikzpicture}

:::