---
title: Manifolds 
todos: true
book: true
---

These are notes live-tex'd from a graduate course in Smooth Manifolds taught by David Gay at the University of Georgia in Fall 2020.
As such, any errors or inaccuracies are almost certainly my own.

\medskip
\begin{flushright}
  D. Zack Garza, \today \\
  \currenttime
\end{flushright}

# Thursday, August 20

:::{.exercise}
Show that \( \theset{(\RR^1, \id), (\RR^1, x\mapsto x^3)} \) is *not* a smooth atlas.
:::

:::{.exercise}
Let $S^1\da\ts{(x, y) \in \RR^2 \st x^2 + y^2 = 1}$ with charts given by stereographic projection from $(0, 1)$ and $(0, -1)$ on $U = S^1\sm\ts{(0, 1)}\to \RR$ and $V = S^1\sm\ts{(0, -1)}\to \RR$.

Show that this gives a smooth atlas.
:::

:::{.exercise}
Write down a smooth atlas on the unit square.
:::



# Tuesday, August 25

## Submanifolds

:::{.exercise}
Prove that charts on a manifold are smooth maps.

> Hint: use the identity smooth structure on $\RR^n$.

:::

:::{.exercise}
Show that open subsets of manifolds are again manifolds in a canonical way.
:::

:::{.exercise}
Show that $S^1$ is a manifold.
:::

:::{.example}
Prove that a submanifold is again a manifold.
:::




# Thursday, September 24

:::{.exercise title="?"}
Write down an explicit diffeomorphism between $\CP^1$ and $S^2$.
:::

:::{.exercise title="?"}
Show that the map
\[  
\RP^n &\to \CP^n \\
[x_0: \cdots :x_n] &\mapsto [x_0 + 0i: \cdots :x_n + 0i]
\]
is an *embedding*, i.e. a differentiable map whose image is a submanifold, which is a diffeomorphism onto its image.
:::


:::{.exercise title="?"}
Define a vector field $V = -x_1 \del_{x_1} + x_2 \del_{x_2}$ on $M = (-1, 1)^2$.
Find the best possible $\eps: M \to (0, \infty]$, i.e. for each $p$, $\sup \ts{t>0 \st \Phi(t, p) \text{ is defined}}$.
:::

# Tuesday, September 29

Questions to look at for next Tuesday:

:::{.exercise title="?"}
Show that the 3 natural coordinate charts on $\CP^2$ given by e.g. $\phi_{U_0}\qty{\thevector{z_0: z_1: z_2}} = \thevector{{z_1 \over z_0}, {z_2\over z_0}}$ yield a smooth atlas.
:::

:::{.exercise title="?"}
Consider the map
\[  
\pi: \CP^2 &\to \RR^2 \\
\thevector{z_0: z_1 : z_2} &\mapsto \thevector{
{\abs z_1^2 \over \abs z_0^2 + \abs z_1^2 + \abs z_2^2},
{\abs z_2^2 \over \abs z_0^2 + \abs z_1^2 + \abs z_2^2}
}.\]


- Show that $\im \pi = \ts{p_1, p_2 \geq 0, p_1 + p_2 \leq 1}$.

- Show that $\pi$ is smooth 

![O](figures/image_2020-09-29-12-59-37.png)

- If $\thevector{p_1, p_2} \in T^\circ$ is in the interior of the above triangle, then $\pi^{-1}(p_1, p_2) \cong S^1 \cross S^1$ is diffeomorphic to a torus.
- If the point is on an edge, the fiber is diffeomorphic to $S^1$,
- If the point is on a vertex, the fiber is a single point.
:::

:::{.exercise title="?"}
Find a vector field $V$ on some maximal subset of $\CP^2$ such that $D\pi(V) = p_1 \del_{p_1} + p_2 \del_{p_2}$ (the radial vector field).

I.e., for all $q\in \CP^2$, we have a map 
\[  
D_1 \pi : T_1 \CP^2 &\to T_{\pi(q)} \RR^2
\]
and $V(q) \in T_q \CP^2$, so we want $D_q \pi(V(q)) = p_1 \del_{p_1} + p_2 \del_{p_2}$.

> Note that there will be a problem defining $V$ on the fiber over the hypotenuse of $T$.
:::



:::{.theorem title="Collar Neighborhood"}
For all manifolds with boundary $X$, there exists an open neighborhood $N$ of $\bd X$ which is diffeomorphic to $(-\eps, 0] \cross \bd X$.
:::

Proof strategy: construct a vector field pointing outward and flow it backward.
Construct by forming local vector fields on open sets, then patch together using a partition of unity.


:::{.definition title="Partition of Unity"}
A collection $\ts{\phi_i: M\to \RR\st i\in I}$ such that

1. $\ts{\supp \phi_i}$ is locally finite, i.e. for all $p$, we have $\abs{\ts{i \st p\in \supp(\phi_i)}} < \infty$.
2. $\phi(p) \geq 0$ for all $p\in X$
3. For all $p\in X$, the sum $\sum_{i\in I}\phi_i(p) = 1$.
:::