Lecture Notes

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====== 2020-01-24, Friday ======


$$ \gdef\In\subset $$

===== Smooth Structure =====

Recall that an atlas for a topological manifold $M$ is a collection of coordinate charts $\{(U_\alpha, \varphi_\alpha)\}$ such that $M = \cup_\alpha U_\alpha$. And a smooth atlas is an atlas such that the transition functions between charts
$$ g_{\alpha\beta}:= \varphi_\alpha \circ \varphi_\beta^{-1}: \varphi_\beta(U_\alpha \cap U_\beta) \to 
\varphi_\alpha(U_\alpha \cap U_\beta) $$
are diffeomorphism. 

A smooth atlas is **maximal** if it is not contained in any larger smooth atlas. 

**Definition** If $M$ is a topological manifold, a **smooth structure on $M$ ** is a maximal smooth atlas. A smooth manifold is a pair $\gdef\cA{\mathcal A} (M, \cA)$ where $\cA$ is a smooth structure. 

==== Smooth functions and maps ====
Let $M$ be a smooth manifold, $f: M \to \R^k$ any function. We say $f$ is a smooth function on $M$ if for any chart $(U, \varphi)$ on $M$, $f \circ \varphi^{-1}: \varphi(U) \to \R^k$ is a smooth function. 

Similarly, if $M, N$ are smooth manifolds and $f: M \to N$ is any map. We say $f$ is smooth, if for any $x \in M$, there exists coordinate neighborhood $(U, \varphi)$ for $x$ and $(V,\psi)$ for $f(x)$, such that $f(U) \In V$ and 
$$ \psi \circ f \circ \varphi^{-1}: \varphi(U) \to \psi(V) $$
is a smooth function. 


===== Open Cover and Paracompactness =====


** Definitions ** 
  * A collection of subset $\{U_\alpha\}$ of $M$ is a cover of a subset $W \In M$, if $W \subset \cup_\alpha U_\alpha$. It is an open cover if each $U_\alpha$ is open. 
  * A subcollection of the $U_\alpha$ which still covers is called a //subcover//. 
  * A //refinement// $\{V_\beta\}$ of the $U_\alpha$ is a cover such that for each $\beta$ there is a $\alpha$ such that $V_\beta \In U_\alpha$. 
  * A collection $\{A_\alpha\}$ of subsets of $M$ is //locally finite//, if for each point $x \in M$, there exists a neighborhood $W_x$, such that only finitely many $A_\alpha$ satisfies $A_\alpha \cap W_x \neq \emptyset$. 
  * A topological space is //paracompact// if every open cover has an open refinement that is locally finite.


** Lemma **
If a topological space $X$ is a locally compact, Hausdorff and second countable (e.g $X$ is a topological manifold), then $X$ is paracompact. In fact, each open cover has a countable, locally-finite refinement consisting of precompact((precompact subset: a subset whose closure is compact)) open subsets. 

Proof: See [Warner Lemma 1.9], or [Lee, Thm 1.15] for the case $X$ is topological manifold. 


===== Partition of Unity =====
** Definition (Partition of Unity) **: Let $\{U_\alpha, \alpha \in A \}$ be an open cover of $M$.  A smooth partition of unity on $M$ is a collection of smooth $\R$-valued functions $\{\varphi_\alpha: \alpha \in A \}$ such that 
  - $0 \leq \varphi_\alpha \leq 1$ for each $\alpha \in A$. 
  - $\gdef\supp{\rm supp} \supp(\varphi_\alpha) \In U_\alpha$ for each  $\alpha \in A$. (Recall that $supp(f) = \overline {\{x : f(x) \neq 0\}}$)
  - The collection of support  $\{ supp(\varphi_\alpha)\}$ is locally finite. 
  - $\sum_{\alpha \in A} \varphi_\alpha(p) = 1$ for all $p \in M$.





Our goal here is to show the following theorem. 

** Theorem(Existence of Partition of Unity) **
Suppose $M$ is a smooth manifold, and $\{U_\alpha, \alpha \in A\}$ is an open cover of $M$. Then there exists a partition of unity $\{\varphi_i\}$ subordinate to $\{U_\alpha\}$. 

Sketch of proof: \\
1. Existence of smooth cut-off function on $\R$. Define
$$
f(x) = \begin{cases}
e^{-1/x} &\text{if } x > 0 \cr
0 &\text{if } x \leq 0
\end{cases}
$$

Then we can verify $f(x)$ is smooth. Consider the following function (smoothed step function)
$$ g(x) = \frac{ f(x)}{f(x) + f(1-x)} $$
then $g(x)$ is smooth and interpolate from value $0$ on $x<0$ to value $1$ on $x>1$.  

Finally, by splicing $g(x)$, we may build a 'bump function' $h(x)$ that is supported on $[-1, 1]$
$$
h(x) = \begin{cases}
1  &\text{if } |x| \leq 1/2 \cr
1-g(2|x|-1) &\text{if } |x| \in (1/2, 1) \cr
0 &\text{if } |x|  \geq  1
\end{cases}
$$

2. By paracompactness of $X$, we may refine the cover $U_\alpha$ to $V_i$ that is locally compact and the closure of each $V_i$ is compact. We may assume (see Lee for why we may) that each $V_i$ is contained in some coordinate chart $(W_i, \psi_i)$ such that, $\psi_i(V_i)$ is the unit open ball in $\R^n$. Then, we may construct a smooth function $h_i: M \to \R$ such that $supp(h_i) = \overline{V_i}$, e.g., $h_i(p) = h( \| \psi_i(p) \|^2)$, where $\| \|$ is the length of a vector in $\R^n$.

Let $H(p) = \sum_i h_i(p)$ for $p \in M$. Then $H(p) > 0$ everywhere. We can normalize $h_i$ by define $f_i = h_i / H$, thus $\sum_i f_i=1$. Finally, for each $i \in I$, we fix a choice $\alpha(i) \in A$, such that $V_i \In U_{\alpha(i)}$, then we define $\varphi_\alpha = \sum_{i: \alpha(i) = \alpha } h_i$. We can check then $\{\varphi_\alpha\}$ forms a smooth partition of unity subordinate to $\{U_\alpha\}$.
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