Partition of Unity

微分流形初步的单位分解定理, 这个定理本质上是流形paracompactness所带来的结论. 它提供了一种通过乘上partition of unity函数的方式来将locally smooth的函数粘贴成global smooth的函数的方式. 下面的Whitney approximation theorem便是一个直接的例子.

Paracompactness of Manifold

First we give the construction of bump function. Here we use those function $f\in C^{\infty}$ s.t. $f\equiv 1$ on cubic $\bar{C(1)}$ and $f\equiv 0$ outside $C(2)$ .

Consider the function $f(t)=\left\{\begin{aligned} &e^{-\frac{1}{t}}\quad &t>0\\&0\quad &t\leq 0 \end{aligned}\right.$ then $f\in C^{\infty}$ . Take $g(t)=\frac{f(t)}{f(t)+f(-t)}$ , $g\in C^{\infty}$ and $g\equiv 1$ for all $t\geq 1$ , $g=0$ for all $t\leq 0$ . Take $h(t)=g(2+t)g(2-t)$ , then $h\in C^{\infty}$ and $h\equiv 1$ on $[-1,1]$ and $h\equiv 0$ outside $(-2,2)$ . Finally, just take the function to be $\varphi=\prod\limits_{i=1}^{n}h\circ r_{i}$ , where $r_{i}$ are coordinate function.

Lemma 1. Let $X$ be a topological space which is locally compact, Hausdorff and second countable, then $X$ is paracompact.

Proof

Proof. First, we construct a sequence of open sets $G_{i}$ s.t. $\bar{G}_{i}$ compact, $\bar{G}_{i}\subset G_{i+1}$ and $X=\cup_{i=1}^{\infty}G_{i}$ . Let $\{U_{i}\}_{i\in I}$ be a countable basis of $X$ and $\bar{U}_{i}$ compact. This can be shown by following:

We consider the set $\{U_{i}\}_{i\in J}$ with $\bar{U}_{i}$ compact, we show that $\{U_{i}\}_{i\in J}$ also a basis of $X$ . We only need to show $\forall V\in\{U_{i}\}_{i\in I}$ , $V=\cup U_{j}$ with $j\in J$ . $\forall x\in V$ , $\exists C$ compact and $x\in U\subset C$ . Then
$\bar{U}$ compact, then $x\in U\cap V\subset\bar{U\cap V}\subset C$ and $\bar{U\cap V}$ compact.

Then we get such a countable basis: $\{ U_{i}\}_{i\in I}$ and $\bar{U}_{i}$ compact.

Take $G_{1}=U_{1}$ , $G_{k}=U_{1}\cup \cdots\cup U_{j_{k}}$ , let $j_{k+1}$ be the smallest number s.t. $\bar{G_{k}}\subset \cup_{i=1}^{j_{k+1}}U_{i}$ and $G_{k+1}=\cup_{i=1}^{j_{k+1}}U_{i}$ . Then we have a sequence $\{G_{k}\}$ satisfying the condition in the beginning of proof. $\bar{G}_{k}-G_{k-1}$ is a compact set and $\bar{G}_{k}-G_{k-1}\subset G_{k+1}-\bar{G_{k-2}}$ .

For any $k\geq 3$ , for the cover $\{U_{\alpha}\cap (G_{k+1}-\bar{G}_{k-2})\}_{\alpha\in A}$ of $\bar{G}_{k}-G_{k-1}$ , we choose a finite subcover $\{U_{\alpha}\cap (G_{k+1}-\bar{G}_{k-2})\}_{\alpha\in A}$ . This collection is obviously a countable and locally finite refinement of open cover $\{U_{\alpha}\}_{\alpha\in A}$ . ◻

Partition of Unity

Definition 1. A partition of unity on $M$ is a collection $\{\varphi_{i}\}_{i\in I}$ of $C^{\infty}$ functions on $M$ s.t.

1. The collection of $\{Supp \varphi_{i}\}_{i\in I}$ is locally finite;

2. $\sum\limits_{i\in I}\varphi_{i}(p)=1$ for $p\in M$ and $\varphi_{i}(p)\geq 0$ for all $p\in M$ and $i\in I$ .

Now we give the partition of unity theorem.

Theorem 1 (partition of unity). Let $M$ be a smooth manifold and $\{U_{\alpha}\}_{\alpha\in A}$ be an open cover of $M$ . Then there exists a countable partition of unity $\{\varphi_{i}\}$ subordinate to the cover $\{U_{\alpha}\}$ with $Supp \varphi_{i}$ compact for each $i$ .

Proof

Proof. We choose the sequence $\{G_{i}\}$ of $M$ and $G_{0}=\varnothing$ . For $p\in M$ , let $i_{p}$ be the largest integer s.t. $p\in M-\bar{G}_{i_{p}}$ . Choose $\alpha_{p}$ s.t. $p\in U_{\alpha_{p}}$ and let $(V,\tau)$ be a coordinate system centered at $p$ s.t. $V\subset (U_{\alpha_{p}}\cap (G_{i_{p+2}}-\bar{G}_{i_{p}}))$ and $\tau(V)$ contains cube $\bar{C(2)}$ .

Define $\psi_{p}=\left\{\begin{aligned} &\varphi\circ \tau\quad & \text{on }V\\ &0\quad &\text{otherwise} \end{aligned}\right.$ $\varphi$ is the bump function. Then $\psi_{p}$ is a $C^{\infty}$ function on $M$ which has value 1 on some open neighborhood $w_{p}$ of $p$ and has compact support in $V\subset U_{\alpha_{p}}\cap (G_{i_{p+2}}-\bar{G}_{i_{p}})$ . For each $k\geq 1$ , choose a finite set of points $p$ in $M$ s.t. $\{w_{p}\}$ covers $\bar{G}_{i}-G_{i-1}$ , the $Supp \varphi_{i}$ form a locally finite family of subsets of $M$ .

Take $\psi=\sum\limits_{j=1}^{\infty}\psi_{j}$ , $\psi$ is a well-defined $C^{\infty}$ function and $\psi(p)>0$ for each $p\in M$ . For $i=1,2,\dots$ , define $\varphi_{i}=\frac{\psi_{i}}{\psi}$ . Then $\{\psi_{i}\}$ is a partition of unity subordinate $\{U_{\alpha}\}$ with $\{Supp \varphi_{i}\}$ compact. ◻

Theorem 2 (Whitney approximation theorem). Let $M$ be a smooth manifold, and $A \subset M$ a closed subset. Then for any continuous function $g:M\to R$ which is smooth on $A$ and any positive continuous function $\delta:M\to R>0$ , there exists $f\in C^{\infty}(M)$ so that $f(p)=g(p)$ for all $p\in A$ and $\left| f(p)-g(p) \right|<\delta(p)$ for all $p\in M$ .

Here $g$ smooth on $A$ means $\exists U\supset A$ with a function $g_{0}=g$ on $A$ and $g_{0}$ smooth on $U$ .

Proof

Proof. There is an open set $U\supset A$ and a smooth function $g_{0}$ defined on $U$ s.t. $g_{0}=g$ on $A$ . Let $U_{0}=\{p\in U| \left| g_{0}(p)-g(p) \right|<\delta(p) \}$ . Then $U_{0}$ is open in $M$ and $U_{0}\supset A$ .

Now we construct a open cover of $M-A$ , for any $q\in M-A$ , let $U_{q}=\{p\in M-A\left| g(p)-g(q) \right| <\delta(p)\}$ . Then $\{U_{q}|q\in M-A\}$ is an open covering of $M-A$ . Let $\{\varphi_{0}, \varphi_{q}|q\in M\}$ be a partition of unity subordinate the cover $\{U_{0}, U_{q}|q\in M\}$ of $M$ and define a function on $M$ by

f(p)=\varphi_{0}(p)g_{0}(p)+\sum\limits_{q\in M}\varphi_{q}(p)g(q)

Since the support is locally finite, $f$ is smooth, $f=g_{0}$ on $A$ . Moreover, for any $q\in M$ , we have

\begin{aligned} \left| f(p)-g(p) \right| &=\left| \varphi_{0}(p)+g_{0}(p)+\sum_{q}\varphi_{q}(p)g(q)-\varphi_{0}(p)g(p)-\sum_{q}\varphi_{q}(p)g(p) \right|\\&\leq \varphi_{0}(p)\left| g_{0}(p)-g(p) \right| +\sum_{q}\varphi_{q}(p)\left| g(q)-g(p) \right| \\&<\varphi_{0}(p)\delta(p)+\sum_{q}\varphi_{q}(p)\delta(p)\\&=\delta(p)\end{aligned}

◻