17.1 Partitions of unity

中文

Given an open covering $\{U_i|i\in I\}$ of a topological space $S,$ an open covering $\{V_a\}$ of $S$ is called a refinement of $\{U_i\}$ if each $V_a\subseteq U_i$ for some $i\in I$ . The refinement is said to be locally finite if every point $p$ belongs to at most a finite number of $V_a\bar{\mathbf{s}}.$ . The topological space $S$ is said to be paracompact if for every open covering $U_i\left(i\in I\right)$ there exists a locally finite refinement. As it may be shown that every locally compact Hausdorff secondcountable space is paracompact [12], we will from now on restrict attention to manifolds that are paracompact topological spaces.

Given a locally finite open covering $\{V_a\}$ of a manifold $M$, a partition of unity subordinate to the covering $\{V_a\}$ consists of a family of differentiable functions $g_a:M\to$ $R$ such that

(1) $0\le g_a\le 1$ on $M$ for all $\alpha ,$ ,

(2) $g_a(p)=0$ for all $p\notin V_a.$

$$(3)\sum _a{g}_a(p)=1\text{ for all }p\in M.$$

It is important that the covering be locally finite, so that the sum in (3) reduces to a finite sum.

Theorem 17.1 · Existence of partitions of unity

For every locally finite covering $\{V_a\}$ of a paracompact manifold $M$ there exists a partition of unity $\{g_a\}$ subordinate to this refinement.

Proof

For each $p\in M$ let $B_p$ be an open neighbourhood of $p$ such that its closure $\overline{B_p}$ is compact and contained in some $V_a-$ for example, take $B_p$ to be the inverse image of a small coordinate ball in ${\mathbb{R}}^n$ . As the sets $\{B_p\}$ form an open covering of $M,$ they have a locally finite refinement $\{B_{\alpha }^{\prime }\}$ . For each $\alpha$ let $V_a^{\prime }$ be the union of all $B_{\alpha }^{\prime }$ whose closure $\overline{B_{\alpha }^{\prime }}\subset V_a$ . Since every $\overline{B_{\alpha }^{\prime }}\subset \overline{B_p}\subset V_a$ for some $p$ and $a,$ , it follows that the sets $V_a^{\prime }$ are an open covering of $M$ . For each $\alpha$ the closure of $V_a^{\prime }$ is compact and, by the local finiteness of the covering $\{B_{\alpha }^{\prime }\}$

$$\overline{V_a^{\prime }}=\bigcup \overline{B_{\alpha }^{\prime }}\subset V_a.$$

As seen in Lemma 16.1 for any point $p\in \overline{V_a^{\prime }}$ it is possible to find a differentiable function $h_p:M\to$ $R$ such that $h_p(p)=1$ and $h_p=0$ on $M-V_a$ . For each point $p\in \overline{V_a^{\prime }}$ let $U_p$ be the open neighbourhood $\{q\in V_a\mid f_p(q)>\frac{1}{2}\}$ . Since $\overline{V_a^{\prime }}$ is compact, there exists a finite subcover $\{U_{p_1},\dots ,U_{p_k}\}$ . The function $h_a=h_{p1}+\cdot \cdot \cdot +h_{p_k}$ has the following three properties: $(\mathrm{i})h_a\ge 0\mathrm{on}M,(\mathrm{ii})h>0$ on $\overline{V_a^{\prime }}$ , and (iii) $h_a=0$ outside $V_{a\cdot }\mathrm{As}\left\{V_a\right\}$ is a locally finite covering of $M$, the function $h={\sum }_a{h}_a$ is well-defined, and positive everywhere on $M$ . The functions $g_a=h_a/h$ satisfy all requirements for a partition of unity subordinate to $V_a$

We can now construct a non-vanishing $n$-form on a manifold $M$ from any atlas of charts $(U_{\alpha },{\varphi }_{\alpha };x_{\alpha }^i)$ having positive Jacobian determinants on all overlaps. Let $\{V_a\}$ be a locally finite refinement of $\{U_{\alpha }\}$ and $g_a\mathrm{a}$ partition of unity subordinate to $\{V_a\}$ . The charts $(V_a,{\varphi }_a;x_a^i)$ where ${\varphi }_a={\varphi }_{\alpha }{|}_{V_a}$ and $x_a^i=x_{\alpha }^i{|}_{V_a}$ form an atlas on $M$, and

$$\omega =\sum _a{g}_a\mathrm{d}{x}_a^1\wedge \mathrm{d}{x}_a^2\wedge \cdots \wedge \mathrm{d}{x}_a^n$$

is a differential $n$-form on $M$ that nowhere vanishes.

Example 17.1

The Möbius band can be thought of as a strip of paper with the two ends joined together after giving the strip a twist, as shown in Fig. 17.1. For exam ple, let $M=\{(x,y)|-2\le x\le 2,-1<y<1\}$ where the end edges are identified in opposing directions, $(2,y)\equiv (-2,-y)$ . This manifold can be covered by two charts

Figure 17.1 Möbius band

$$(U=\{(x,y)|-2<x<2,-1<y<1\},\varphi ={\mathrm{id}}_U)\text{ and }(V=V_1\cup V_2,\psi )\text{ where }$$ $$V_1=\{(x,y)|-2\le x<-1,-1<y<1\},$$ $$V_2=\{(x,y)|1\le x<2,-1<y<1\},$$ $$(x^{\prime },y^{\prime })=\psi (x,y)=\{\begin{array}{ll}(x+2,y)& \text{ if }(x,y)\in V_1,\\ (x-2,-y)& \text{ if }(x,y)\in V_2.\end{array}$$

The Jacobian is 1 on $U\cap V_1$ and 1 on $U\cap V_2$ , so these two charts do not have the same orientation everywhere. The Möbius band is non-orientable, for if there existed a non-vanishing 2-form $\omega$, we would have $\omega =f\mathrm{d}x\wedge \mathrm{d}y$ with $f(x,y)>0$ or $f(x,y)<0$ everywhere on $U$. Setting $\omega =f^{\prime }\mathrm{d}{x}^{\prime }\wedge \mathrm{d}{y}^{\prime }$ we have $f^{\prime }=f\mathrm{on}{V}_1$ and $f^{\prime }=-f$ on $V_2$ Hence $f^{\prime }(x^{\prime },y^{\prime })$ must vanish on the line $x=\pm 2$ , which contradicts $\omega$ being non-vanishing everywhere.