Existence of atlas with certain properties

Question

I have a question about the following definition from Gilkey's book, "Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem" (p. 31):

(Let $M$ be a closed manifold.)

Definition: An atlas $\mathcal{U}$ is a collection $\{U_i,h_i,\phi_i\}$ where the $U_i$ are a finite open cover of $M$, $(U_i,h_i)$ is a coordinate chart, and the $\phi_i$ are a partition of unity subordinate to the cover $U_i$. We assume that any two points of $M$ belong to at least one of the $U_i$. We assume that for any pair of indices $i$ and $j$ there exists a coordinate chart $(O_{ij},h_{ij})$ so that $\bar U_i\cup\bar U_j\subseteq O_{ij}$.

The first condition I think follows from Hausdorffness of $M$, so we can put enough charts into the collection (which might be rather large) to make this happen.

Question: How do we know that a collection $\mathcal{U}$ exists that satisfies the second condition, $\bar U_i\cup\bar U_j\subseteq O_{ij}$?

Answer 1

Let $M$ be a smooth compact connected $n$-dimensional manifold. Let $g$ be a Riemannian metric on $M$, $d$ the corresponding Riemannian distance function on $M$. Let $R$ denote the injectivity radius of $(M,g)$ and $r:= R/6$. Then the products $B(x,R)\times B(y,R)$, $(x,y)\in M^2=M\times M$, form an open cover of $M^2$. We let $\{x_1,...,x_N\}$ denote a finite subset of $M$ such that $$ M^2= \bigcup_{1\le i, j\le N} B(x_i,r)\times B(x_j,r) $$

Thus, for every pair of points $x, y\in M$ there exist $i, j$ such that $x\in B(x_i,r)$, $y\in B(x_j,r)$. Set $U_{ij}:= B(x_i,r)\cup B(x_j,r)$, $1\le i\le j\le N$. Then we clearly have that any two points in $M$ belong to one of the subsets $U_{ij}$. Next, consider $U_{ij}, U_{kl}$. We will need to find an open subset $O_{ijkl}\subset M$ containing the closure of $U_{ij}\cup U_{kl}$ such that $O_{ijkl}$ is diffeomorphic to an open subset of $\mathbb R^n$. There are several cases which may occur, I will analyze just two and leave you the rest to work out:

1. All four closed balls $\bar{B}(x_i,r), \bar{B}(x_j,r), \bar{B}(x_k,r), \bar{B}(x_l,r)$ are pairwise disjoint. Then there exists $\epsilon>0$ such that $r+\epsilon<R$ and the open balls ${B}(x_i,r+\epsilon)$, ${B}(x_j,r+\epsilon)$, ${B}(x_k,r+\epsilon)$, ${B}(x_l,r+\epsilon)$ are still pairwise disjoint. Since $r+\epsilon<R$, the disjoint union $O_{ijkl}$ of the balls ${B}(x_i,r+\epsilon), {B}(x_j,r+\epsilon), {B}(x_k,r+\epsilon), {B}(x_l,r+\epsilon)$ is diffeomorphic to an open subset of $\mathbb R^n$.

2. All three intersections $\bar{B}(x_i,r)\cap \bar{B}(x_j,r)$, $ \bar{B}(x_j,r)\cap \bar{B}(x_k,r)$, $ \bar{B}(x_k,r)\cap \bar{B}(x_l,r)$ are nonempty. Then $$ \bar{B}(x_i,r)\cup \bar{B}(x_j,r)\cup \bar{B}(x_k,r)\cup \bar{B}(x_l,r) \subset \bar{B}(x_j, 5r)\subset B(x_j, R). $$ In this case I will take $O_{ijkl}:= B(x_j, R)$, it is an open subset of $M$ diffeomorphic to an open subset of $\mathbb R^n$.