Exercise 1.43 (Smooth manifold chart lemma for manifolds with boundary)

Show that the smooth manifold chart lemma (Lemma 1.35) holds with “ ${ℝ}^n$” replaced with “ ${ℝ}^n$ or ${ℍ}^n$” and “smooth manifold” replaced by “smooth manifold with boundary”.

Lemma 1.35 (Smooth Manifold Chart Lemma). Let $M$ be a set, and suppose we are given a collection $\left\{U_{\alpha }\right\}$ of subsets of $M$ together with maps ${\phi }_{\alpha }:U_{\alpha }\to {ℝ}^n$, such that the following properties are satisfied:

(i)

For each $\alpha ,{\phi }_{\alpha }$ is a bijection between $U_{\alpha }$ and an open subset ${\phi }_{\alpha }\left(U_{\alpha }\right)\subseteq {ℝ}^n$.

(ii)

For each $\alpha$ and $\beta$, the sets ${\phi }_{\alpha }\left(U_{\alpha }\cap U_{\beta }\right)$ and ${\phi }_{\beta }\left(U_{\alpha }\cap U_{\beta }\right)$ are open in ${ℝ}^n$.

(iii)

Whenever $U_{\alpha }\cap U_{\beta }\ne \varnothing$, the map ${\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}:{\phi }_{\alpha }\left(U_{\alpha }\cap U_{\beta }\right)\to {\phi }_{\beta }\left(U_{\alpha }\cap U_{\beta }\right)$ is smooth.

(iv)

Countably many of the sets $U_{\alpha }$ cover $M$.

(v)

Whenever $p,q$ are distinct points in $M$, either there exists some $U_{\alpha }$ containing both $p$ and $q$ or there exist disjoint sets $U_{\alpha },U_{\beta }$ with $p\in U_{\alpha }$ and $q\in U_{\beta }$.

Then $M$ has a unique smooth manifold structure such that each $\left(U_{\alpha },{\phi }_{\alpha }\right)$ is a smooth chart.