Boundary and interior of a manifold with boundary

Definition

Let $M$ be an $n$ -dimensional smooth manifold with boundary. A point $p\in M$ is a boundary point if a boundary chart $\varphi:U\to V\subseteq\mathbb H^n$ sends it to the bounding hyperplane $x^n=0$ . It is an interior point if $\varphi(p)$ has $x^n>0$ . These conditions do not depend on the chart. The boundary and interior are

\partial M=\{p\in M:p\text{ is a boundary point}\}, \qquad \operatorname{Int}M=M\setminus\partial M.

Here $\partial M$ is intrinsic to the manifold-with-boundary structure, not the boundary of $M$ inside an unspecified ambient space.

Intrinsic structure

The invariance-of-boundary theorem makes the chart-based definition intrinsic; see Lee, Chapter 1, “Smooth Manifolds”. The interior is an open $n$ -dimensional smooth manifold without boundary. When $\partial M$ is nonempty, it inherits a canonical smooth structure of dimension $n-1$ by restricting boundary charts to $x^n=0$ . In particular, the boundary of a smooth manifold with boundary is itself a smooth manifold without boundary.

Every diffeomorphism of manifolds with boundary preserves the two strata: it sends $\partial M$ to the target boundary and $\operatorname{Int}M$ to the target interior.

Examples and contrasts

For the closed ball $B^n$ , the intrinsic boundary is $S^{n-1}$ and the interior is the open ball. For the interval $[0,1]$ , the boundary is $\{0,1\}$ , a $0$ -manifold. A manifold without boundary is allowed: then $\partial M=\varnothing$ and $\operatorname{Int}M=M$ .

Intrinsic boundary need not agree with an ambient topological boundary. The embedded circle $S^1\subset\mathbb R^2$ has ambient boundary $S^1$ as a subset of $\mathbb R^2$ , but as a $1$ -manifold it has empty intrinsic boundary.

Conventions and scope

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. Publisher record. Relevant: Chapter 1, “Smooth Manifolds,” especially invariance of the boundary and the induced boundary structure.
Loring W. Tu, An Introduction to Manifolds, 2nd ed., Universitext, Springer, 2011. Publisher record. Relevant: the chapter “Manifolds with Boundary.”