Smooth manifold with boundary - Imported Knowlpedia Content

Definition

An $n$ -dimensional smooth manifold with boundary is a Hausdorff, second-countable space $M$ equipped with a maximal smooth atlas of charts onto relatively open subsets of

\mathbb{H}^n=\{(x^1,\ldots,x^n)\in\mathbb{R}^n:x^n\geq0\}.

Coordinate changes must extend smoothly to open Euclidean neighborhoods of their domains. The separation and countability axioms are the same as for a topological manifold, but the local model is the half-space. A point belongs to $\partial M$ when one, and hence every, chart sends it to $x^n=0$ ; all remaining points form the interior $\operatorname{Int}M$ .

Intrinsic boundary and interior

The invariance of the boundary under coordinate changes is a theorem, not an additional chart choice. The interior is an open smooth $n$ -manifold without boundary. In the induced charts, $\partial M$ is a smooth $(n-1)$ -manifold without boundary. A diffeomorphism of manifolds with boundary necessarily maps boundary to boundary and interior to interior.

Smooth maps

A map $F:M\to N$ is smooth when its coordinate representatives extend smoothly to maps on open subsets of Euclidean space. This definition permits a smooth map to send interior points to boundary points. A smooth vector field at a boundary point is a vector in the full $n$ -dimensional tangent space; it need not be tangent to $\partial M$ .

Examples and scope

The closed ball, the closed upper half-space, and a compact interval are manifolds with boundary. Their boundaries are respectively a sphere, a Euclidean hyperplane, and two points. A product of two manifolds with nonempty boundary naturally has corner points, so its product atlas belongs to the broader theory of manifolds with corners rather than to this half-space definition.

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: Chapter 1 and the treatment of manifolds with boundary.
M. W. Hirsch, Differential Topology, Springer, 1976. DOI record. Relevant: Chapter 1.