Boundary chart - Imported Knowlpedia Content

Definition

Let $M$ be an $n$ -dimensional smooth manifold with boundary, and write

\mathbb H^n=\{(x^1,\ldots,x^n)\in\mathbb R^n:x^n\geq 0\}.

A boundary chart on $M$ is a pair $(U,\varphi)$ in its smooth atlas for which $U\subseteq M$ is open and $\varphi:U\to V$ is a homeomorphism onto a relatively open subset $V\subseteq\mathbb H^n$ . Its transition maps with overlapping boundary charts extend smoothly to maps between open subsets of $\mathbb R^n$ . It is centered at $p$ when $p\in U$ ; points represented on $\partial\mathbb H^n$ are the boundary points of $M$ .

Why relative openness matters

A neighborhood of a point on $\partial\mathbb H^n$ is not open in $\mathbb R^n$ , but it is open in the subspace topology of $\mathbb H^n$ . Requiring relative openness therefore gives boundary points genuine manifold neighborhoods while retaining ordinary Euclidean neighborhoods at points with $x^n>0$ . Smoothness is tested by extensions to Euclidean-open sets because ordinary derivatives are not intrinsically defined on a one-sided domain. This is the half-space chart convention used in Lee, Chapter 1, “Smooth Manifolds”.

Basic consequences

The topological invariance of the half-space boundary implies that if one boundary chart sends $p$ to $x^n=0$ , then every overlapping boundary chart does. Thus the distinction between boundary and interior points does not depend on coordinates.

At an interior point, restricting $V$ to $x^n>0$ turns a boundary chart into an ordinary smooth chart. Near a boundary point, the first $n-1$ coordinates restrict to coordinates on $\partial M$ , while $x^n$ is an inward half-space coordinate.

Examples and non-examples

The identity map from $\mathbb H^n$ to itself is the standard boundary chart. A hemisphere of the sphere cut out by a closed half-space admits boundary charts obtained by flattening its equator.

The map from a neighborhood of $0$ in $[0,\infty)$ onto an interval $(-\varepsilon,\varepsilon)$ is not a boundary chart: its image uses a full Euclidean neighborhood rather than a relatively open half-space neighborhood compatible with the one-sided local topology.

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 smooth manifolds with boundary.