Manifolds 45 | Manifolds with Boundary

Manifolds with boundary extend generalized surfaces by allowing “edge pieces” that behave like a cropped Euclidean space, a move designed to make the generalized Stokes theorem work. Instead of requiring every point to look locally like an open set in 2, the construction permits charts whose images lie in a half-space. Concretely, a two-dimensional surface in three-dimensional space can include its boundary circles; once those edge components are included, the object no longer looks locally like 2 everywhere, so the usual manifold definition needs a boundary-aware version.

The key local model is the half-space in 2: take 2n and restrict the first coordinate so that x1  0 (equivalently, x1  0 depending on convention). This half-space comes with its own (n 1)-dimensional boundary, consisting of points where x1 equals 0, which can be written as {0}  2n 1. The half-space inherits a topology from 2n via the subspace topology: a set is open in the half-space exactly when it equals the intersection of the half-space with some open set in 2n. That matters because open sets in the half-space can include points on the “edge,” meaning they may fail to be open in the ambient 2n.

With this local geometry in place, charts split into two types. For interior points, charts map to 2n as usual. For boundary points, charts map to the half-space. Transition maps between overlapping charts must then be smooth in the boundary-aware sense: differentiability on the half-space is defined by extending the function across the boundary to an open set in 2n and requiring ordinary differentiability there. A map between chart domains is a diffeomorphism when it is bijective and both the map and its inverse admit such differentiable extensions (and similarly for higher regularity Ck).

A smooth manifold with boundary is then defined like a smooth manifold, but with an atlas that allows boundary charts. The underlying topological space must still be Hausdorff and second countable, and the atlas must be maximal with compatible Ck transition maps. The manifold boundary, denoted M (written as M in the transcript), is defined using the charts: it consists of points p in M for which some chart maps p into the half-space boundary (i.e., the image lies where x1 = 0). This “manifold boundary” is not the same as the topological boundary of the underlying space; it is a geometric boundary determined by the atlas.

Dimensionally, the boundary typically behaves like an (n 1)-dimensional manifold: a 2D smooth manifold with boundary has a 1D boundary. The framework also includes the possibility of having no boundary at all, in which case the definition reduces to the standard smooth manifold. The payoff is a consistent boundary calculus, setting up later work on integration and the generalized Stokes theorem.