Smooth map between manifolds with boundary - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be smooth manifolds with boundary of dimensions $m$ and $n$ . A map $F:M\to N$ is smooth if, for every $p\in M$ , there are boundary charts $(U,\varphi)$ at $p$ and $(V,\psi)$ at $F(p)$ , with $F(U)\subseteq V$ , such that

\psi\circ F\circ\varphi^{-1}:\varphi(U)\longrightarrow\psi(V)

locally extends around $\varphi(p)$ to a smooth map between open subsets of $\mathbb R^m$ and $\mathbb R^n$ . This condition is independent of the chosen charts. It is the boundary analogue of a smooth map between manifolds without boundary.

Equivalent local tests

It suffices to verify the extension condition in one pair of compatible charts around each point: smooth transition maps and their local Euclidean extensions transfer it to every other pair. Equivalently, each component of the coordinate representative is the restriction of an ordinary smooth real-valued function near every point of its half-space domain. These formulations are developed in Lee, Chapter 2, “Smooth Maps”.

For real-valued functions on $M$ , no target boundary is involved: a function is smooth precisely when its expression in every boundary chart extends smoothly across the hyperplane $x^m=0$ .

Structure and consequences

Identity maps are smooth, and composites of smooth maps between manifolds with boundary are smooth. Consequently these objects and maps form a category. The differential $dF_p:T_pM\to T_{F(p)}N$ is defined using an extension and is independent of that extension. At a boundary point, $T_pM$ is still an $m$ -dimensional vector space rather than a half-space.

Smoothness alone imposes no boundary-preservation condition. A smooth map may send an interior point to the boundary or a boundary point to the interior.

Examples and non-examples

The inclusion $[0,\infty)\hookrightarrow\mathbb R$ , $x\mapsto x$ , and the function $[0,\infty)\to\mathbb R$ , $x\mapsto\sqrt{1+x}$ , are smooth because they extend across $0$ .

The function $x\mapsto\sqrt{x}$ on $[0,\infty)$ is continuous and smooth on the interior, but it is not smooth as a map of manifolds with boundary: no smooth Euclidean extension near $0$ can have its unbounded one-sided derivative.

Conventions and scope

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. Publisher record. Relevant: Chapter 2, “Smooth Maps,” and the section on manifolds with boundary.
Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. Publisher record. Relevant: Chapter 1, differentiable manifolds and maps on half-spaces.