Boundary defining function - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold with boundary. A boundary defining function is a smooth function $\rho:M\to[0,\infty)$ such that

\rho^{-1}(0)=\partial M \qquad\text{and}\qquad d\rho_p\neq0\quad\text{for every }p\in\partial M.

Equivalently, $0$ is a regular value of $\rho$ and its zero set is exactly the boundary. A local boundary defining function on an open set $U\subseteq M$ satisfies the same conditions with $\partial M\cap U$ in place of $\partial M$ . Nonnegativity fixes which side of the regular hypersurface belongs to $M$ .

Local normal form

The regular-value condition implies that near every boundary point there are boundary coordinates

(x^1,\ldots,x^{n-1},r),\qquad r\geq0,

in which $\rho=r$ . Consequently $d\rho$ spans the conormal line of the boundary. Conversely, the final coordinate in any boundary chart is a local defining function. This local normal form is the reason the nonvanishing differential, rather than mere set-theoretic vanishing, belongs in the definition.

Existence and comparison

Every smooth manifold with boundary admits a global boundary defining function. One construction starts from a collar, uses its inward coordinate near the boundary, and extends it to a positive function on the remaining interior. If $\rho$ and $\rho'$ are two defining functions, then near the boundary

\rho'=a\rho

for a smooth positive function $a$ . Thus their first-order vanishing agrees up to a positive scale, while their extensions deep in the interior may be unrelated. This comparison is standard in Melrose, Chapter 1.

Examples and non-examples

On the half-space $\mathbb H^n$ , the last coordinate $x^n$ is a boundary defining function. On the closed unit ball, $\rho(x)=1-\lVert x\rVert^2$ is one because its differential is nonzero on the sphere.

The function $(x^n)^2$ has the correct zero set on $\mathbb H^n$ but is not a defining function: its differential vanishes along the boundary. A signed defining function on a larger manifold containing the boundary hypersurface is also not nonnegative on both sides; restricting it to the chosen side recovers the convention used here.

References

Richard B. Melrose, The Atiyah–Patodi–Singer Index Theorem, A K Peters/CRC Press, 1993. Publisher DOI record. Relevant: Chapter 1, boundary defining functions and product structures.
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. Springer DOI record. Relevant: manifolds with boundary, regular level sets, and collar neighborhoods.