TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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Smooth invariance of the manifold boundary

Statement

A smooth diffeomorphism between relatively open half-space sets carries face points to face points and relative-interior points to relative-interior points; consequently $\partial M$ and $Int M$ are intrinsic.

Facts & Assumptions

Given: Relatively open sets $U, V \subseteq H^{n}$ and a smooth diffeomorphism $f : U \to V$ .

[L1]

In dimension zero the model face is empty and every point of $H^{0}$ is a relative-interior point (Interior and boundary of a manifold with boundary).

[L2]

Smooth half-space maps satisfy the intrinsic chain-rule formula (Chain rule for smooth half-space maps).

Proof

technique · direct

1.1

If $n = 0$ , [L1] makes the claim immediate. Assume $n \geq 1$ . At any $p \in U$ , choose Euclidean extensions of $f$ and $f^{- 1}$ near $p$ and $f (p)$ . Since both half-space composites are the identity, [L2] gives $D (f^{- 1})_{f (p)} D f_{p} = I$ and $D f_{p} D (f^{- 1})_{f (p)} = I$ .

givenL1L2cases

2.1

For $n \geq 1$ , if a face point $p$ mapped to the relative interior, then on a Euclidean neighbourhood of $f (p)$ contained in $V$ , the last coordinate of an extension of $f^{- 1}$ would be nonnegative and would vanish at the interior point $f (p)$ . Its differential there would therefore be zero, contradicting the invertibility from step 1.1. Applying the same argument to $f^{- 1}$ proves the converse. Hence face and relative-interior points are preserved in every dimension, making the manifold boundary and interior intrinsic.

givenstep 1.1algebra∎

Depends on

Used by

Diffeomorphisms preserve interior and boundary Corollary
Inward, outward, and boundary-tangent vectors Definition
The closed half-space as a manifold with boundary Example
Can a smooth chart turn a boundary point into an interior point? False statement
Empty boundary is equivalent to being boundaryless Proposition
The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold Theorem
The interior is an open smooth n-manifold Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)