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

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Morse-Sard for maps from manifolds with boundary

Statement

Assume ${A C}_{ω}$ . Let $F : M^{m} \to N^{n}$ be $C^{r}$ in the local-extension sense, where $N$ is boundaryless and $r > \max {m - n, 0}$ . The union of critical values of $F ∣_{Int M}$ and $F ∣_{\partial M}$ is null; values regular for both restrictions are dense.

Facts & Assumptions

Given: ${A C}_{ω}$ , a $C^{r}$ map $F : M^{m} \to N^{n}$ in the local-extension sense, a boundaryless target $N$ , and $r > \max {m - n, 0}$ .

[L1]

The interior is a boundaryless smooth $m$ -manifold and, for $m \geq 1$ , the boundary is a boundaryless smooth $(m - 1)$ -manifold (The interior is an open smooth n-manifold; The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold).

[L2]

A $C^{r}$ Euclidean map from dimension $d$ to positive dimension $n$ , with $r > \max {d - n, 0}$ , has a null critical-value set (Morse-Sard for Euclidean maps).

[L3]

Under ${A C}_{ω}$ , countable unions and subsets of manifold null sets are null (Countable unions and subsets of manifold null sets are null).

Proof

technique · direct

1.1

If $n = 0$ , every differential to the zero tangent space is surjective, so both critical-value sets are empty. Suppose $n \geq 1$ . By [L1] and second countability, choose countable coordinate covers of $Int M$ and, when $m \geq 1$ , of $\partial M$ , refining them so each image lies in a target chart. Each coordinate representative has a $C^{r}$ Euclidean extension. Since $r > \max {m - n, 0}$ also implies $r > \max {(m - 1) - n, 0}$ , [L2] makes every chartwise critical-value set null.

givenL1L2caseschoose

2.1

By [L3], each restriction-critical-value set and their union are null. A manifold null set has empty interior, so its complement is dense; that complement consists exactly of the values regular for both restrictions. Together with the $n = 0$ case of step 1.1, this proves the claim.

L3step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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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)