TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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For a compact manifold embedded in Euclidean space, the squared-distance function from a generic center is Morse

Statement

Let $M \subseteq R^{N}$ be a compact embedded smooth manifold. Then there is a null subset $E \subseteq R^{N}$ such that for every $p \in R^{N} ∖ E$ , the squared-distance function $d_{p} : M \to R, d_{p} (x) = ∥ x - p ∥^{2}$ is Morse.

Facts & Assumptions

Given: A compact embedded smooth manifold $M \subseteq R^{N}$ .

[F1]

A smooth function is Morse exactly when its differential section is transverse to the zero section (A smooth function is Morse if and only if its differential section is transverse to the zero section).

[L1]

Parametric transversality makes the set of bad parameters null once the total family is transverse to the target submanifold (Parametric transversality).

[L2]

The zero section of the cotangent bundle is an embedded submanifold (The zero section is a smooth embedding).

[A1]

For fixed $p \in R^{N}$ and $x \in M$ , the differential of $d_{p}$ at $x$ is the cotangent vector $v \mapsto 2 (x - p) \cdot v$ on $T_{x} M$ . Varying the parameter $p$ changes this differential by $v \mapsto - 2 w \cdot v$ , and as $w$ ranges over $R^{N}$ these restrictions realize every cotangent vector on $T_{x} M$ .

Proof

technique · direct

1.1

Define a smooth family of cotangent sections by $D : M \times R^{N} \to T^{*} M, D (x, p) = d (d_{p})_{x} .$ By [L2], the zero section is an embedded submanifold of the target bundle. At any zero $(x, p)$ , the parameter-derivative description in [A1] spans the full fibre $T_{x}^{*} M$ , so $D$ is transverse to the zero section.

L2A1givenconstruct

2.1

Apply [L1] to the family $D$ . The bad centers $p$ for which $d (d_{p})$ is not transverse to the zero section form a null subset $E \subseteq R^{N}$ . For every $p \notin E$ , [F1] turns this transversality conclusion into the statement that $d_{p}$ is Morse.

F1L1step 1.1

3.1

Therefore squared-distance functions are Morse for generic centers in the ambient Euclidean space.

step 2.1∎

Depends on

Used by

Squared distance to a circle is Morse for centers off the medial axis and degenerate at the center Example

Dependency tree · two levels

12 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

Marco Gualtieri, Topology I: Smooth Manifolds, Part 11 (standard reference, not scraped)
Shintaro Fushida-Hardy, Morse theory (standard reference, not scraped)