CorollaryStatement: AI-adaptedProof: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The critical level is a quadratic cone in Morse coordinates

Statement

Let $f : M \to R$ be smooth, let $p$ be a nondegenerate critical point of index $λ$ on an $n$ -manifold, and choose Morse coordinates around $p$ . Then near $p$ the critical level set is

$f^{- 1} (f (p)) = {\sum_{i = 1}^{λ} (x^{i})^{2} = \sum_{i = λ + 1}^{n} (x^{i})^{2}} .$

If $λ = 0$ or $λ = n$ , this local level set is just the point $p$ .

Facts & Assumptions

Given: A smooth function $f : M \to R$ and a nondegenerate critical point $p$ of index $λ$ .

[L1]

In Morse coordinates, $f - f (p) = - \sum_{i = 1}^{λ} (x^{i})^{2} + \sum_{i = λ + 1}^{n} (x^{i})^{2} .$ (Morse lemma)

Proof

technique · normal form reading

1.1L1givenalgebra

By [L1], the equation $f = f (p)$ becomes $\sum_{i = 1}^{λ} (x^{i})^{2} = \sum_{i = λ + 1}^{n} (x^{i})^{2}$ .

2.1step 1.1algebra

If $λ = 0$ , the left-hand side is the empty sum $0$ , so the equation is $0 = \sum_{i = 1}^{n} (x^{i})^{2}$ , which forces $x = 0$ . If $λ = n$ , the right-hand side is the empty sum and the same conclusion follows.

3.1step 1.1step 2.1∎

Therefore the local critical level is the stated quadratic cone, with the index- $0$ and index- $n$ cases collapsing to the single critical point.

Depends on

Morse lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)
Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)