DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nondegenerate critical points, nullity, index, and coindex

Definition

Let $f : M \to R$ be smooth and let $p$ be a critical point of $f$ . The Hessian ${Hess}_{p} (f)$ is a symmetric bilinear form on the finite-dimensional real vector space $T_{p} M$ (The intrinsic Hessian of a smooth function at a critical point).

The nullity of $p$ for $f$ is ${null}_{p} (f) : = \dim \ker ({Hess}_{p} (f)) .$
The critical point $p$ is nondegenerate when ${null}_{p} (f) = 0$ .
The index of $p$ for $f$ is the largest dimension of a subspace of $T_{p} M$ on which ${Hess}_{p} (f)$ is negative definite.
The coindex of $p$ for $f$ is the largest dimension of a subspace of $T_{p} M$ on which ${Hess}_{p} (f)$ is positive definite.

The positivity and negativity conventions are those of Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form.

Depends on

Used by

Index and coindex swap under negation Corollary
A degenerate critical set can be nonisolated Counterexample
Morse functions and excellent Morse functions Definition
A standard torus height function has four critical points Example
The height function on the sphere is Morse and excellent Example
The standard quadratic form realizes every Morse index Example
Nondegenerate critical points are isolated Lemma
Sylvester inertia makes the Morse index intrinsic Lemma
The zero-dimensional Morse convention Remark
Morse lemma Theorem

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

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