CorollaryStatement: Literature-sourcedProof: Literature-sourcedaudited 2026-09-04

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Index and coindex swap under negation

Statement

Let $f : M \to R$ be smooth and let $p$ be a critical point of $f$ . Then $p$ is also a critical point of $- f$ , the nullity is unchanged, and the index and coindex are exchanged:

${null}_{p} (- f) = {null}_{p} (f), {ind}_{p} (- f) = {coind}_{p} (f), {coind}_{p} (- f) = {ind}_{p} (f) .$

Facts & Assumptions

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

[F1]

Nullity, index, and coindex are defined from the Hessian by kernel, negative-definite subspaces, and positive-definite subspaces (Nondegenerate critical points, nullity, index, and coindex).

Proof

technique · direct sign reversal

1.1F1givenalgebra

Because differentiation is linear, ${Hess}_{p} (- f) = - {Hess}_{p} (f)$ .

2.1F1step 1.1

Multiplication by $- 1$ does not change the kernel of a bilinear form, so the nullity is unchanged.

2.2F1step 1.1

A subspace is negative definite for $- {Hess}_{p} (f)$ exactly when it is positive definite for ${Hess}_{p} (f)$ , and similarly with "positive" and "negative" exchanged. Therefore the index and coindex swap by [F1].

3.1step 2.1step 2.2∎

Hence negating $f$ preserves nullity and exchanges index with coindex.

Depends on

Nondegenerate critical points, nullity, index, and coindex

Used by

A Morse gradient zero contributes (-1)^λ to the index Corollary

Dependency tree · two levels

3 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)