ExampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-04

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The standard quadratic form realizes every Morse index

Example

Fix $n \geq 0$ and $0 \leq λ \leq n$ . On $R^{n}$ , the quadratic form

$q_{λ} (x_{1}, \dots, x_{n}) : = - \sum_{i = 1}^{λ} x_{i}^{2} + \sum_{i = λ + 1}^{n} x_{i}^{2}$

has a unique critical point at the origin, and that critical point has Morse index $λ$ .

Facts & Assumptions

Given: Integers $n \geq 0$ and $0 \leq λ \leq n$ , and the quadratic form $q_{λ}$ above.

[F1]

Index and nondegeneracy are read from the Hessian (Nondegenerate critical points, nullity, index, and coindex).

[L1]

The Morse normal form is exactly the displayed signed quadratic form (Morse lemma).

Verification

technique · direct computation

1.1givenalgebra

The partial derivatives satisfy $\partial q_{λ} / \partial x_{i} = - 2 x_{i}$ for $1 \leq i \leq λ$ and $\partial q_{λ} / \partial x_{i} = 2 x_{i}$ for $λ < i \leq n$ , so all first derivatives vanish exactly at $x = 0$ .

2.1F1step 1.1algebra

The Hessian matrix at the origin is $diag (\underset{λ}{\underset{⏟}{- 2, \dots, - 2}}, \underset{n - λ}{\underset{⏟}{2, \dots, 2}})$ , so the critical point is nondegenerate and has exactly $λ$ negative directions. By [F1], its index is $λ$ .

3.1L1step 2.1∎

The displayed formula is already the Morse normal form from [L1], including the endpoint cases $λ = 0$ and $λ = n$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)