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

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The intrinsic Hessian of a smooth function at a critical point

Definition

Let $f : M \to R$ be smooth, and let $p \in M$ be a critical point of $f$ (Critical points and critical values of a smooth function).

Choose a smooth chart $x = (x^{1}, \dots, x^{n})$ around $p$ , write $a : = x (p)$ , and let $g : = f \circ x^{- 1}$ . The matrix

$H_{x} (f, p) : = {(\frac{\partial^{2} g}{\partial x^{i} \partial x^{j}} (a))}_{1 \leq i, j \leq n}$

is symmetric, and it defines a bilinear form on $T_{p} M$ by

${Hess}_{p} (f) ⁣ (\sum_{i} u^{i} \frac{\partial}{\partial x^{i}} ∣_{p}, \sum_{j} v^{j} \frac{\partial}{\partial x^{j}} ∣_{p}) : = \sum_{i, j} u^{i} v^{j} \frac{\partial^{2} g}{\partial x^{i} \partial x^{j}} (a) .$

By Critical-point Hessian matrices transform by congruence under chart changes ↗, this bilinear form is independent of the chosen chart. It is the Hessian of $f$ at $p$ .

Depends on

Critical points and critical values of a smooth function

Used by

A degenerate critical set can be nonisolated Counterexample
An isolated critical point can be degenerate Counterexample
Nondegenerate critical points, nullity, index, and coindex Definition
At a critical point, the intrinsic Hessian agrees with the Levi-Civita Hessian Lemma
Critical-point Hessian matrices transform by congruence under chart changes Lemma
Nondegenerate critical points are isolated Lemma
Splitting one Morse coordinate preserves the residual Hessian Lemma

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)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)