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

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Critical-point Hessian matrices transform by congruence under chart changes

Statement

Let $f : M \to R$ be smooth, let $p$ be a critical point of $f$ , and let $x$ and $y$ be smooth charts around $p$ . If

$H_{x} (f, p) = (\frac{\partial^{2} (f \circ x^{- 1})}{\partial x^{i} \partial x^{j}} (x (p))), H_{y} (f, p) = (\frac{\partial^{2} (f \circ y^{- 1})}{\partial y^{i} \partial y^{j}} (y (p))),$

and if $J = D (x \circ y^{- 1})_{y (p)}$ , then

$H_{y} (f, p) = J^{T} H_{x} (f, p) J .$

In particular the two Hessian matrices are congruent.

Facts & Assumptions

Given: A smooth function $f : M \to R$ , a critical point $p$ , and two charts $x$ and $y$ around $p$ .

[F1]

The critical-point Hessian is defined from the second partial derivatives of a coordinate representative at the critical point (The intrinsic Hessian of a smooth function at a critical point).

Proof

technique · coordinate computation

1.1

Write $g : = f \circ x^{- 1}$ , $h : = f \circ y^{- 1}$ , $a : = x (p)$ , $b : = y (p)$ , and $ϕ : = x \circ y^{- 1}$ . Then $h = g \circ ϕ$ , and the matrix $J$ in the statement is $D ϕ_{b}$ .

F1givenconstruct

2.1

Because $p$ is critical, the first derivative of $g$ at $a$ is zero, so differentiating $h = g \circ ϕ$ twice at $b$ gives $D^{2} h_{b} (u, v) = D^{2} g_{a} (J u, J v) + D g_{a} (D^{2} ϕ_{b} (u, v)) = D^{2} g_{a} (J u, J v)$ for all $u, v \in R^{n}$ .

step 1.1algebra

3.1

Step 2.1 is exactly the matrix identity $H_{y} (f, p) = J^{T} H_{x} (f, p) J$ , so the two chart Hessians are congruent.

F1step 2.1algebra∎

Depends on

The intrinsic Hessian of a smooth function at a critical point

Used by

Sylvester inertia makes the Morse index intrinsic Lemma

Cited to discharge well-definedness by The intrinsic Hessian of a smooth function at a critical point.

Dependency tree · two levels

2 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

Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)
Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)