LemmaStatement: Literature-sourcedProof: Literature-sourcedjudge 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.

At a critical point, the intrinsic Hessian agrees with the Levi-Civita Hessian

Statement

Let $f : M \to R$ be smooth, let $p$ be a critical point of $f$ , and let $g$ be any Riemannian metric on $M$ . If $\nabla$ is the Levi-Civita connection of $g$ , then

$(\nabla^{2} f)_{p} = {Hess}_{p} (f) .$

Facts & Assumptions

Given: A smooth function $f : M \to R$ , a critical point $p$ , a Riemannian metric $g$ , and its Levi-Civita connection $\nabla$ .

[F1]

The intrinsic critical-point Hessian is the bilinear form represented in a chart by the second partial derivatives of the coordinate representative (The intrinsic Hessian of a smooth function at a critical point).

[F2]

The covariant Hessian satisfies $(\nabla^{2} f)_{i j} = \frac{\partial^{2} (f \circ x^{- 1})}{\partial x^{i} \partial x^{j}} - \sum_{k} Γ_{i j}^{k} \frac{\partial (f \circ x^{- 1})}{\partial x^{k}}$ in any chart (Riemannian metrics, symmetric cotangent-bundle connections, and covariant Hessians).

[L1]

In a chart $x = (x^{1}, \dots, x^{n})$ around $p$ , $d f_{p} = \sum_{i} \frac{\partial (f \circ x^{- 1})}{\partial x^{i}} (x (p)) d x_{p}^{i} .$ (Coordinate formula for the differential of a function)

Proof

technique · coordinate computation

1.1F2given

Choose a smooth chart $x$ around $p$ , put $a : = x (p)$ and $h : = f \circ x^{- 1}$ . By [F2], the coordinate matrix of $(\nabla^{2} f)_{p}$ is $(\partial_{i j} h (a) - \sum_{k} Γ_{i j}^{k} (a) \partial_{k} h (a))_{i j}$ .

1.2L1givenalgebra

Since $p$ is critical, $d f_{p} = 0$ , and [L1] therefore forces every coefficient $\partial_{k} h (a)$ to vanish.

2.1F1step 1.1step 1.2∎

Step 1.1 reduces to the matrix $(\partial_{i j} h (a))_{i j}$ , which is exactly the matrix of ${Hess}_{p} (f)$ by [F1]. Therefore $(\nabla^{2} f)_{p} = {Hess}_{p} (f)$ .

Depends on

Used by

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

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

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)