LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

The Riemannian gradient vanishes exactly at the critical points

Statement

For a Riemannian metric $g$ , a smooth $f : M \to R$ , and $x \in M$ , ${grad}_{g} f (x) = 0$ if and only if $x$ is a critical point of $f$ .

Facts & Assumptions

Given: A Riemannian metric $g$ , a smooth function $f : M \to R$ , and $x \in M$ .

[F1]

The gradient is characterized by $g_{x} ({grad}_{g} f (x), v) = d f_{x} (v)$ for every $v \in T_{x} M$ (The Riemannian gradient is the metric dual of the differential).

[F2]

A point is critical exactly when its differential is the zero map (Critical points and critical values of a smooth function).

Proof

technique · direct

1.1F1given

If ${grad}_{g} f (x) = 0$ , then [F1] gives $d f_{x} (v) = g_{x} (0, v) = 0$ for every $v$ , so $d f_{x} = 0$ .

1.2F1given

Conversely, if $d f_{x} = 0$ , then [F1] gives $g_{x} ({grad}_{g} f (x), v) = 0$ for every $v$ . Taking $v = {grad}_{g} f (x)$ and using positive definiteness gives ${grad}_{g} f (x) = 0$ .

2.1F2step 1.1step 1.2∎

By [F2], the two implications say exactly that ${grad}_{g} f (x) = 0$ if and only if $x$ is critical.

Depends on

Used by

Dependency tree · two levels

6 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, §13.1 (standard reference, not scraped)