DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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 is the metric dual of the differential

Definition

Let $g$ be a Riemannian metric on a smooth manifold $M$ and let $f : M \to R$ be smooth. The Riemannian gradient of $f$ is the smooth vector field ${grad}_{g} f$ characterized by

$g_{x} (({grad}_{g} f)_{x}, v) = d f_{x} (v) for every x \in M and v \in T_{x} M .$

Pointwise, it is the inverse metric-dual of $d f_{x}$ . In a local frame with metric matrix $(g_{i j})$ and inverse $(g^{i j})$ , it is

${grad}_{g} f = \sum_{i, j} g^{i j} \frac{\partial f}{\partial x^{j}} \partial_{x^{i}};$

the displayed coefficients are smooth, so this pointwise definition is a smooth vector field.

Depends on

Dependency tree · 2 levels

Sources

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, §13.1