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

Used by

Finite-budget retraction Armijo backtracking Algorithm
Finite-budget Riemannian trust-region method Algorithm
A Morse gradient zero contributes (-1)^λ to the index Corollary
Closed odd-dimensional manifolds have zero Euler characteristic Corollary
The Euler number of the tangent bundle is the Euler characteristic Corollary
The Morse critical-point sum is the Euler characteristic Corollary
A regular continuation datum between Morse--Smale pairs Definition
Negative-gradient trajectories of a Morse function Definition
A negative-gradient trajectory satisfies the energy identity Lemma
Adapted descending field near a compact morse band Lemma
Normalized gradient crosses a compact regular band in controlled time Lemma
The continuation energy identity Lemma
The Riemannian gradient vanishes exactly at the critical points Lemma
Gradient of a unitary-invariant Hermitian projector energy Proposition
Projected Euclidean gradients are the Stiefel and Grassmann gradients Proposition
Converse Poincare-Hopf for nowhere-zero fields Theorem
Poincare-Hopf for closed manifolds Theorem
Stable and unstable manifolds for a Morse gradient with an arbitrary smooth metric Theorem
Uniform pullback smoothness guarantees an Armijo step Theorem

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, §13.1 (standard reference, not scraped)