DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Riemannian metric and riemannian manifold

Definition

A Riemannian metric on a Hausdorff second-countable smooth manifold $M$ is a smooth symmetric covariant two-tensor $g$ such that $g_{p} (v, v) > 0$ for every point $p$ and every nonzero $v \in T_{p} M$ . A Riemannian manifold is the pair $(M, g)$ .

This is a A smooth tensor field giving a Smooth bundle metrics on $T M$ . Dimension zero is allowed: its zero bilinear form is positive definite because there are no nonzero vectors. The empty manifold has its unique empty metric. Boundaries are allowed where stated, with smoothness understood up to the boundary.

Source locator

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 13, pp.328–332 and 341–342.

Depends on

Used by

Conformal equivalence of riemannian metrics Definition
Musical isomorphisms Definition
Pointwise norm and angle from a riemannian metric Definition
Pullback of a riemannian metric as a tensor Definition
Parallel transport on the round sphere along the equator Example
Coordinate criterion for a riemannian metric Proposition
Every smooth manifold admits a riemannian metric Theorem

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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry, September 2025 (standard reference, not scraped)