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.

Pointwise norm and angle from a riemannian metric

Definition

The pointwise norm is $∣ v ∣_{g} = \sqrt{g (v, v)}$ . For nonzero $v, w$ in the same tangent space, the angle is the unique $θ \in [0, π]$ with $\cos θ = g (v, w) / (∣ v ∣_{g} ∣ w ∣_{g})$ .

Positive definiteness in Riemannian metric and riemannian manifold makes both denominators positive. Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors places the quotient in $[- 1, 1]$ , on which the inverse of cosine restricted to $[0, π]$ is defined. Angles $0$ and $π$ are included. The norm of zero is zero; no angle is assigned when either vector is zero.

Source locator

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

Depends on

Used by

Riemannian speed and length Definition
Local comparison of a riemannian metric with the euclidean metric Lemma

Dependency tree · two levels

9 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)