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.

Pointwise norm and angle from a riemannian metric

Definition

Source locator

Depends on

Dependency tree · 2 levels

Sources