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

Polar integration may discard the cut locus Corollary
Comparison triangle in the two dimensional space form Definition
Hermitian metric and L² pairing on a compact Riemann surface Definition
Riemannian speed and length Definition
Cut locus of a point on a flat circle Example
Cut locus on a flat rectangular torus from the Dirichlet cell Example
Distance hessian in euclidean space Example
Toponogov comparison on a round sphere Example
A section curvature lower bound makes triangles thinner than the model False statement
A complete no-return extension of a strictly convex manifold Lemma
Local comparison of a riemannian metric with the euclidean metric Lemma
The pointwise norm on a tangent space is smooth off the zero vector Lemma
Unit spheres in finite-dimensional normed spaces are sequentially compact Lemma
Distance between corresponding side points in toponogov comparison Proposition
Gradient of the distance is the outward unit radial field off the base point and the cut locus Proposition
Cut locus of a point has riemannian volume zero Theorem
Distance from p is smooth off p and the cut locus Theorem
The cut locus of a point is closed Theorem
The exponential map is a diffeomorphism on the open tangent cut domain Theorem
Toponogov hinge comparison Theorem
Toponogov triangle comparison Theorem

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)