DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

Normal neighborhood and normal coordinate chart

Definition

Assume ${A C}_{ω}$ . A normal neighbourhood centred at $p$ is an open set $U \subseteq M$ for which there is an open star-shaped neighbourhood $D \subseteq T_{p} M$ of $0_{p}$ such that $D \subseteq E_{p}$ and $\exp_{p} : D \to U$ is a diffeomorphism.

Given a supplied ordered basis $e = (e_{1}, \dots, e_{n})$ of $T_{p} M$ , let $E_{e} : R^{n} \to T_{p} M$ be $E_{e} (a^{1}, \dots, a^{n}) = \sum_{i} a^{i} e_{i}$ . The associated normal coordinate chart is $x_{e} = E_{e}^{- 1} \circ \exp_{p}^{- 1} : U ⟶ E_{e}^{- 1} (D) \subseteq R^{n} .$ When $e$ is orthonormal, these are orthonormal normal coordinates.

Facts & Assumptions

Given: A boundaryless Riemannian manifold, a point $p$ , and, for the coordinate clause, a supplied ordered basis $e$ of $T_{p} M$ .

[F1]

Under The Axiom of Countable Choice ( ${A C}_{ω}$ ), Existence of normal neighborhoods supplies at least one such star-shaped exponential diffeomorphism at every point.

Verification

1.1

Because $\exp_{p} ∣_{D}$ is a diffeomorphism, its inverse is a well-defined smooth map $U \to D$ . The supplied basis makes $E_{e}$ a specified linear isomorphism, so $E_{e}^{- 1} (D)$ is open and $x_{e}$ is a diffeomorphism onto it. Thus the intrinsic normal neighbourhood does not depend on coordinates, while the displayed coordinate list records exactly its dependence on the supplied basis.

F1given

2.1

Star-shaped means that $v \in D$ implies $t v \in D$ for every $0 \leq t \leq 1$ , including both endpoints. The zero vector belongs to $D$ and maps to $p$ . In dimension zero, the ordered basis is the empty tuple, $R^{0}$ , $T_{p} M$ , $D$ , and $U$ are singletons, and the formula gives the unique chart; dimension one is literal. If $M$ is empty there is no centre $p$ . The basis is supplied rather than chosen, so the only choice principle is the stated ${A C}_{ω}$ inherited through [F1].

F1step 1.1∎

Depends on

Used by

Local formula for distance from the centre of a normal neighbourhood Corollary
Polar form of the metric in normal coordinates Corollary
Sufficiently short geodesic segments are uniquely minimizing Corollary
Normal coordinates on the round sphere Example
Normal coordinates make the metric Euclidean throughout the chart False statement
Properties of normal coordinates at the center Proposition
Existence of geodesically convex neighborhoods Theorem

Dependency tree · two levels

12 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

Ved Datar, Lectures on Riemannian Geometry, Definition 17.2.1, p.130 (standard reference, not scraped)