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.

Domain and exponential map of a connection

Definition

Assume ${A C}_{ω}$ . Let $M$ be a smooth manifold without boundary with an affine connection. For $v \in T_{p} M$ , let $γ_{p, v} : I_{p, v} \to M$ be its unique maximal geodesic. The domain of the exponential map is $E = {v \in T M : 1 \in I_{p, v} for p = π (v)} .$ The exponential map and its fibrewise restrictions are $\exp : E ⟶ M, \exp (v) = γ_{p, v} (1), \exp_{p} = \exp ∣_{E_{p}} : E_{p} ⟶ M,$ where $E_{p} = E \cap T_{p} M$ .

Facts & Assumptions

Given: A boundaryless smooth manifold $M$ with an affine connection, and the bundle projection $π : T M \to M$ .

[F1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the assumed ${A C}_{ω}$ , and Existence uniqueness and smooth dependence of geodesics supplies, for each $v \in T_{p} M$ , the unique maximal geodesic $γ_{p, v}$ on an open interval $I_{p, v}$ containing zero.

Verification

1.1

Every tangent vector $v \in T M$ has the unique base point $p = π (v)$ , and [F1] uniquely determines both $I_{p, v}$ and $γ_{p, v}$ . Thus membership in $E$ and the value $γ_{p, v} (1)$ are well-defined. The definition only evaluates curves whose maximal interval actually contains $1$ and therefore does not presume geodesic completeness.

F1given

2.1

The zero vector $0_{p}$ gives the constant geodesic on all of $R$ , so $0_{p} \in E_{p}$ and $\exp_{p} (0_{p}) = p$ . In dimension zero all tangent vectors are zero; if $M$ is empty, then $T M$ and $E$ are empty and the displayed map is the unique empty function. Because $I_{p, v}$ is open, $1 \in I_{p, v}$ is an interior-time condition rather than an included-endpoint convention; $E$ may still be a proper subset of $T M$ . The only choice principle used is the stated ${A C}_{ω}$ inherited through [F1].

F1step 1.1∎

Depends on

Used by

Hopf–Rinow on a flat cylinder Example
Normal coordinates on the round sphere Example
The exponential map of a flat torus is not injective Example
The exponential map is always defined on all of TM False statement
The exponential map scales geodesic time Proposition
Hopf–Rinow theorem Theorem
The exponential domain is open and the exponential map is smooth Theorem

Dependency tree · two levels

11 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 17.1.2, pp.127--128 (standard reference, not scraped)