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

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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.
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Geodesically complete Riemannian manifold

Definition

Assume ${A C}_{ω}$ . A Riemannian manifold without boundary is geodesically complete when, for every initial vector $v \in T_{p} M$ , the unique maximal geodesic has domain $I_{p, v} = R .$ For a disconnected manifold this condition is componentwise. The zero initial vector is included.

Facts & Assumptions

Given: A boundaryless Riemannian manifold $M$ .

[F1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ) is the assumed ${A C}_{ω}$ , and Existence uniqueness and smooth dependence of geodesics then supplies the unique maximal interval $I_{p, v}$ for every initial vector.

Verification

1.1

The definition is intrinsic because [F1] makes $I_{p, v}$ unique. A geodesic remains in the connected component of its initial point, since the continuous image of its interval is connected; therefore requiring all $I_{p, v} = R$ is equivalent to requiring the same condition separately on every component.

F1given

2.1

The zero vector gives the constant geodesic and already has domain $R$ . In dimension zero every vector is zero, so every boundaryless zero-manifold is geodesically complete; the empty manifold satisfies the universal condition vacuously. Failure means one explicitly existing initial vector has a finite end in its maximal open interval, so no included-endpoint ambiguity occurs. The only choice principle is the stated ${A C}_{ω}$ inherited from the construction of the maximal geodesics; the universal quantifier itself selects nothing.

F1step 1.1∎

Depends on

Used by

A local isometry from a complete connected manifold has geodesically complete target image Corollary
Compact Riemannian manifolds are geodesically complete Corollary
A complete manifold with zero global injectivity radius Counterexample
Hopf–Rinow on a flat cylinder Example
Hyperbolic space is complete Example
The punctured Euclidean plane is geodesically incomplete Example
A Riemannian product is complete iff each factor is complete Proposition
Incompleteness is finite-time geodesic escape Proposition
Hopf–Rinow theorem Theorem
Metric completeness implies geodesic completeness 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 19.2.1, p.141 (standard reference, not scraped)