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

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Injectivity radius at a point and of a manifold

Definition

Assume ${A C}_{ω}$ . For $p \in M$ , put $R_{p} = {r > 0 : B_{r} (0_{p}) \subseteq E_{p} and \exp_{p} ∣_{B_{r} (0_{p})} is a diffeomorphism onto its image} .$ The injectivity radius at $p$ and the injectivity radius of $M$ are the extended nonnegative numbers $inj (p) = \sup R_{p} \in (0, + \infty], inj (M) = \inf_{p \in M} inj (p) \in [0, + \infty] .$ For the empty manifold, the second infimum is defined to be $+ \infty$ .

Facts & Assumptions

Given: A boundaryless Riemannian manifold $M$ and, for the pointwise clause, $p \in M$ .

[F1]

Under The Axiom of Countable Choice ( ${A C}_{ω}$ ), Existence of normal neighborhoods supplies an open star-shaped exponential-diffeomorphism domain about $0_{p}$ .

[F2]

The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined supplies $+ \infty$ and the extended order used by the supremum and infimum conventions.

Verification

1.1F1F2

The set $R_{p}$ is nonempty: by [F1], an open exponential-diffeomorphism domain contains some ball $B_{r} (0_{p})$ with $r > 0$ , and restricting a diffeomorphism to that ball remains a diffeomorphism onto its open image. It is downward closed among positive radii. Hence its supremum is a well-defined element of $(0, + \infty]$ ; it is $+ \infty$ exactly when admissible radii are unbounded.

2.1F1F2step 1.1∎

If $M$ is nonempty, the set of positive pointwise radii has an infimum in $[0, + \infty]$ ; it may be zero even though every term is positive. For $M = \emptyset$ , the stated empty-infimum convention gives $inj (M) = + \infty$ . In dimension zero, $T_{p} M = {0_{p}}$ and every positive-radius ball is that singleton, so $R_{p} = (0, \infty)$ and $inj (p) = + \infty$ ; dimension one uses the displayed formula unchanged. The balls are open, so their sphere endpoints are not included, while $0_{p}$ is always included. ${A C}_{ω}$ is used only through [F1], not in taking the uniquely determined supremum or infimum.

Depends on

Used by

A complete manifold with zero global injectivity radius Counterexample
Injectivity radius at each point is positive Proposition
Injectivity radius is the infimum of cut times Proposition
Round sphere model geometry Proposition

Dependency tree · two levels

19 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 23.3.3, pp.171--172 (standard reference, not scraped)