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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The exponential map scales geodesic time

Statement

Assume ${A C}_{ω}$ . For $v \in T_{p} M$ and $t \in R$ , $t \in I_{p, v} ⟺ t v \in E_{p},$ and whenever these equivalent conditions hold, $\exp_{p} (t v) = γ_{p, v} (t) .$ In particular, if $v \in E_{p}$ , then $s v \in E_{p}$ for every $s \in [0, 1]$ .

Facts & Assumptions

Given: A boundaryless smooth manifold with an affine connection, $v \in T_{p} M$ , and $t \in R$ .

[F1]

Under The Axiom of Countable Choice ( ${A C}_{ω}$ ), Geodesic scaling identity gives $γ_{p, a v} (u) = γ_{p, v} (a u)$ and, for $a \neq 0$ , $I_{p, a v} = a^{- 1} I_{p, v}$ ; for $a = 0$ the scaled geodesic is constant on $R$ .

[F2]

Domain and exponential map of a connection says $w \in E_{p}$ exactly when $1 \in I_{p, w}$ and then $\exp_{p} (w) = γ_{p, w} (1)$ .

Proof

technique · direct

1.1F1F2

Suppose first that $t \neq 0$ . By [F1], $1 \in I_{p, t v}$ iff $t \in I_{p, v}$ , and on that domain $γ_{p, t v} (1) = γ_{p, v} (t)$ . Applying [F2] proves both the domain equivalence and the exponential identity.

1.2F1F2

If $t = 0$ , then $0 \in I_{p, v}$ , while [F1] makes $γ_{p, 0}$ the constant curve on $R$ ; hence $0 v \in E_{p}$ and $\exp_{p} (0) = p = γ_{p, v} (0)$ . Thus the equivalence and identity also hold at zero.

2.1F1F2step 1.1step 1.2

Let $v \in E_{p}$ and $s \in [0, 1]$ . Then $1 \in I_{p, v}$ by [F2]. Since $I_{p, v}$ is an interval containing $0$ , it contains $s$ , so step 1.1 or 1.2 gives $s v \in E_{p}$ . Thus every fibre domain is star-shaped about zero. Negative parameters are covered by step 1.1 whenever the corresponding geodesic time lies in the maximal interval.

3.1F1F2step 1.1step 1.2step 2.1∎

On an empty manifold there are no initial vectors. In dimension zero only step 1.2 occurs, and in dimension one the proof is unchanged. Zero velocity and zero scale were handled explicitly; membership is always at the interior time $1$ of an open maximal interval, including when the original time is a finite endpoint candidate. The stated ${A C}_{ω}$ is inherited through [F1]--[F2], and no additional selection is made.

Depends on

Used by

Sufficiently short geodesic segments are uniquely minimizing Corollary
A conjugate point at which there are many geodesics Counterexample
A clean framed Whitney bigon has an adapted tube Lemma
Arcs joining two points of a connected submanifold avoiding finitely many points Lemma
Embedded bands joining two framed spheres exist Lemma
Local length comparison for a conjugate-free geodesic Lemma
Radial geodesics from one point reach every point under global exponential domain Lemma
At a conjugate endpoint the index form is degenerate Proposition
Gradient of the distance is the outward unit radial field off the base point and the cut locus Proposition
Hessian of distance in terms of radial jacobi fields Proposition
Injectivity radius is the infimum of cut times Proposition
Properties of normal coordinates at the center Proposition
Cut time is positive and continuous Theorem
Differential of the exponential map in terms of Jacobi fields Theorem
Gauss lemma Theorem
The differential of exp at zero is the identity Theorem
The exponential map is a diffeomorphism on the open tangent cut domain Theorem

Dependency tree · two levels

10 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, Proposition 17.1.4(2)--(3), p.128 (standard reference, not scraped)