LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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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.
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.

Minimizing along a geodesic is an initial interval property

Statement

Assume countable choice through the declared dependencies. Let $(M, g)$ be a complete connected boundaryless Riemannian manifold, $p \in M$ , $v \in T_{p} M$ a unit vector, and $γ (t) = \exp_{p} (t v)$ for $t \geq 0$ . The set $A_{p} (v) = {t \geq 0 : d_{g} (p, γ (t)) = t}$ is an initial interval: if $T \in A_{p} (v)$ and $0 \leq s \leq T$ , then $s \in A_{p} (v)$ . Moreover, if the cut time $c_{p} (v)$ is finite, then $c_{p} (v) \in A_{p} (v)$ . Equivalently, once an earlier segment fails to minimize, no later segment along this ray minimizes.

Facts & Assumptions

Given: The complete ray $γ$ and the fixed initial point $p$ .

[A1]

The setup carries ${A C}_{ω}$ through The Axiom of Countable Choice ( ${A C}_{ω}$ ) and the cut-time supplier. This proof makes no use of full AC.

[F1]

The cut time is the supremum of the positive minimizing times, as defined in Cut time in a unit tangent direction.

[F2]

Riemannian distance is the infimum of lengths of piecewise $C^{1}$ curves, by Riemannian distance on a connected manifold.

Proof

technique · direct competitor concatenation and an epsilon argument

1.1F2given

Put $m (t) = d_{g} (p, γ (t))$ and $A = {t \geq 0 : m (t) = t}$ . The ray segment from $0$ to $t$ has unit speed and length $t$ , so [F2] gives $m (t) \leq t$ ; also $0 \in A$ . Thus membership in $A$ is exactly the assertion that the segment to $γ (t)$ minimizes.

1.2F2given

For any $s, t \geq 0$ and $ε > 0$ , [F2] gives one curve from $p$ to $γ (s)$ of length less than $m (s) + ε$ ; appending the ray segment gives $m (t) \leq m (s) + ε + ∣ t - s ∣$ . Reversing $s, t$ and letting $ε ↓ 0$ proves $∣ m (t) - m (s) ∣ \leq ∣ t - s ∣$ . This uses one near-minimizer at a time.

2.1F2step 1.1

If $T \in A$ and $0 < s < T$ were not in $A$ , then $m (s) < s$ ; by [F2] there is a piecewise $C^{1}$ curve from $p$ to $γ (s)$ of length strictly less than $s$ . Concatenating it with $γ ∣_{[s, T]}$ gives a curve to $γ (T)$ of length less than $s + (T - s) = T$ , contradicting $T \in A$ . The case $s = 0$ is already in [1.1], so $A$ is an initial interval.

2.2F1F2step 1.2

Let $c = c_{p} (v)$ from [F1]. If $c < \infty$ , then for every sufficiently small $ε > 0$ the supremum property gives $s \in A$ with $c - ε < s \leq c$ ; by the Lipschitz estimate [1.2], $m (c) \geq m (s) - ∣ c - s ∣ = 2 s - c > c - 2 ε$ , while the ray segment gives $m (c) \leq c$ . Hence $m (c) = c$ , so every finite cut-time endpoint minimizes; for $c = + \infty$ there is no finite endpoint to check.

3.1

At $T = 0$ the segment is constant and minimizing. The empty manifold has no point $p$ ; in dimension zero there are no unit directions, and in dimension one the two directions are handled separately. The finite endpoint argument uses one existential near-minimizer for each arbitrary $ε$ , not a selected sequence, so it spends no countable-choice instance beyond the inherited ${A C}_{ω}$ setup [A1]. The contrapositive is the equivalent formulation in the Statement; both implications are established. [A1, F1, step 2.1, step 2.2] $□$

Depends on

Used by

A conjugate point at which there are many geodesics Counterexample
Cut point and cut locus of a point Definition
Injectivity radius is the infimum of cut times Proposition
Characterization of a cut point Theorem
Cut time is positive and continuous Theorem
Distance from p is smooth off p and the cut locus Theorem
The cut locus of a point is closed Theorem
The exponential map is a diffeomorphism on the open tangent cut domain Theorem

Dependency tree · two levels

13 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

John M. Lee, Riemannian Manifolds: An Introduction to Curvature (1997), Chapter 10 (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (2025), Lectures 21–24 (standard reference, not scraped)