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

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Affine reparametrization of a geodesic is a geodesic

Statement

If $γ : I \to M$ is an affinely parametrized geodesic and $ℓ (t) = a t + b$ maps an interval $J$ with nonempty interior into $I$ , then $γ \circ ℓ$ is a geodesic. For any supplied Riemannian metric, its speed at $t$ is $∣ a ∣$ times the speed of $γ$ at $a t + b$ .

Facts & Assumptions

Given: The geodesic $γ$ , constants $a, b$ , and intervals in the statement.

[F1]

Geodesic of an affine connection defines the geodesic equation by $D_{s} γ^{'} = 0$ .

[F2]

Riemannian speed and length defines speed as the Riemannian norm of the velocity.

Proof

1.1

Put $\tilde{γ} = γ \circ ℓ$ . The ordinary and covariant chain rules give ${\tilde{γ}}^{'} (t) = a γ^{'} (a t + b)$ and $D_{t} {\tilde{γ}}^{'} = a^{2} (D_{s} γ^{'}) (a t + b) = 0$ by [F1]. Hence $\tilde{γ}$ is geodesic.

F1givenalgebra

2.1

By homogeneity of the norm in [F2], $∣ {\tilde{γ}}^{'} (t) ∣ = ∣ a ∣ ∣ γ^{'} (a t + b) ∣$ . If $a = 0$ , the reparametrized curve is constant and both formulas give zero; negative $a$ reverses the parameter and uses the absolute value. Dimensions zero and one require no change, empty manifolds have no curves, and included endpoints use the corresponding one-sided chain rule. The constants and curve are supplied, so no choice principle is used.

F2step 1.1∎

Depends on

Used by

Every affinely reparametrized geodesic remains unit speed False statement
Geodesic scaling identity Lemma
Incompleteness is finite-time geodesic escape Proposition
Length minimizers are constant-speed geodesics up to reparametrization Theorem
Metric completeness implies geodesic completeness Theorem

Dependency tree · two levels

7 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, Remark 15.1.2, pp.113–114 (standard reference, not scraped)