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

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Geodesics have constant speed for a metric-compatible connection

Statement

Let $γ : I \to M$ be a geodesic for a metric-compatible affine connection on a Riemannian manifold. Then $g (γ^{'}, γ^{'})$ and the speed $∣ γ^{'} ∣$ are constant on $I$ .

Facts & Assumptions

Given: The geodesic, metric, and compatible connection in the statement.

[F1]

Geodesic of an affine connection gives $D_{t} γ^{'} = 0$ .

[F2]

Metric compatible connection on a riemannian vector bundle gives the product rule for differentiating the metric pairing along a curve.

[F3]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant makes a differentiable real function with zero derivative on an interval constant.

Proof

1.1F1F2given

Metric compatibility and the symmetry of $g$ give $\frac{d}{d t} g (γ^{'}, γ^{'}) = g (D_{t} γ^{'}, γ^{'}) + g (γ^{'}, D_{t} γ^{'}) = 2 g (D_{t} γ^{'}, γ^{'}) = 0$ by [F1] and [F2].

2.1F3step 1.1∎

By [F3], $g (γ^{'}, γ^{'})$ is constant on the interval. It is nonnegative, so its nonnegative square root $∣ γ^{'} ∣$ is constant as well. This includes the zero-speed constant geodesics and shows that a nonconstant geodesic never has zero velocity. In dimension zero the constant is zero; empty manifolds give no curves. Included parameter endpoints follow by continuity from the interior, and no choices are made.

Depends on

Used by

A geodesic segment before its first conjugate point is locally energy minimizing Corollary
Polar integration may discard the cut locus Corollary
Squared distance is strictly convex along geodesics in a hadamard manifold Corollary
A complete manifold with zero global injectivity radius Counterexample
A conjugate point at which there are many geodesics Counterexample
Conjugate points along a geodesic and their multiplicity Definition
Conjugate antipodes on the round sphere Example
Great circles as round-sphere geodesics Example
No conjugate points in nonpositive constant curvature Example
Arcs joining two points of a connected submanifold avoiding finitely many points Lemma
H-one local length comparison for a conjugate-free geodesic Lemma
Local length comparison for a conjugate-free geodesic Lemma
Radial geodesics from one point reach every point under global exponential domain Lemma
Hessian of distance in terms of radial jacobi fields Proposition
Incompleteness is finite-time geodesic escape Proposition
Injectivity radius is the infimum of cut times Proposition
Round sphere model geometry Proposition
Tangential jacobi fields are affine multiples of the velocity Proposition
A geodesic does not minimize past its first conjugate point Theorem
Cut time is positive and continuous Theorem
Existence of geodesically convex neighborhoods Theorem
Gauss lemma Theorem
Length minimizers are constant-speed geodesics up to reparametrization Theorem
Metric completeness implies geodesic completeness Theorem
No conjugate points under nonpositive sectional curvature Theorem
The exponential map is a diffeomorphism on the open tangent cut domain Theorem

Dependency tree · two levels

18 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)