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

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Geodesic of an affine connection

Definition

Let $M$ be a smooth manifold without boundary with affine connection $\nabla$ , and let $I \subseteq R$ be an interval with nonempty interior. A smooth curve $γ : I \to M$ is an affinely parametrized geodesic when $D_{t} γ^{'} (t) = 0 (t \in I),$ with one-sided interpretation at an included endpoint. Constant curves are geodesics. Unless another parametrization is explicitly stated, “geodesic” means affinely parametrized geodesic.

Facts & Assumptions

Given: The manifold, affine connection, interval, and smooth curve in the definition.

[F1]

Boundaryless convention for geodesic flow and Hopf–Rinow fixes the boundaryless convention for this page.

[F2]

Affine connection on a smooth manifold makes $\nabla$ a connection on $T M$ , and Covariant derivative along a curve defines $D_{t}$ on sections of $γ^{*} T M$ , with one-sided endpoint values and zero derivative for the zero section.

Verification

1.1F2given

The velocity $γ^{'}$ is a section of $γ^{*} T M$ , so [F2] makes $D_{t} γ^{'}$ well defined and intrinsic. The equation therefore compares vectors in $T_{γ (t)} M$ and is independent of any chart or extension of the velocity field.

2.1F1F2step 1.1∎

If $γ (t) = p$ is constant, then $γ^{'}$ is the zero section and [F2] gives $D_{t} γ^{'} = 0$ , so constant curves are included. On a zero-dimensional manifold every smooth curve on an interval is locally constant and hence has zero velocity; the empty manifold has no such curves. A singleton parameter interval is excluded because [F2] supplies no derivative operator there. Included interval endpoints use the one-sided convention, and no point, chart, or curve is selected from a family, so no choice principle is used.

Depends on

Used by

Gauss-Bonnet for a geodesic polygon Corollary
Geodesics are exactly critical points of energy with fixed endpoints Corollary
Polar integration may discard the cut locus Corollary
A conjugate point at which there are many geodesics Counterexample
Conjugate points along a geodesic and their multiplicity Definition
Geodesic triangulation with prescribed boundary arcs Definition
Injective geodesic ray transform on a transversal manifold Definition
Conjugate antipodes on the round sphere Example
Cut locus of a point on a round sphere Example
Great circles as round-sphere geodesics Example
Jacobi fields in constant sectional curvature Example
Killing jacobi fields from rotations Example
No conjugate points in nonpositive constant curvature Example
Geodesic curvature need not equal ambient curve curvature False statement
Geodesic triangles need not have Euclidean angle sum False statement
CTA Gaussian-beam quasimodes and geodesic concentration Lemma
Integration by parts for the index form Lemma
Local isometries send geodesics to geodesics Lemma
Local length comparison for a conjugate-free geodesic Lemma
Uniform short-geodesic scale on a compact surface Lemma
Affine reparametrization of a geodesic is a geodesic Proposition
Compact Lie groups admit bi-invariant metrics Proposition
Coordinate geodesic equation Proposition
Geodesics have constant speed for a metric-compatible connection Proposition
Hessian of distance in terms of radial jacobi fields 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
Bonnet conjugate radius theorem Theorem
Differential of the exponential map in terms of Jacobi fields Theorem
Equivalent characterizations of a totally geodesic submanifold Theorem
Every Jacobi field is induced by a geodesic variation Theorem
Gauss-Bonnet for a geodesic triangle Theorem
Length minimizers are constant-speed geodesics up to reparametrization Theorem
Metric completeness implies geodesic completeness 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, Definition 15.1.1, p.113 (standard reference, not scraped)