DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Covariant derivative along a curve

Definition

For a given connection on $E$ and a smooth curve on an interval with nonempty interior, the covariant derivative along the curve is $D_{t} V = (γ^{*} \nabla)_{\partial / \partial t} V .$ Here $V$ is a section of the pullback bundle as in Vector field and section along a smooth curve, and the pullback connection exists by Pullback connection is well defined and functorial. It is real-linear and satisfies $D_{t} (f V) = f^{'} V + f D_{t} V$ for functions of the parameter.

At an included endpoint the smooth up-to-boundary coefficients have their one-sided derivative; equivalently any smooth local extension of the coefficients gives this value, because extensions agreeing on a one-sided interval have the same derivative there. For a piecewise smooth curve the derivative is defined on each piece, with separate one-sided derivatives at corners; those derivatives are not required to agree. A singleton interval carries no derivative operator under this convention and will have identity transport by definition. A zero section or a rank-zero bundle gives zero covariant derivative. Constant base curves can have sections with nonzero coefficient derivative.

Depends on

Used by

Polar integration may discard the cut locus Corollary
The space of Jacobi fields along a geodesic has dimension two n Corollary
Conjugate points along a geodesic and their multiplicity Definition
Geodesic of an affine connection Definition
Index form of a geodesic segment Definition
Jacobi field Definition
Radial Jacobi tensor Definition
Signed geodesic curvature Definition
Conjugate antipodes on the round sphere Example
Great circles as round-sphere geodesics Example
Index form in constant curvature Example
Jacobi fields in constant sectional curvature Example
Jacobi fields in euclidean space Example
Model jacobi fields in positive zero and negative curvature Example
No conjugate points in nonpositive constant curvature Example
A geodesic stops minimizing exactly at its first conjugate point False statement
Every vector field along a geodesic is a Jacobi field False statement
Normal coordinates make the metric Euclidean throughout the chart False statement
Covariant derivatives commute up to curvature in a two parameter variation Lemma
Integration by parts for the index form Lemma
Local isometries send geodesics to geodesics Lemma
Wronskian of two jacobi fields is constant Lemma
Hessian of distance in terms of radial jacobi fields Proposition
Jacobi fields are the null solutions of the index form with fixed endpoints Proposition
Tangential jacobi fields are affine multiples of the velocity Proposition
A geodesic does not minimize past its first conjugate point Theorem
Conjugate points are critical values of the exponential map along the geodesic Theorem
Covariant derivative along a curve is independent of frame and extension 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
Existence and uniqueness of jacobi fields from initial data Theorem
First variation formula for energy Theorem
First variation formula for length Theorem
Index lemma Theorem
Radial riccati equation Theorem
Second variation formula for energy Theorem
Variation field of a geodesic variation is a Jacobi field Theorem

Dependency tree · two levels

9 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 (standard reference, not scraped)