Riemannian geodesic

测地线

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A smooth parametrized curve on a Riemannian manifold whose velocity is parallel along the curve with respect to the Levi–Civita connection, so its covariant acceleration is zero; the parameter is affine.

黎曼流形上的光滑参数曲线，其速度关于 Levi-Civita 联络沿曲线平行，即协变加速度为零；参数采用仿射参数。

Example

On the unit sphere, $\gamma(t)=(\cos t,\sin t,0)$ has ambient acceleration $-\gamma(t)$, normal to the sphere, and hence zero tangential covariant acceleration.

Usage and distinctions

A geodesic need not minimize length between distant endpoints. A full great-circle traversal is a geodesic although its endpoints can coincide; arbitrary reparametrization may destroy the equation.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§4.1.2, Definition 4.1.10 and geodesic equation (4.1.5) · Checked 2026-10-14

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要二：测地线；PDF p. 77, printed p. 74 · Checked 2026-10-14

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