Levi–Civita connection

Levi-Civita 联络

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

Other names / variants: 黎曼联络

Translation evidence: Literature usage

Meaning in context

For a smooth Riemannian manifold, the unique connection on its tangent bundle that preserves the metric and has zero torsion: its covariant derivatives satisfy the metric product rule and their antisymmetric difference equals the vector-field Lie bracket.

光滑黎曼流形切丛上唯一的保度量、无挠联络；协变导数满足度量的乘积法则，其反对称部分等于向量场的李括号。

Example

For $g=dr^2+r^2d\theta^2$, $\nabla_{\partial_\theta}\partial_\theta=-r\partial_r$ and $\nabla_{\partial_r}\partial_\theta=r^{-1}\partial_\theta$ when $r>0$.

Usage and distinctions

Metric compatibility alone does not determine a connection. Christoffel symbols may be nonzero in curvilinear coordinates even for a flat metric.

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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, Proposition 4.1.9 and equations (4.1.3)–(4.1.4) · Checked 2026-10-14

Translation sources

1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要二：Riemann 流形的 Levi-Civita 联络；PDF p. 77, printed p. 74 · Checked 2026-10-14
2. Zhang Sirong — Beihang University, 微分流形与黎曼几何, Chapter 3 slidesChapter 3，黎曼联络的存在唯一，slide 52/54; PDF p. 13 · Checked 2026-10-14

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