Riemann curvature tensor

曲率张量

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

Other names / variants: Riemannian curvature tensor

Translation evidence: Literature usage

Meaning in context

For a smooth Riemannian metric with Levi–Civita connection, the tensor R(X,Y)Z = ∇X∇Y Z − ∇Y∇X Z − ∇[X,Y]Z; pairing its output with the metric gives its covariant four-tensor form.

光滑黎曼流形的 Levi-Civita 联络所定义的张量 $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$；用度量与其输出配对可得到四阶协变形式。

Example

On the unit round sphere, $R(u,v)w=g(v,w)u-g(u,w)v$, so $g(R(u,v)v,u)=1$ for orthonormal $u,v$.

Usage and distinctions

The Chinese source uses 曲率张量 in the Levi–Civita setting; the word also occurs for general connections. This entry fixes the Riemannian scope and a sign convention, rather than identifying curvature with scalar curvature or with Christoffel symbols.

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Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Eckhard Meinrenken — Riemannian Geometry§18, Definition 18.1, Theorem 18.2 and the covariant curvature tensor before Theorem 18.5, printed pp. 51–52 · Checked 2026-10-15

Translation sources

1. Wang Wenlong — 黎曼几何, Nankai UniversityChapter 4, §4.1 曲率张量, Definition 4.1, PDF p. 21, printed p. 18 · Checked 2026-10-15

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