Cotton tensor

Builds on Pseudo-Riemannian manifold Levi-Civita connection Ricci curvature Scalar curvature

Definition

Let $(M, g)$ be a pseudo-Riemannian manifold of dimension $n \ge 3$ , with Levi-Civita connection $\nabla$ , Ricci curvature $\operatorname{Ric}$ and scalar curvature $S$ . Let $P = \frac{1}{n - 2} \left( \operatorname{Ric} - \frac{S}{2(n - 1)} g \right)$ be the Schouten tensor. The Cotton tensor of $(M, g)$ is the $(0,3)$ -tensor $C(X, Y, Z) = (\nabla_Z P)(X, Y) - (\nabla_Y P)(X, Z),$ for vector fields $X, Y, Z$ ; in index notation $C_{ijk} = \nabla_k P_{ij} - \nabla_j P_{ik}$ .

Remarks

Example: if $(M, g)$ is Einstein with $\operatorname{Ric} = \lambda g$ for a constant $\lambda$ and $n \ge 3$ , then $S = n\lambda$ is constant and $P$ is a constant multiple of $g$ . Since $\nabla g = 0$ , also $\nabla P = 0$ , so $C = 0$ .

Non-example: for $n = 2$ the factor $1/(n - 2)$ is not defined, so there is no Cotton tensor of a surface in this sense.

$C$ is antisymmetric in its last two arguments, and its trace over the first two arguments vanishes by the contracted Bianchi identity. For $n = 3$ , $C = 0$ if and only if $g$ is locally conformally flat. For $n \ge 4$ , the divergence of the Weyl tensor is $(n - 3)$ times $C$ , up to a sign fixed by the conventions. Some texts use the opposite sign or a different order of indices.

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Tag	`83ZA89PAGK`
URI	`m2://83ZA89PAGK`
Kind	definition
Writing	W0 stub
MSC	53C18, 53B20
Defines	Cotton tensor
Uses	Pseudo-Riemannian manifold, Levi-Civita connection, Ricci curvature, Scalar curvature
Changed	2 November 2026

Cotton tensor

Definition

Remarks

Reading path

Further reading

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