---
tag: 10CQ5007PH
kind: concept
name: Cotton tensor
msc: [53C18, 53B20]
sources:
  - url: https://en.wikipedia.org/wiki/Cotton_tensor
same_as:
  wikidata: Q3518152
  wikipedia: Cotton tensor
---

## Summary

The Cotton tensor is defined on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension at least $3$.
It is a $3$-index tensor built from the covariant derivatives of the Schouten tensor, a modified [[Ricci curvature|4GTNM0WHDH]].
In dimension $3$ the [[Weyl tensor|SBHY31HESP]] is always zero, and the Cotton tensor takes over its role.
A $3$-dimensional metric is locally conformally flat if and only if its Cotton tensor vanishes.
In higher dimensions it is, up to a constant factor, the divergence of the Weyl tensor.

## Explanations

### L1 human

In three dimensions all of the curvature is determined by the Ricci curvature, so the usual test for being conformally flat, the Weyl tensor, says nothing.
The Cotton tensor is the replacement test.
It looks at how the Ricci curvature changes from point to point, through its derivatives.
A space where the Ricci curvature is a constant multiple of the metric, such as the round $3$-sphere, has zero Cotton tensor.