---
tag: SBHY31HESP
kind: concept
name: Weyl tensor
msc: [53B20, 53C18]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl_tensor
same_as:
  wikidata: Q869108
  wikipedia: Weyl tensor
---

## Summary

The Weyl tensor is defined on a [[pseudo-Riemannian manifold|GH6CXWFNDJ]] of dimension at least $3$.
It is the part of the [[Riemann curvature tensor|JA29ETCM02]] left after removing all that the [[Ricci curvature|4GTNM0WHDH]] determines.
It is unchanged, as a $(1,3)$-tensor, when the metric is multiplied by a positive function, so it measures conformal geometry.
In dimension at least $4$, it vanishes exactly when the metric is locally conformally flat.
In general relativity it describes tidal forces and gravitational waves in vacuum.

## Explanations

### L1 human

Curvature in several dimensions has many components.
Some of them say how volumes change, and these are captured by the Ricci curvature.
The rest say how shapes are distorted without changing volume, and that rest is the Weyl tensor.
A round sphere has zero Weyl tensor, because it can be mapped to flat space by a map that keeps angles, the stereographic projection.