Weyl tensor

Summary

The Weyl tensor is defined on a pseudo-Riemannian manifold of dimension at least $3$. It is the part of the Riemann curvature tensor left after removing all that the Ricci curvature 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.