In differential geometry, the Weyl curvature tensor, named after Hermann Weyl,[1] is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold.
The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force.
In general relativity, the Weyl curvature is the only part of the curvature that exists in free space—a solution of the vacuum Einstein equation—and it governs the propagation of gravitational waves through regions of space devoid of matter.
[2] More generally, the Weyl curvature is the only component of curvature for Ricci-flat manifolds and always governs the characteristics of the field equations of an Einstein manifold.
[2] In dimensions 2 and 3 the Weyl curvature tensor vanishes identically.
This fact was a key component of Nordström's theory of gravitation, which was a precursor of general relativity.
The decomposition (1) expresses the Riemann tensor as an orthogonal direct sum, in the sense that This decomposition, known as the Ricci decomposition, expresses the Riemann curvature tensor into its irreducible components under the action of the orthogonal group.
[3] In dimension 4, the Weyl tensor further decomposes into invariant factors for the action of the special orthogonal group, the self-dual and antiself-dual parts C+ and C−.
The Weyl tensor has the special property that it is invariant under conformal changes to the metric.
It follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish.
In dimension 3 the vanishing of the Cotton tensor is a necessary and sufficient condition for the Riemannian manifold being conformally flat.
Any 2-dimensional (smooth) Riemannian manifold is conformally flat, a consequence of the existence of isothermal coordinates.
Indeed, the existence of a conformally flat scale amounts to solving the overdetermined partial differential equation In dimension ≥ 4, the vanishing of the Weyl tensor is the only integrability condition for this equation; in dimension 3, it is the Cotton tensor instead.