Weyl tensor is then given by contracting the above with the inverse of the metric. The decomposition () 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 . 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 can also be expressed using the Schouten tensor, which is a trace-adjusted multiple of the Ricci tensor, Then In indices, where is the Riemann tensor, is the Ricci tensor, is the Ricci scalar (the scalar curvature) and brackets around indices refers to the antisymmetric part. Equivalently, where S denotes the Schouten tensor. Properties Conformal rescaling The Weyl tensor has the special property that it is invariant under conformal changes to the metric. That is, if for some positive scalar function then the (1,3) valent Weyl tensor satisfies . For this reason the Weyl tensor is also called the conformal tensor. It follows that a necessary condition for a Riemannian manifold to be conformally flat is that the Weyl tensor vanish. In dimensions ≥ 4 this condition is sufficient as well. 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. Symmetries The Weyl tensor has the same symmetries as the Riemann tensor. This includes: In addition, of course, the Weyl tensor is trace free: for all u, v. In indices these four conditions are Bianchi identity Taking traces of the usual second Bianchi identity of the Riemann tensor eventually shows that where S is the Schouten tensor. The valence (0,3) tensor on the right-hand side is the Cotton tensor, apart from the initial factor. See also Curvature of Riemannian manifolds Christoffel symbols provides a coordinate expression for the Weyl tensor. Lanczos tensor Peeling theorem Petrov classification Plebanski tensor Weyl curvature hypothesis Weyl scalar Notes