In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry. Definition of the decomposition Let (M,g) be a Riemannian or pseudo-Riemannian n-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention written multilinearly, this is the convention With this convention, the Ricci tensor is a (0,2)-tensor field defined by Rjk=gilRijkl and the scalar curvature is defined by R=gjkRjk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci tensor with the opposite sign. Under that more common convention, the signs of the Ricci tensor and scalar must be changed in the equations below.) Define the traceless Ricci tensor and then define three (0,4)-tensor fields S, E, and W by The "Ricci decomposition" is the statement As stated, this is vacuous since it is just a reorganization of the definition of W. The importance of the decomposition is in the properties of the three new tensors S, E, and W. Terminological note. The tensor W is called the Weyl tensor. The notation W is standard in mathematics literature, while C is more common in physics literature. The notation R is standard in both, while there is no standardized notation for S, Z, and E. Basic properties Properties of the pieces Each of the tensors S, E, and W has the same algebraic symmetries as the Riemann tensor. That is: together with The Weyl tensor has the additional symmetry that it is completely traceless: Hermann Weyl showed that W has the remarkable property of measuring the deviation of a Riemannian or pseudo-Riemannian manifold from local conformal flatness; if it is zero, then M can be covered by charts relative to which g has the form gij=efδij for some function f defined chart by chart. Properties of the decomposition One may check that the Ricci decomposition is orthogonal in the sense that recalling the general definition This has the consequence, which could be proved directly, that Terminological note. It would be symbolically clean to present this orthogonality as saying together with However, there is an