Ricci curvature

of a Riemannian manifold at a point

A number corresponding to each one-dimensional subspace of the tangent space by the formula

where is the Ricci tensor, is a vector generating the one-dimensional subspace and is the metric tensor of the Riemannian manifold . The Ricci curvature can be expressed in terms of the sectional curvatures of . Let be the sectional curvature at the point in the direction of the surface element defined by the vectors and , let be normalized vectors orthogonal to each other and to the vector , and let be the dimension of ; then

For manifolds of dimension greater than two the following proposition is valid: If the Ricci curvature at a point has one and the same value in all directions , then the Ricci curvature has one and the same value at all points of the manifold. Manifolds of constant Ricci curvature are called Einstein spaces. The Ricci tensor of an Einstein space is of the form , where is the Ricci curvature. For an Einstein space the following equality holds:

where , are the covariant and contravariant components of the Ricci tensor, is the dimension of the space and is the scalar curvature of the space.

The Ricci curvature can be defined by similar formulas also on pseudo-Riemannian manifolds; in this case the vector is assumed to be anisotropic.

From the Ricci curvature the Ricci tensor can be recovered uniquely:

References

[1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2]	A.Z. Petrov, "Einstein spaces" , Pergamon (1969) (Translated from Russian)

References

[a1]	N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a2]	A.L. Besse, "Einstein manifolds" , Springer (1987)