Ricci curvature

of a Riemannian manifold $ M $ at a point $ p \in M $

A number corresponding to each one-dimensional subspace of the tangent space $ M _ {p} $ by the formula

$$ r ( v) = \ \frac{( c R ) ( v , v ) }{g ( v , v ) } , $$

where $ c R $ is the Ricci tensor, $ v $ is a vector generating the one-dimensional subspace and $ g $ is the metric tensor of the Riemannian manifold $ M $. The Ricci curvature can be expressed in terms of the sectional curvatures of $ M $. Let $ K _ {p} ( \alpha , \beta ) $ be the sectional curvature at the point $ p \in M $ in the direction of the surface element defined by the vectors $ \alpha $ and $ \beta $, let $ l _ {1} \dots l_{n-1} $ be normalized vectors orthogonal to each other and to the vector $ v $, and let $ n $ be the dimension of $ M $; then

$$ r(v) = \sum_{i=1}^{n-1} K_p (v, l_i) . $$

For manifolds $ M $ of dimension greater than two the following proposition is valid: If the Ricci curvature at a point $ p \in M $ has one and the same value $ r $ in all directions $ v $, then the Ricci curvature has one and the same value $ r $ 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 $ c R = r g $, where $ r $ is the Ricci curvature. For an Einstein space the following equality holds:

$$ n R _ {ij} R ^ {ij} - s ^ {2} = 0 , $$

where $ R _ {ij} $, $ R ^ {ij} $ are the covariant and contravariant components of the Ricci tensor, $ n $ is the dimension of the space and $ s $ 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:

$$ ( c R ) ( u , v ) = $$

$$ = \ \frac{1}{2} [ r ( u + v ) g ( u + v , u + v ) - r ( u) g ( u , u ) - r ( v) g ( v , v ) ] . $$

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