# Ricci curvature

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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)