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: