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Ricci curvature
In
differential geometry
, the
Ricci curvature tensor
, named after
Gregorio Ricci-Curbastro
, represents the amount by which the
volume
of a
geodesic
ball
in a curved
Riemannian manifold
deviates from that of the standard ball in
Euclidean space
. As such, it provides one way of measuring the degree to which the geometry determined by a given
Riemannian metric
might differ from that of ordinary Euclidean
n
-space. The Ricci tensor is defined on any
pseudo-Riemannian manifold
, as a
trace
of the
Riemann curvature tensor
. Like the metric itself, the Ricci tensor is a
symmetric bilinear form
on the
tangent space
of the manifold .
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