An Einstein manifold is a Riemannian manifold whose Ricci tensor is a constant multiple
of its metric tensor, so 
. In dimension greater
than two, the contracted Bianchi identity forces 
to be constant on each connected component. Ricci-flat
manifolds are Einstein manifolds with 
.
Einstein Manifold
See also
Metric Tensor, Ricci Curvature Tensor, Riemannian Manifold, Scalar CurvatureExplore with Wolfram|Alpha
References
Besse, A. L. Einstein Manifolds. Berlin, Germany: Springer-Verlag, 1987.Cite this as:
Weisstein, Eric W. "Einstein Manifold." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EinsteinManifold.html