Einstein manifold

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (with cosmological constant), although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

If M is the underlying n-dimensional manifold and g is its metric tensor the Einstein condition means that

for some constant k, where Ric denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds.

In local coordinates the condition that (M, g) be an Einstein manifold is simply

Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by

where n is the dimension of M.