Einstein manifold

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Consider an m-dimensional manifold M with metric tensor g. The pair (M,g) is said to be an Einstein manifold if the Ricci tensor on M is proportional to the metric:

$\mathrm{Ric} = k\,g,$

for some constant k. If one introduces a coordinate chart, the condition that (M,g) be an Einstein manifold is then simply

$R_{ab} = k\,g_{ab}.$

By tensor multiplying both sides of this equation by $g a b$ one shows that the constant of proportionality, k, for Einstein manifolds is related to the scalar curvature by

$k = \frac{1}{m}R,$

where R is the scalar curvature. Einstein manifolds with k = 0 are also often referred to as Ricci-flat manifolds.

In general relativity, Einstein's equations relate the curvature of a space-time to the energy-momentum tensor which describes the matter distribution in the space-time according to

$R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab},$

where we have used geometrized units G = c = 1. Therefore, in a spacetime one for which $T a b = 0$ we can regard an Einstein manifold as a vacuum solution to Einstein's equations with a cosmological constant proportional to k.