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 Lorentzian manifolds (including the four-dimensional Lorentzian manifolds usually studied in general relativity). Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons. If M is the underlying n-dimensional manifold, and g is its metric tensor, the Einstein condition means that R i c = k g {\displaystyle \mathrm {Ric} =kg} for some constant k, where Ric denotes the Ricci tensor of g. Einstein manifolds with k = 0 are called Ricci-flat manifolds.