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. Einstein manifolds in four Euclidean dimensions are studied as gravitational instantons. 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. 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 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. In general relativity, Einstein's equation with a cosmological constant Λ is written in geometrized units with G = c = 1. The stress–energy tensor Tab gives the matter and energy content of the underlying spacetime. In vacuum (a region of spacetime devoid of matter) Tab = 0, and Einstein's equation can be rewritten in the form (assuming that n > 2): Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.