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.
is its metric tensor, the Einstein condition means that for some constant
be an Einstein manifold is simply Taking the trace of both sides reveals that the constant of proportionality
for Einstein manifolds is related to the scalar curvature
In general relativity, Einstein's equation with a cosmological constant
However, this necessary condition is very far from sufficient, as further obstructions have been discovered by LeBrun, Sambusetti, and others.
Four dimensional Riemannian Einstein manifolds are also important in mathematical physics as gravitational instantons in quantum theories of gravity.
The term "gravitational instanton" is sometimes restricted to Einstein 4-manifolds whose Weyl tensor is anti-self-dual, and it is very often assumed that the metric is asymptotic to the standard metric on a finite quotient Euclidean 4-space (and are therefore complete but non-compact).
In differential geometry, simply connected self-dual Einstein 4-manifolds are coincide with the 4-dimensional, reverse-oriented hyperkähler manifolds in the Ricci-flat case, but are sometimes called quaternion Kähler manifolds otherwise.
Hyperkähler and quaternion Kähler manifolds (which are special kinds of Einstein manifolds) also have applications in physics as target spaces for nonlinear σ-models with supersymmetry.
Compact Einstein manifolds have been much studied in differential geometry, and many examples are known, although constructing them is often challenging.
Compact Ricci-flat manifolds are particularly difficult to find: in the monograph on the subject by the pseudonymous author Arthur Besse, readers are offered a meal in a starred restaurant in exchange for a new example.