Einstein manifold

An Einstein manifold is a Riemannian manifold whose Ricci tensor is proportional to the metric, making it a fixed point of the Ricci flow. These spaces generalize the vacuum solutions of general relativity and serve as the geometric equilibrium states toward which curvature-driven evolution tends. The search for Einstein metrics on compact manifolds — the Yamabe problem and its higher-dimensional extensions — remains one of the central programs in differential geometry.