Let denote a connected manifold with Riemannian metric , its Ricci tensor, its scalar curvature. Then is Einstein if , with constant; and is quasi-Einstein if there exist functions , and a 1-form such that
whereever the equation fails.
Quasi-Einstein space-times are used during the study of exact solutions of the Einstein field equations and model that third stage in the evolution of the universe where effects of viscosity and heat flux have become negligible and the matter content of the universe may be assumed to be a perfect fluid.
The authors classify quasi-Einstein space-times in two important subclasses introduced by Gray, namely (i) manifolds whose Ricci tensor is cyclic parallel; and (ii) manifolds whose Ricci tensor is of Codazzi type. They derive further basic properties of quasi-Einstein space-times by considering the energy-momentum tensor, Killing vector fields, integral curves of the flow vector field, isotropic fluid pressure, vorticity and shear, etc. They also give a number of interesting examples of quasi-Einstein manifolds.