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On quasi-Einstein spacetimes. (English) Zbl 1213.53027

Let ${M}_{n}$ denote a connected manifold with Riemannian metric $g$ $\left(>2\right)$, $S$ its Ricci tensor, $r$ its scalar curvature. Then ${M}^{n}$ is Einstein if $S-\frac{r}{n}=0$, with $r$ constant; and ${M}^{n}$ is quasi-Einstein if there exist functions $\alpha$, $\beta$ and a 1-form $A$ such that

$S-\alpha g=\left(\beta A\right)\otimes A,$

whereever the equation $S-\frac{r}{n}g=0$ 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.

##### MSC:
 53B30 Lorentz metrics, indefinite metrics 53B50 Applications of local differential geometry to physics 83D05 Relativistic gravitational theories other than Einstein’s