Curvature tensor for the spacetime of general relativity

Abstract

In the differential geometry of certain [Formula: see text]-structures, the role of [Formula: see text]-curvature tensor is very well known. A detailed study of this tensor has been made on the spacetime of general relativity. The spacetimes satisfying Einstein field equations with vanishing [Formula: see text]-tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing [Formula: see text]-tensor have also been considered. The divergence of [Formula: see text]-tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved [Formula: see text]-tensor represents either an Einstein space or a Friedmann-Robertson-Walker cosmological model.