In the differential geometry of certain F -structures, the importance of concircular curvature tensor is very well known. The relativistic significance of this tensor has been explored here. The spacetimes satisfying Einstein field equations and with vanishing concircular curvature tensor are considered and the existence of Killing and conformal Killing vectors have been established for such spacetimes. Perfect fluid spacetimes with vanishing concircular curvature tensor have also been considered. The divergence of concircular curvature tensor is studied in detail and it is seen, among other results, that if the divergence of the concircular tensor is zero and the Ricci tensor is of Codazzi type then the resulting spacetime is of constant curvature. For a perfect fluid spacetime to possess divergence-free concircular curvature tensor, a necessary and sufficient condition has been obtained in terms of Friedmann-Robertson-Walker model.
The main objective of the present paper is to investigate the curvature properties of generalized pp-wave metric. It is shown that generalized pp-wave spacetime is Ricci generalized pseudosymmetric, 2-quasi-Einstein and generalized quasi-Einstein in the sense of Chaki. As a special case it is shown that pp-wave spacetime is semisymmetric, semisymmetric due to conformal and projective curvature tensors, R-space by Venzi and satisfies the pseudosymmetric type condition P · P = − 1 3 Q(S, P ). Again we investigate the sufficient condition for which a generalized pp-wave spacetime turns into pp-wave spacetime, pure radiation spacetime, locally symmetric and recurrent.Finally, it is shown that the energy-momentum tensor of pp-wave spacetime is parallel if and only if it is cyclic parallel. And the energy momentum tensor is Codazzi type if it is cyclic parallel but the converse is not true as shown by an example. Finally we make a comparison between the curvature properties of the Robinson-Trautman metric and generalized pp-wave metric.
: In the differential geometry of certain F -structures, the role of W -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 W -tensor have been considered and the existence of Killing and conformal Killing vector fields has been established. Perfect fluid spacetimes with vanishing W -tensor have also been considered. The divergence of W -tensor is studied in detail and it is seen, among other results, that a perfect fluid spacetime with conserved W -tensor represents either an Einstein space or a Friedmann-RobertsonWalker cosmological model.
The general theory of relativity is a theory of gravitation in which gravitation emerges as the property of the space-time structure through the metric tensor g ij . The metric tensor determines another object (of tensorial nature) known as Riemann curvature tensor. At any given event this tensorial object provides all information about the gravitational field in the neighbourhood of the event. It may, in real sense, be interpreted as describing the curvature of the space-time. The Riemann curvature tensor is the simplest non-trivial object one can build at a point; its vanishing is the criterion for the absence of genuine gravitational fields and its structure determines the relative motion of the neighbouring test particles via the equation of geodesic deviation.The above discussion clearly illustrates the importance of the Riemann curvature tensor in general relativity and it is for these reasons that a study of this curvature tensor has been made here. §2. The decomposition of Riemann tensor Using the decomposition of Riemann tensor and the invariants of Riemann tensor, a criterion for the existence of gravitational radiation has been given.The Riemann curvature tensor R k ijl is defined, for a covariant vector field A k , through the Ricci identitywhereThe Riemann curvature tensor can be decomposed aswhere C ijkl is the Weyl tensor, E ijkl = − 1 2 (g ik S jl + g jl S ik − g il S jk − g jk S il ), is the Einstein curvature tensor and S ij ≡ R ij − 1 4 g ij R, G ijkl ≡ − R 12 (g ik g jl − g il g
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