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Journal ArticleDOI

Concircular Curvature Tensor and Fluid Spacetimes

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TLDR
In this paper, the relativistic significance of concircular curvature tensors has been explored and the existence of Killing and conformal Killing vectors has been established for spacetimes satisfying Einstein field equations.
Abstract
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.

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Citations
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Ricci soliton and geometrical structure in a perfect fluid spacetime with torse-forming vector field

TL;DR: In this paper, geometrical aspects of perfect fluid spacetime with torse-forming vector field are described and conditions for the Ricci soliton to be expanding, steady or shrinking are also given.
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General Relativistic Space-Time with η1-Einstein Metrics

TL;DR: In this article , the authors studied the relativistic space-time with a torse-forming potential vector field, and evaluated the characterization of the metrics when the space time with a semi-symmetric energy-momentum tensor admits an η1-Einstein soliton, whose potential field is torseforming.
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Spacetimes admitting W2-curvature tensor

TL;DR: In this article, it was shown that a W2-flat spacetime is conformally flat and hence it is of Petrov type O, and if the perfect fluid spacetime with vanishing W 2-curvature tensor obeys Einstein's field equation without cosmological constant, then the spacetime has vanishing acceleration vector and expansion scalar and the ideal fluid always behaves as a cosmologically constant.
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A spacetime with pseudo-projective curvature tensor

TL;DR: In this paper, it was shown that a pseudo-projectively flat spacetime with vanishing pseudoprojective curvature tensor obeys Einstein's field equation without cosmological constant is an Euclidean space.
References
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Journal ArticleDOI

Relativistic Cosmology. I

TL;DR: In this paper, the authors present general relations obtaining in relativistic cosmology and show that a simple change over to anisotropy without the introduction of spin does not solve any of the outstanding difficulties of isotropic cosmological models.
Book

Structures on manifolds

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Complete Riemannian manifolds and some vector fields

TL;DR: In this article, a nonconstant scalar field p in an n-dimensional Riemannian manifold with metric tensor field (1) g is defined as a concircular scalar fields.
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