# Space-times with covariant-constant energy-momentum tensor

01 May 1996-International Journal of Theoretical Physics (Kluwer Academic Publishers-Plenum Publishers)-Vol. 35, Iss: 5, pp 1027-1032

TL;DR: In this paper, it was shown that a general relativistic space-time with covariant-constant energy-momentum tensors is Ricci symmetric, and two particular types of such general space-times were considered and determined.

Abstract: It is shown that a general relativistic space-time with covariant-constant energy-momentum tensor is Ricci symmetric. Two particular types of such general relativistic space-times are considered and the nature of each is determined.

##### Citations

More filters

••

01 Jan 2019

TL;DR: The theory of manifolds with affine connection has been studied in this paper, where the authors deal with the theory of conformal, geodesic, and projective mappings and transformations.

Abstract: The monograph deals with the theory of conformal, geodesic,
holomorphically projective, F-planar and others mappings and
transformations of manifolds with affine connection,
Riemannian, Kahler and Riemann-Finsler manifolds. Concretely,
the monograph treats the following: basic concepts of
topological spaces, the theory of manifolds with affine
connection (particularly, the problem of semigeodesic
coordinates), Riemannian and Kahler manifolds (reconstruction
of a metric, equidistant spaces, variational problems in
Riemannian spaces, SO(3)-structure as a model of statistical
manifolds, decomposition of tensors), the theory of
differentiable mappings and transformations of manifolds (the
problem of metrization of affine connection, harmonic
diffeomorphisms), conformal mappings and transformations
(especially conformal mappings onto Einstein spaces, conformal
transformations of Riemannian manifolds), geodesic mappings
(GM; especially geodesic equivalence of a manifold with affine
connection to an equiaffine manifold), GM onto Riemannian
manifolds, GM between Riemannian manifolds (GM of equidistant
spaces, GM of Vn(B) spaces, its field of symmetric linear
endomorphisms), GM of special spaces, particularly Einstein,
Kahler, pseudosymmetric manifolds and their generalizations,
global geodesic mappings and deformations, GM between
Riemannian manifolds of different dimensions, global GM,
geodesic deformations of hypersurfaces in Riemannian spaces,
some applications of GM to general relativity, namely three
invariant classes of the Einstein equations and geodesic
mappings, F-planar mappings of spaces with affine connection,
holomorphically projective mappings (HPM) of Kahler manifolds
(fundamental equations of HPM, HPM of special Kahler manifolds,
HPM of parabolic Kahler manifolds, almost geodesic mappings,
which generalize geodesic mappings, Riemann-Finsler spaces and
their geodesic mappings, geodesic mappings of Berwald spaces
onto Riemannian spaces.

114 citations

### Cites background from "Space-times with covariant-constant..."

...In addition, the seven class is defined by the following condition ∇T = 0, see [327]....

[...]

••

TL;DR: In this article, the curvature tensors of Ricci solitons in a perfect fluid spacetime are described in terms of different curvatures tensors and conditions for the Ricci Solitons to be steady, expanding or shrinking are also given.

Abstract: Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and η-Ricci and η-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector field ξ of the soliton is of gradient type, ξ:= grad(f), we derive a Poisson equation from the soliton equation.

40 citations

•

TL;DR: In this article, the curvature tensors of Ricci solitons in a perfect fluid spacetime are described in terms of different curvatures tensors and conditions for the Ricci Solitons to be steady, expanding or shrinking are also given.

Abstract: Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and $\eta$-Ricci and $\eta$-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector field $\xi$ of the soliton is of gradient type, $\xi:=grad(f)$, we derive a Poisson equation from the soliton equation.

31 citations

••

TL;DR: In this article, the authors introduced spacetimes with semisymmetric energy-momentum tensors and characterized the perfect fluid spacetime with semi-measure tensors.

Abstract: The object of the present paper is to introduce spacetimes with semisymmetric energy-momentum tensor. At first we consider the relation R(X,Y)⋅T=0, that is, the energy-momentum tensor T of type (0,2) is semisymmetric. It is shown that in a general relativistic spacetime if the energy-momentum tensor is semisymmetric, then the spacetime is also Ricci semisymmetric and the converse is also true. Next we characterize the perfect fluid spacetime with semisymmetric energy-momentum tensor. Then, we consider conformally flat spacetime with semisymmetric energy-momentum tensor. Finally, we cited some examples of spacetimes admitting semisymmetric energy-momentum tensor.

26 citations

••

TL;DR: In this paper, a conformally flat almost pseudo-Ricci symmetric spacetime is considered, and it is shown that the energy density and the isotropic pressure are not constants.

Abstract: We consider a conformally flat almost pseudo-Ricci symmetric spacetime. At first we show that a conformally flat almost pseudo-Ricci symmetric spacetime can be taken as a model of the perfect fluid spacetime in general relativity and cosmology. Next we show that if in a conformally flat almost pseudo-Ricci symmetric spacetime the matter distribution is perfect fluid whose velocity vector is the vector field corresponding to 1-form B of the spacetime, the energy density and the isotropic pressure are not constants. We also show that a conformally flat almost pseudo-Ricci symmetric spacetime is the Robertson-Walker spacetime. Finally we give an example of a conformally flat almost pseudo-Ricci symmetric spacetime with non-zero non-constant scalar curvature admitting a concircular vector field.

24 citations

### Cites background from "Space-times with covariant-constant..."

...In the paper [4] Chaki and Ray showed that a general relativistic spacetime with covariant-constant energy-momentum tensor is Ricci symmetric, that is, ∇S = 0, where S is the Ricci tensor of the spacetime....

[...]

...This requirement is satisfied if the energy-momentum tensor is covariant-constant [4]....

[...]

##### References

More filters

•

08 Mar 1996

TL;DR: In this paper, the Riemannian themes in Lorentzian geometry connections and curvature of curvature are discussed and the splitting problem in global geodesic geometry is discussed.

Abstract: Introduction - Riemannian themes in Lorentzian geometry connections and curvature Lorentzian manifolds and causality Lorentzian distance examples of space-times completness and extendibility stability of completeness and incompleteness maximal geodesics and causally disconnected space-times the Lorentzian cut locus Morse index theory on Lorentzian manifolds some results in global Lorentzian geometry singularities gravitational plane wave space-times the splitting problem in global Lorentzian geometry. Appendices: Jacobi Fields and Toponogov's theorem for Lorentzian manifolds from the Jacobi, to a Riccati, to the Raychaudhuri equation - Jacobi Tensor Fields and the exponential map revisited.

1,198 citations

••

TL;DR: In this article, a Ricci recurrent space-time with covariantly constant stress tensor is an Einstein space time, and this result was extended to Ricci Riemann-Cartan space-times with spin density.

Abstract: A Ricci recurrent space-time with covariantly constant stress tensor is an Einstein space-time. We extend this result to Ricci recurrent space-times with torsion. The result is applied to the case of Riemann-Cartan space-times with spin density.

1 citations