Showing papers by "Uday Chand De published in 2016"

A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time

Abstract: A perfect-fluid space-time of dimension n ≥ 4, with (1) irrotational velocity vector field and (2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with an Einstein fiber. Condition (1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson-Walker space-time with null divergence of the Weyl tensor is a perfect-fluid space-time.

A note on generalized Robertson–Walker space-times

Abstract: A generalized Robertson–Walker (GRW) space-time is the generalization of the classical Robertson–Walker space-time. In the present paper, we show that a Ricci simple manifold with vanishing divergence of the conformal curvature tensor admits a proper concircular vector field and it is necessarily a GRW space-time. Further, we show that a stiff matter perfect fluid space-time or a mass-less scalar field with time-like gradient and with divergence-free Weyl tensor are GRW space-times.

A spacetime with pseudo-projective curvature tensor

Abstract: The object of the present paper is to study spacetimes admitting pseudo-projective curvature tensor. At first we prove that a pseudo-projectively flat spacetime is Einstein and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying Einstein’s field equation with cosmological constant is covariant constant. Next, we prove that if the perfect fluid spacetime with vanishing pseudo-projective curvature tensor obeys Einstein’s field equation without cosmological constant, then the spacetime has constant energy density and isotropic pressure, and the perfect fluid always behaves as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field U. Moreover, it is shown that a pseudo-projectively flat spacetime satisfying Einstein’s equation without cosmological constant for a purely electromagnetic distribution is an Euclidean space. We also prove that under certain conditions a perfect fluid spacetime with divergence-free pseudo-projective curvature is a Robertson-Walker spacetime and the possible local cosmological structure of such a spacetime is of type I, D or O. We also study dust-like fluid spacetime with vanishing pseudo-projective curvature tensor.

Trans-Sasakian manifolds homothetic to Sasakian manifolds

Abstract: In this paper, it is shown that for a 3-dimensional compact simply connected trans-Sasakian manifold of type \({(\alpha,\beta)}\), the smooth functions \({\alpha,\beta}\) satisfy the Poisson equations \({\Delta \alpha = \beta}\), \({\Delta \alpha = \alpha ^{2}\beta}\) and \({\Delta \beta = \alpha ^{2}\beta}\), respectively, if and only if it is homothetic to a Sasakian manifold. We also find a necessary and sufficient condition for a connected 3-dimensional trans-Sasakian manifold of type \({(\alpha,\beta)}\) in terms of a differential equation satisfied by the smooth function \({\alpha}\) to be homothetic to a Sasakian manifold.

A special type of semi-symmetric non-metric connection on a riemannian manifold

Abstract: The object of the present paper is to study a Riemannian manifold admitting a type of semi-symmetric non-metric connection whose torsion tensor is pseudo symmetric.

On k-contact einstein manifolds

Abstract: In this paper, we investigate K-contact Einstein manifolds satisfying the conditions RC = Q(S,C), where C is the conformal curvature tensor and R the Riemannian curvature tensor. Next we consider K-contact Einstein manifolds satisfying the curvature condition C.S = 0, where S is the Ricci tensor. Also we study K-contact Einstein manifolds satisfying the condition S.C = 0. Finally, we consider K-contact Einstein manifolds satisfying Z .C = 0, where Z is the concircular curvature tensor.

On a Class of Generalized quasi-Einstein Manifolds with Applications to Relativity

Abstract: Quasi Einstein manifold is a simple and natural generalization of Einstein manifold. The object of the present paper is to study some properties of generalized quasi Einstein manifolds. We also discuss $G(QE)_{4}$ with space-matter tensor and some properties related to it. Two non-trivial examples have been constructed to prove the existence of generalized quasi Einstein spacetimes.

Spacetimes with Pseudosymmetric Energy-momentum Tensor

Abstract: The object of the present paper is to introduce spacetimes with pseudosymmetricenergy-momentum tensor. In this paper at first we consider the relation \(R(X,Y)\cdot T=fQ(g,T)\), that is, the energy-momentumtensor \(T\) of type (0,2) is pseudosymmetric. It is shown that in a general relativistic spacetimeif the energy-momentum tensor is pseudosymmetric, then the spacetime is also Ricci pseudosymmetricand the converse is also true. Next we characterize the perfect fluid spacetimewith pseudosymmetric energy-momentum tensor. Finally, we consider conformally flat spacetime withpseudosymmetric energy-momentum tensor.

On weakly cyclic Z symmetric spacetimes

Abstract: The object of the present paper is to study weakly cyclic Z symmetric spacetimes At first we prove that a weakly cyclic Z symmetric spacetime is a quasi Einstein spacetime Then we study $${{(WCZS)}_{4}}$$ spacetimes satisfying the condition div $${C=0}$$ Next we consider conformally flat $${{(WCZS)}_{4}}$$ spacetimes Finally, we characterise dust fluid and viscous fluid $${{(WCZS)}_{4}}$$ spacetimes

On some classes of invariant submanifolds of lorentzian para-sasakian manifolds

Abstract: The object of the present paper is to study invariant submanifolds of Lorenzian Para-Sasakian manifolds. We consider the recurrent and bi-recurrent invariant submanifolds of Lorentzian para-Sasakian manifolds and pseudo-parallel and generalized Ricci pseudo-parallel invariant submanifolds of Lorentzian para-Sasakian manifolds. Also we search for the conditions $\mathcal{Z}(X,Y)\cdot\alpha=fQ(g,\alpha)$ and $\mathcal{Z}(X,Y)\cdot\alpha=fQ(S,\alpha)$ on invariant submanifolds of Lorentzian para-Sasakian manifolds, where $\mathcal{Z}$ is the concircular curvature tensor. Finally, we construct an example of an invariant submanifold of Lorentzian para Sasakian manifold.

On a Semi-symmetric Metric Connection in an Almost Kenmotsu Manifold with Nullity Distributions

Abstract: We consider a semisymmetric metric connection in an almost Kenmotsu manifold with its characteristic vector field ξ belonging to the (k, μ)′-nullity distribution and (k, μ)-nullity distribution respectively. We first obtain the expressions of the curvature tensor and Ricci tensor with respect to the semisymmetric metric connection in an almost Kenmotsu manifold with ξ belonging to (k, μ)′and (k, μ)-nullity distribution respectively. Then we characterize an almost Kenmotsu manifold with ξ belonging to (k, μ)′-nullity distribution admitting a semisymmetric metric connection.

Spacetimes admitting quasi-conformal curvature tensor

Abstract: ‎The object of the present paper is to study spacetimes admitting‎ ‎quasi-conformal curvature tensor‎. ‎At first we prove that a quasi-conformally flat spacetime is Einstein‎ ‎and hence it is of constant curvature and the energy momentum tensor of such a spacetime satisfying‎ ‎Einstein's field equation with cosmological constant is covariant constant‎. ‎Next‎, ‎we prove that if the perfect fluid pacetime with‎ ‎vanishing quasi-conformal curvature tensor obeys Einstein's field equation without cosmological constant‎, ‎then the spacetime has constant energy density and isotropic pressure and the perfect fluid always behave‎ ‎as a cosmological constant and also such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field $U$‎. ‎Moreover‎, ‎it is shown that in a purely electromagnetic distribution the spacetime with vanishing quasi-conformal curvature tensor is filled with‎ ‎radiation and extremely hot gases‎. ‎We also study dust-like fluid spacetime with vanishing quasi-conformal curvature tensor.

Para-Sasakian manifolds satisfying certain curvature conditions

Abstract: In this paper, we investigate P-Sasakian manifolds satisfying the conditions R(X, ξ) · C = 0 and $$C \cdot \widetilde Z = 0$$ , where C and $$\widetilde Z$$ are the Weyl conformal curvature tensor and the concircular curvature tensor respectively. Next, we study 3-dimensional P-Sasakianmanifolds. Finally, we give an example of a 3-dimensional P-Sasakian manifold.

Certain semisymmetry properties of (, )-contact metric manifolds

Abstract: The object of the present paper is to characterize (κ, μ)contact metric manifolds whose concircular curvature tensor satisfies certain semisymmetry conditions. We also verify that the result holds by a concrete example.

Projective Curvature Tensorin 3-dimensional Connected Trans-Sasakian Manifolds

Abstract: The object of the present paper is to study $\xi $-projectively flat and $\phi $-projectively flat 3-dimensional connected trans-Sasakian manifolds. Also we study the geometric properties of connected trans-Sasakian manifolds when it is projectively semi-symmetric. Finally, we give some examples of a 3-dimensional trans-Sasakian manifold which verifies our result.