Arindam Bhattacharyya

Abstract: The present paper deals with Lorentzian para-Sasakian (briefly LP-Sasakian) manifolds with conformally flat and quasi conformally flat cur- vature tensor. It is shown that in both cases, the manifold is locally isometric with a unit sphere S n (1). Further it is shown that an LP-Sasakian manifold with R(X,Y ).C = 0 is locally isometric with a unit sphere S n (1).

Abstract: In this paper we extend the notion of generalized quasi-Einstein manifold and name it mixed generalized quasi-Einstein manifold(MG(QE)n). We prove the existence of such manifolds. We also introduce the notion of generalized quasi umbilical hypersurface of a Riemannian manifold and show that such a manifold is a mixed generalized quasi Einstein manifold. Finally, we obtain the relation between the manifolds with mixed gener- alized quasi constant curvature and the mixed generalized quasi-Einstein quasi conformally ∞at manifolds.

Abstract: In this paper, mixed super quasi-Einstein manifolds (MS(QE)n) have been defined. The existence theorem and an example have been provided and the relations between the associated scalars have been established. As well, manifolds of mixed super quasi-constant curvature are defined, and it is shown that quasi conformally flat, conformally flat, conharmonically flat and projectively flat MS(QEn) are manifolds of mixed super quasi-constant curvature. Further, super quasi-umbilical hypersurfaces of Riemannian manifolds have been defined and it is proved that a super quasi-umbilical hypersurface of a Euclidean space is MS(QE)n. M.S.C. 2000: 53C25.

Abstract: This paper is based on contact CR-submanifolds of an indefinite trans-Sasakian manifold of type (α, β). Here some properties of contact CR-submanifolds of an indefinite trans-Sasakian manifold have been studied and also the sectional curvatures of contact CR-submanifolds of an indefinite trans-Sasakian space form are discussed.

Abstract: This paper characterizes the warping functions for a multiply generalized Robertson–Walker space-time to get an Einstein space M with a quarter-symmetric connection for different dimensions of M (i.e. (1). dim M = 2, (2). dim M ≥ 3) when all the fibers are Ricci flat. Then we have also computed the warping functions for a Ricci flat Einstein multiply warped product spaces M with a quarter-symmetric connection for different dimensions of M (i.e. (1). dim M = 2, (2). dim M = 3, (3). dim M ≥ 4) and all the fibers are Ricci flat. In the last section, we have given two examples of multiply generalized Robertson–Walker space-time with respect to quarter-symmetric connection.

Abstract: We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0 and S · R(ξ,X) = 0. We prove that on a Lorentzian para-Sasakian manifold (M, φ, ξ, η, 1), if the Ricci curvature satisfies one of the previous conditions, the existence of η-Ricci solitons implies that (M, 1) is Einstein manifold. We also conclude that in these cases there is no Ricci soliton on M with the potential vector field ξ. On the other way, if M is of constant curvature, then (M, 1) is elliptic manifold. Cases when the Ricci tensor satisfies different other conditions are also discussed.

Abstract: The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study quasi-Einstein spacetimes. Some basic geometric properties of such a spacetime are obtained. The applications of quasi-Einstein spacetimes in general relativity and cosmology are investigated. Finally, the existence of such spacetimes are ensured by several interesting examples.