Uday Chand De

On quasi Einstein manifolds

In this paper we give some examples of a quasi Einstein manifold (QE)n. Next we prove the existence of (QE)n manifolds. Then we study some properties of a quasi Einstein manifold. Finally the hypersurfaces of a Euclidean space have been studied.

On pseudo symmetric spaces

where 21 is a non-zero vector and comma denotes covariant differentiation with respect to the metric tensor of the space. Such a space was called by him a pseudo symmetric space and 2t was called its associated vector. An n-space of this kind was denoted by (PS),. The name pseudo symmetric was chosen, because if in (1) 2t is taken as zero, then the equation (1) takes the form Rhijk.t=0 and the space reduces to a symmetric space in the sense of Caftan. In the present paper some results on a (PS), are established. In Section 3 it is shown that if a (PS), is a decomposable space V~• (r_->2, n-r>=2), then one of the decomposition spaces is flat and the other is a pseudo symmetric space. In Section 4 the Ricci-associate of the vector field 21 is defined and some theorems relating to it are proved. Section 5 deals with (PS), (n>3) having cyclic Ricci tensor. The last section is concerned with (PS), admitting a concurrent or a recurrent vector field [4].

On the Projective Curvature Tensor of Generalized Sasakian-Space-Forms

Abstract The object of the present paper is to study the nature of generalized Sasakian-space-forms under some conditions regarding projective curvature tensor. All the results obtained in this paper are in the form of necessary and sufficient conditions.

On Weakly Symmetric Spaces

The object of the present paper is to study the semi-decomposibility of a weakly symmetric space introduced by Tamássy and Binh [1]. Also a weakly symmetric space admitting a concurrent or a recurrent vector field has been studied.

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

A perfect-fluid space-time of dimension n>3 with 1) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with 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.

ON QUASI EINSTEIN MANIFOLDS

The object of the present paper is to study some properties of a quasi Einstein manifold. A non-trivial concrete example of a quasi Einstein manifold is also given.

On Conformally at Almost Pseudo Ricci Symmetric Mani-folds

The object of the present paper is to study conformally at almost pseudo Ricci symmetric manifolds. The existence of a conformally at almost pseudo Ricci symmetric manifold with non-zero and non-constant scalar curvature is shown by a non-trivial example. We also show the existence of an n-dimensional non-conformally at almost pseudo Ricci symmetric manifold with vanishing scalar curvature.

A note on generalized Robertson–Walker space-times

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.

Some curvature properties of generalized Sasakian-space-forms

The object of the present paper is to study generalized Sasakian-space-forms with vanishing quasi-conformal curvature tensor. The space-forms satisfying ▿S = 0 and R.S = 0 are also considered.

ON A CLASS OF N(κ)-QUASI EINSTEIN MANIFOLDS

The object of the present paper is to study N()-quasi Einstein manifolds. Existence of N()-quasi Einstein manifolds are proved. Physical example of N()-quasi Einstein manifold is also given. Finally, Weyl-semisymmetric N()-quasi Einstein manifolds have been considered.