Showing papers by "Uday Chand De published in 2015"
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, the authors obtained a necessary condition for a three dimensional invariant submanifold of a Kenmotsu manifold to be totally geodesic, where S, R are the Ricci tensor and curvature tensor respectively and α is the second fundamental form.
Abstract: The object of the present paper is to obtain a necessary condition for a three dimensional invariant submanifold of a Kenmotsu manifold to be totally geodesic. Besides this we study an invariant submanifold of Kenmotsu manifolds satisfying Q(α, R) = 0 and Q(S, α) = 0, where S, R are the Ricci tensor and curvature tensor respectively and α is the second fundamental form. Finally, we construct an example to verify our results.
16 citations
TL;DR: In this article, the authors studied ϕ -Weyl semisymmetric and ϕ-projectively semisymmetric generalized Sasakian space-forms and illustrative examples are given.
15 citations
TL;DR: In this article, a perfect-fluid space-time of dimension n>3 with irrotational velocity vector field and null divergence of the Weyl tensor is defined.
Abstract: 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.
15 citations
TL;DR: In this article, a new type of warped products called as sequential warped products are introduced to cover a wider variety of exact solutions to Einstein's equation, and the geometry of two classes of sequential warped product space-time models are studied.
Abstract: In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein's equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consider the geometry of two classes of sequential warped product space-time models which are sequential generalized Robertson-Walker spacetimes and sequential standard static spacetimes.
14 citations
TL;DR: In this article, a sufficient condition for a weakly cyclic Z symmetric manifold to be a quasi-Einstein manifold has been obtained and the equivalence of semisymmetry and Weyl-semisymmetric in a (WCZS)4.
Abstract: The object of the present paper is to study weakly cyclic Z symmetric manifolds. Some geometric properties have been studied. We obtain a sufficient condition for a weakly cyclic Z symmetric manifold to be a quasi Einstein manifold. Next we consider conformally flat weakly cyclic Z symmetric manifolds. Then we study Einstein (WCZS)
n
(n > 2). Also we study decomposable (WCZS)
n
(n > 2). We prove the equivalency of semisymmetry and Weyl-semisymmetry in a (WCZS)
n
(n > 2). Finally, we give an example of a (WCZS)4.
12 citations
03 Feb 2015
TL;DR: In this paper, the authors considered pseudosymmetric and pseudoprojectively at $N(k)$-contact metric manifolds with curvature conditions on the projective curvature tensor.
Abstract: The object of the present paper is to classify $N(k)$-contact metric manifolds satisfying certain curvature conditions on the projective curvature tensor. Projectively pseudosymmetric and pseudoprojectively at $N(k)$-contact metric manifolds are considered. Beside these we also study $N(k)$-contact metric manifolds satisfying $\tilde(Z)\dot P = 0$, where \tilde(Z)$ and $P$ denote respectively the concircular and projective curvature tensor. Finally, we give an example of a $N(k)$-contact metric manifold.
8 citations
TL;DR: In this article, the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection is investigated and the curvature tensor and Ricci metric connection are verified.
Abstract: Abstract In this paper, we consider a quarter-symmetric metric connection in a P-Sasakian manifold. We investigate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection. We consider semisymmetric P-Sasakian manifold with respect to the quarter- symmetric metric connection. Furthermore, we consider generalized recurrent P-Sasakian manifolds and prove the non-existence of recurrent and pseudosymmetric P-Sasakian manifolds with respect to the quarter-symmetric metric connection. Finally, we construct an example of a 5-dimensional P-Sasakian manifold admitting quarter-symmetric metric connection which verifies Theorem 4.1.
7 citations
TL;DR: In this article, the existence of a generalized quasi-Einstein manifold has been proved by several non-trivial examples, and some geometric properties of such a manifold have been studied.
Abstract: The object of the present paper is to study some geometric properties of a generalized quasi-Einstein manifold. The existence of such a manifold have been proved by several non-trivial examples.
3 citations
TL;DR: In this article, the authors studied certain curvature conditions on generalized Sasakian space-forms and classified them into P · P = 0, P · Z˜= 0, Z˜ · P=0, Z` · Z´ · Z` = 0.
Abstract: The object of the present paper is to study certain curvature conditions on generalized Sasakian space-forms. We classify generalized Sasakian space-forms which satisfy P · P=0, P · Z˜=0, Z˜ · P=0, Z˜ · Z˜=0, where P is the projective curvature tensor and Z˜ is the concircular curvature tensor.
2 citations
TL;DR: In this article, the authors obtain a necessary and sufficient condition for a 3-dimensional generalized contact metric manifold to be locally φ-symmetric in the sense of Takahashi and the condition is verified by an example.
Abstract: The object of the present paper is to obtain a necessary and sufficient condition for a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold to be locally $\phi $-symmetric in the sense of Takahashi and the condition is verified by an example. Next we characterize a $3$-dimensional generalized $(\kappa ,\mu )$-contact metric manifold satisfying certain curvature conditions on the concircular curvature tensor. Finally, we construct an example of a generalized $(\kappa,\mu)$-contact metric manifold to verify Theorem $1$ of our paper.
01 Dec 2015
TL;DR: In this article, it was shown that a non-cosymplectic manifold is Ricci semisymmetric if and only if it is a Ricci Ricci tensor.
Abstract: Let \(M\) be a \(3\)-dimensional almost contact metric manifold satisfying \((*)\) condition. We denote such a manifold by \(M^{*}\). At first we study symmetric and skew-symmetric parallel tensor of type \((0,2)\) in \(M^{*}\). Next we prove that a non-cosymplectic manifold \(M^{*}\) is Ricci semisymmetric if and only if it is Einstein. Also we study locally \(\phi \)-symmetry and \(\eta \)-parallel Ricci tensor of \(M^{*}\). Finally, we prove that if a non-cosymplectic \(M^{*}\) is Einstein, then the manifold is Sasakian.
01 Mar 2015
TL;DR: In this article, the authors studied pseudosymmetric, Weyl-pseudosymmetric and Ricci-pseudo-ymmetric Sasakian manifolds and showed that Ricci and Weyl are pseudosymptotics.
Abstract: The object of the present paper is to study pseudosymmetric,Weyl-pseudosymmetric and Ricci-pseudosymmetric \(LP\)-Sasakian manifolds.