Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

Integral formulas in Riemannian geometry

Structures on Manifolds

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.

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η-Ricci solitons on Lorentzian para-Sasakian manifolds

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.

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On quasi-Einstein spacetimes

Abstract The object of the present paper is to study a quarter-symmetric metric connection in a Lorentzian α-Sasakian manifold. We study some curvature properties of Lorentzian α-Sasakian manifold with respect to quarter-symmetric metric connection. We investigate quasi-projectively at, ϕ-symmetric, ϕ-projectively at Lorentzian α-Sasakian manifolds with respect to quartersymmetric metric connection. We also discuss Lorentzian α-Sasakian manifold admitting quarter-symmetric metric connection satisfying P̃.S̃ = 0, where P̃ denote the projective curvature tensor with respect to quarter-symmetric metric connection.

Some Classes of Lorentzian Alpha-Sasakian Manifolds with Respect to Quarter-Symmetric Metric Connection

Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. This paper deals with the study of conformal Ricci solitons within the framework of warped product manifolds which extends the notion of usual Riemannian product manifolds. First, we prove that if a warped product manifold admits conformal Ricci soliton then the base and the fiber also share the same property. In the next section the characterization of conformal Ricci solitons on warped product manifolds in terms of Killing and conformal vector fields has been studied. Next, we prove that a warped product manifold admitting conformal Ricci soliton with concurrent potential vector field is Ricci flat. Finally, an application of conformal Ricci soliton on a class of warped product spacetimes namely, generalized Robertson-Walker spacetimes has been discussed.

Characteristics of conformal Ricci soliton on warped product spaces.

By Geometric element transformation method (GETMe) always we get a new element In this paper, we investigate the regularization of heptahedra using GETMe Energy function is a cost function for heptahedra which is also applicable for octahedra, decahedra, hexahedra etc is defined by a particular process, which we call base diagonal apex method (BDAMe) We also try to find the characterization of different cost function using BDAMe when we transform a heptahedra by GETMe

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Regularization of heptahedra using geometric element transformation method

Einstein manifolds form a natural subclass of the class of quasi-Einstein manifolds and plays an important role in geometry as well as in general theory of relativity. In this work, we investigate conformal mappings of generalized quasi-Einstein manifolds. We consider a conformal mapping between two generalized quasi-Einstein manifolds Vn and Vn. We also find some properties of this transformation from Vn to Vn and some theorems are proved. Considering this mapping, we examine some properties of these manifolds. After that, we also study some special vector fields under this mapping on these manifolds and some theorems about them are proved.

Conformal mappings of generalized quasi-Einstein manifolds admitting special vector fields

In this paper we have found that if a Kenmotsu manifold with a Killing vector field satisfies gradient Ricci soliton equation then the smooth function �� is either constant or ���� is orthogonal to the Killing vector field. Mathematics Subject classification 2000: Primary 53C44 ; Secondary 53D10, 53C25.

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Arindam Bhattacharyya

Papers

Some Classes of Lorentzian Alpha-Sasakian Manifolds with Respect to Quarter-Symmetric Metric Connection

Characteristics of conformal Ricci soliton on warped product spaces.

Regularization of heptahedra using geometric element transformation method

Conformal mappings of generalized quasi-Einstein manifolds admitting special vector fields

Gradient Ricci Soliton in Kenmotsu Manifold