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

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).

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On lorentzian para-sasakian manifolds

Conformal Ricci soliton and quasi-Yamabe soliton on generalized Sasakian space form

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.

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On mixed generalized quasi-Einstein manifolds

The object of the present paper is to study some properties of (LCS)$_n$-manifolds whose metric is Yamabe soliton. We establish some characterization of (LCS)$_n$-manifolds when the soliton becomes steady. Next we have studied some certain curvature conditions of (LCS)$_n$-manifolds admitting Yamabe solitons. Lastly we construct a 3-dimensional (LCS)$_n$-manifold satisfying the results.

Yamabe Solitons on (LCS)$_n$-manifolds

The object of the present paper is to study some properties of Kenmotsu manifold whose metric is conformal $\eta$-Einstein soliton. We have studied certain properties of Kenmotsu manifold admitting conformal $\eta$-Einstein soliton. We have also constructed a 3-dimensional Kenmotsu manifold satisfying conformal $\eta$-Einstein soliton.

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Arindam Bhattacharyya

Papers

On lorentzian para-sasakian manifolds

Conformal Ricci soliton and quasi-Yamabe soliton on generalized Sasakian space form

On mixed generalized quasi-Einstein manifolds

Yamabe Solitons on (LCS)$_n$-manifolds

A Kenmotsu metric as a conformal $\eta$-Einstein soliton