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

In this paper we prove some properties of the indefinite Lorentzian para-Sasakian manifolds. Section 1 is introductory. In Sec- tion 2 we define D-totally geodesic and D ? -totally geodesic contact CR- submanifolds of an indefinite Lorentzian para-Sasakian manifold and de- duce some results concerning such a manifold. In Section 3 we state and prove some results on mixed totally geodesic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold. Finally, in Section 4 we obtain a result on the anti-invariant distribution of totally umbilic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian man- ifold.

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Contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold

The objective of the present paper is to study N(k)-mixed generalized quasi-Einstein manifolds. We prove the existence of these manifolds. Later we establish some curvature properties of N(k)-mixed generalized quasi-Einstein manifolds under certain conditions. In the last section, we give two examples of N(k)-mixed generalized quasi-Einstein manifolds.

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On a class of N(k)-mixed generalized quasi-Einstein manifolds

We study nearly quasi-Einstein warped product manifolds for arbitrary dimension n ≥ 3. In the last section we also give an example of warped product on nearly quasi-Einstein manifold. AMS Mathematics Subject Classification (2010): 53C25; 53B30; 53C15.

On nearly quasi-Einstein warped products

In this paper, we study ∗-Conformal η-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting ∗-Conformal η-Ricci soliton. We obtain some sign...

∗-Conformal η-Ricci soliton on Sasakian manifold

The goal of this paper is to study conformal Yamabe soliton and $*$-Yamabe soliton, whose potential vector field is torse forming. Here, we have characterized conformal Yamabe soliton admitting potential vector field as torse forming with respect to Riemannian connection, semi-symmetric metric connection and projective semi-symmetric connection on Riemannian manifold. We have also shown the nature of $*$-Yamabe soliton with torse forming vector field on Riemannian manifold admitting Riemannian connection. Lastly we have developed an example to corroborate some theorems regarding Riemannian connection on Riemannian manifold.

Arindam Bhattacharyya

Papers

Contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold

On a class of N(k)-mixed generalized quasi-Einstein manifolds

On nearly quasi-Einstein warped products

∗-Conformal η-Ricci soliton on Sasakian manifold

Conformal Yamabe soliton and $*$-Yamabe soliton with torse forming potential vector field.