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 goal of the present paper is to deliberate certain types of metric such as $*$-$\eta$-Ricci-Yamabe soliton on $\alpha$-Cosymplectic manifolds with respect to quarter-symmetric metric connection. Further, we have proved some curvature properties of $\alpha$-Cosymplectic manifolds admitting quarter-symmetric metric connection. Here, we have shown the characteristics of the soliton when the manifold satisfies quarter-symmetric metric connection on $\alpha$-Cosymplectic manifolds. Later, we have acquired Laplace equation from $*$-$\eta$-Ricci-Yamabe soliton equation when the potential vector field $\xi$ of the soliton is of gradient type in terms of quarter-symmetric metric connection. Next, we have developed the nature of the soliton when the vector field is conformal killing admitting quarter-symmetric metric connection. Finally, we present an example of a 5-dimensional $\alpha$-cosymplectic metric as a $*$-$\eta$-Ricci-Yamabe soliton with respect to a quarter-symmetric metric connection to prove our results.

$\ast$-$\eta$-Ricci-Yamabe solitons on $\alpha$-Cosymplectic manifolds with a quarter-symmetric metric connection.

In this paper we shall try to investigate the stability of a living cell at an optimum point, their biological significance with programming in C-language.

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Stability of a Cell at Optimum Point

Homology classes were first defined rigorously by Henri Poincar´e in his seminal paper “Analysis situs” in 1895 referring to the work of Riemann,Betti and von Dyck. The homology group was further developed by Emmy Noether and, independently, by Leopold Vietoris and Walther Mayer in the period 1925-28.

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Homology of a Type of Octahedron

In this note we define multitime Ricci flow, studied it on Einstein, quasi Einstein and Ricci-recurrent quasi Einstein manifolds. We have also obtained some inequalities regarding scalar curvature on closed manifold under multitime Ricci flow applying weak maximum and minimum principle. Mathematics Subject Classification: 53C44, 53C21, 58J35, 78A25, 37J20

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A Type of Ricci Flow on Riemannian Manifolds 1

It is well-known that Einstein manifolds play an important role in geometry as well as in general relativity. Einstein manifolds form a natural subclass of the class of quasi-Einstein manifolds. Th...

Arindam Bhattacharyya

Papers

$\ast$-$\eta$-Ricci-Yamabe solitons on $\alpha$-Cosymplectic manifolds with a quarter-symmetric metric connection.

Stability of a Cell at Optimum Point

Homology of a Type of Octahedron

A Type of Ricci Flow on Riemannian Manifolds 1

A Non-Flat Riemannian Manifold Admitting Certain Vectors Fields