Ricci flow with surgery on three-manifolds

A Riemannian manifold (M,g) is called a quasi Einstein manifold if for any coordinate system in M, its Ricci tensor S satisfes S_(ij) = ag_(ij) + bA_iA_j for some scalars a and b, where A(X)=g(X, p) for some unit vector p. This class of manifolds is a generalization of Einstein manifolds which are quasi Einstein manifolds whose b=0. In this paper, we will give examples of these manifolds and we will show that on each coordinate system, a and b are unique.

On Quasi-Einstein Manifolds Which Are Not Einstein

We prove that every Einstein compact almost C-manifold M 2n+s whose Reeb vector fields are Killing is a C-manifold. Then we extend this result considering some generalizations of the Einstein condition (η-Einstein, generalized quasi Einstein, etc.). Moreover, we find some topological properties of compact almost C-manifolds under the assumption that the Ricci tensor is transversally positive definite and the Reeb vector fields are Killing, namely we prove that the first Betti number is s and the first fundamental group is isomorphic to Z s . Finally, a splitting theorem for cosymplectic manifolds is found.

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Einstein-like conditions and cosymplectic geometry

The present paper deals with a study of quasi-conformally ∞at quasi-Einstein spaces and obtained several necessary and su-cient condi- tions for such a space to be of semi-symmetric (resp. projectively semi- symmetric, concircularly semi-symmetric and conformally semi-symmetric). The existence of such space is ensured with a proper example.

On quasi-conformally ∞at quasi-Einstein spaces

This paper deals with generalized quasi-Einstein manifold satisfying certain conditions on conharmonic curvature tensor. Here we study some geometric properties of generalized quasi-Einstein manifold and obtain results which reveal the nature of its associated 1-forms.

https://downloads.hindawi.com/archive/2014/563803.pdf

Some Results on Generalized Quasi-Einstein Manifolds

In this paper, mixed super quasi-Einstein manifolds (MS(QE)n) have been defined. The existence theorem and an example have been provided and the relations between the associated scalars have been established. As well, manifolds of mixed super quasi-constant curvature are defined, and it is shown that quasi conformally flat, conformally flat, conharmonically flat and projectively flat MS(QEn) are manifolds of mixed super quasi-constant curvature. Further, super quasi-umbilical hypersurfaces of Riemannian manifolds have been defined and it is proved that a super quasi-umbilical hypersurface of a Euclidean space is MS(QE)n. M.S.C. 2000: 53C25.

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On mixed super quasi-Einstein manifolds

Within the framework of mixed super quasi-Einstein mani- folds, are proved three theorems: the flrst one shows that the compact orientable manifolds MS(QE)n (n ‚ 3) without boundary do not admit non-isometric conformal vector flelds, the second one provides a su-ciency condition for non-existence of nontrivial Killing vector flelds, and the last one characterizes the harmonic vector flelds under certain assumptions.

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Some global properties of mixed super quasi-Einstein manifolds

In this paper we have introduced the notion of $*-$ Ricci flow and shown that $*-$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $*-$ Ricci flow. We have also find the deformation of geometric curvature tensors under $*-$ Ricci flow. In the last two section of the paper, we have found the $\Im$-functional and $\omega-$ functional for $*-$ Ricci flow respectively.

Some Results on $*-$Ricci Flow.

In this paper, we discuss about a set of geometric and physical properties of hyper-generalized quasi-Einstein spacetime. In the beginning, we discuss about pseudosymmetry over a hyper-generalized ...

Dipankar Debnath

Papers

On mixed super quasi-Einstein manifolds

Some global properties of mixed super quasi-Einstein manifolds

Some Results on $*-$Ricci Flow.

A study on some geometric and physical properties of hyper-generalized quasi-Einstein Spacetime

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