Integral formulas in Riemannian geometry

We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named weakly Z-symmetric and is denoted by (WZS) n .I f theZ tensor is singular we give condi- tions for the existence of a proper concircular vector. For non singular Z tensors, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally har- monic case, and the general form of the Ricci tensor. For conformally flat (WZS) n manifolds, we derive the local form of the metric tensor.

http://wwwteor.mi.infn.it/~molinari/PUBBLICAZIONI/ActaMathHung2012.pdf

Weakly z symmetric manifolds

The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular f-Kenmotsu framework. In this way, the Olszack-Rosca example of Einstein manifolds provided by f-Kenmotsu manifolds via locally symmetric Ricci tensors is recovered as well as a case of Killing vector fields. Some other classes of Einstein-Kenmotsu manifolds are presented. Our result is interpreted in terms of Ricci solitons and special quadratic first integrals.

/pdf/from-the-eisenhart-problem-to-ricci-solitons-in-f-kenmotsu-2x8pxq6m65.pdf

From the Eisenhart problem to Ricci solitons in $f$-Kenmotsu manifolds

We investigate semi-Riemannian manifolds satisfying some curva- ture conditions. Those conditions are strongly related to pseudosymmetry.

/pdf/on-riemann-and-weyl-compatible-tensors-3eggest7fd.pdf

On Riemann and Weyl Compatible Tensors

The object of the present paper is to introduce a non-flat Riemannian manifold called hyper-generalized recurrent manifolds and study its various geometric properties along with the existence of a proper example.

On a generalized class of recurrent manifolds

The object of the present paper is to study weakly symmetric Riemannian manifolds. Among others it is shown that a conformally flat weakly symmetric Riemannian manifold is of hyper quasi-constant curvature which generalizes the notion of quasi-constant curvature and also such a manifold is a quasi-Einstein manifold. Finally several examples of weakly symmetric manifolds of both zero and non-zero scalar curvature are obtained.

On weakly symmetric riemannian manifolds

We introduce a type of non-flat Riemannian manifolds called weakly cyclic Ricci symmetric manifolds and study their geometric properties. The existence of such manifolds is shown by several non-trivial examples.

/pdf/on-weakly-cyclic-ricci-symmetric-manifolds-1htdqbquu5.pdf

On weakly cyclic Ricci symmetric manifolds

The object of the present paper is to study quasi-conformally flat weakly Ricci symmetric manifolds.

/pdf/on-quasi-conformally-flat-weakly-ricci-symmetric-manifolds-2xto86fk2i.pdf

On quasi-conformally flat weakly Ricci symmetric manifolds

where a and b are arbitrary constants not simultaneously zero, C and C are the conformal curvature tensor and the concircular curvature tensor of type (0, 4) respectively. The present chapter deals with a non-quasi-conformally fiat Riemannian manifold (Mn, g) (n > 3) [the condition (n > 3) is assumed throughout this chapter as the conformal curvature tensor vanishes for n = 3] whose quasi-conformal curvature

On weakly quasi-conformally symmetric manifolds

The object of the present paper is to introduce a type of non-flat Riemannian man- ifold called pseudo generalized quasi-Einstein manifold and studied some properties of such a manifold with several non-trivial examples.

/pdf/on-pseudo-generalized-quasi-einstein-manifolds-4t67iq27ff.pdf

Sanjib Kumar Jana

Papers

On weakly symmetric riemannian manifolds

On weakly cyclic Ricci symmetric manifolds

On quasi-conformally flat weakly Ricci symmetric manifolds

On weakly quasi-conformally symmetric manifolds

On pseudo generalized quasi-Einstein manifolds