Structures on Manifolds

The condition for the curvature of a statistical manifold 
to admit a kind of standard hypersurface is given 
as a first step of the statistical submanifold theory. 
A complex version of the notion of statistical structures 
is also introduced.

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Hypersurfaces in Statistical Manifolds

The present paper is to deliberate the class of $ \epsilon $-Kenmotsu manifolds which admits conformal $ \eta $-Ricci soliton. Here, we study some special types of Ricci tensor in connection with the conformal $ \eta $-Ricci soliton of $ \epsilon $-Kenmotsu manifolds. Moving further, we investigate some curvature conditions admitting conformal $ \eta $-Ricci solitons on $ \epsilon $-Kenmotsu manifolds. Next, we consider gradient conformal $ \eta $-Ricci solitons and we present a characterization of the potential function. Finally, we develop an illustrative example for the existence of conformal $ \eta $-Ricci soliton on $ \epsilon $-Kenmotsu manifold.

Conformal $ \eta $-Ricci solitons within the framework of indefinite Kenmotsu manifolds

Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

/pdf/geometry-of-conformal-e-ricci-solitons-and-conformal-e-ricci-2pjjz7h3.pdf

Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry

In this paper we study *-Conformal {\eta}-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting *-Conformal {\eta}-Ricci soliton. We obtain some significant results on *-Conformal {\eta}-Ricci soliton in Sasakian manifolds satisfying R({\xi},X).S = 0, S({\xi},X).R = 0, {\overline}P({\xi},X).S = 0, where {\overline}P is Pseudo-projective curvature tensor.The conditions for *-Conformal {\eta}-Ricci soliton on {\Phi}-conharmonically flat and {\Phi}-projectively flat Sasakian manifolds have been obtained in this article. Lastly we give an example of 5-dimensional Sasakian manifolds satisfying *-Conformal {\eta}-Ricci soliton.

/pdf/conformal-eta-ricci-soliton-on-sasakian-manifold-1mqkfdxpmo.pdf

*-Conformal {\eta}-Ricci Soliton on Sasakian manifold

In this paper, we studied the geometrical aspects of a perfect fluid spacetime in terms of conformal Ricci soliton and conformal η-Ricci soliton with torse-forming vector field ξ. Condition for the...

Conformal Ricci soliton and geometrical structure in a perfect fluid spacetime

The object of the present paper is to study the δ-Lorentzian Trans Sasakian manifolds admitting the conformal η-Ricci Solitons and gradient conformal Ricci soliton. It is shown that a symmetric second order covariant tensor in a δ-Lorentzian Trans Sasakian manifold is a constant multiple of metric tensor. Also an example of conformal η-Ricci soliton in 3-dimensional δ-Lorentzian Trans Sasakian manifold is provided in the region where δ-Lorentzian Trans Sasakian manifold expanding.

http://www.journalmim.com/index.php/ijmm/article/download/13/2

Conformal η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds

In this research article, we establish the geometrical bearing on Riemannian submersions in terms of $\eta$-Ricci-Yamabe Soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an $\eta$-Ricci-Yamabe soliton, $\eta$-Ricci soliton and $\eta$-Yamabe soliton. We also discuss the various conditions for which the target manifold of Riemannian submersion is an $\eta$-Ricci-Yamabe soliton, $\eta$-Ricci soliton, $\eta$-Yamabe soliton and quasi-Yamabe soliton. In a particular case when the potential filed $V$ of the $\eta$-Ricci-Yamabe soliton is of gradient type, we derive a Laplacian equation and providing some examples of an $\eta$-Ricci-Yamabe soliton on a Riemannian submersion. Finally, we study harmonic aspect of $\eta$-Ricci-Yamabe soliton on Riemannian submersions and mention geometrical and physical effects of Ricci-Yamabe solitons.

$\eta$-Ricci-Yamabe Soliton on Riemannian Submersions from Riemannian manifolds

We consider a Kenmotsu manifold immersed in  almost $r$-contact manifold admitting a quarter- symmetric non-metric connection and study semi-invariant submanifolds of an almost $r$-contact  Kenmotsu manifold immersed in almost $r$-contact Riemannian manifold endowed with a quarter- symmetric non- metric connection. We also discuss the integrability conditions of distribution on Kenmotsu manifold.

Semi-invariant submanifolds of Kenmotsu manifold immersed in a generalized almost r-contact structure admitting quarter-symmetric non-metric connection

In the present paper,  we study a new class of submanifolds of a generalized Quasi-Sasakian manifold, called skew semi-invariant submanifold. We obtain integrability conditions of the distributions on a skew semi-invariant submanifold and also find the condition for a skew semi-invariant submanifold  of a generalized Quasi-Sasakian manifold to be mixed totally geodesic. Also it is shown that a  skew semi-invariant submanifold of a generalized Quasi-Sasakian manifold will be anti-invariant if and only if $A_{\xi}=0$; and the submanifold will be skew semi-invariant submanifold if $\nabla w=0$. The equivalence relations for the  skew semi-invariant submanifold of a  generalized Quasi-Sasakian manifold are given. Furthermore, we have proved that a skew semi-invariant $\xi^\perp$-submanifold of a normal almost contact metric manifold and a generalized Quasi-Sasakian manifold with non-trivial invariant distribution is $CR$-manifold. An example of dimension 5 is given to show that a skew semi-invariant $\xi^\perp$ submanifold is a $CR$-structure on the manifold.

Mohd Danish Siddiqi

Papers

Conformal Ricci soliton and geometrical structure in a perfect fluid spacetime

Conformal η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds

$\eta$-Ricci-Yamabe Soliton on Riemannian Submersions from Riemannian manifolds

Semi-invariant submanifolds of Kenmotsu manifold immersed in a generalized almost r-contact structure admitting quarter-symmetric non-metric connection

Skew semi-invariant submanifolds of generalized quasi-Sasakian manifolds