H. Aruna Kumara

Abstract: The aim of this paper, is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also, some suitable examples of Kenmotsu and (κ,μ)′-almost ...

Abstract: In this paper, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation admits a gradient $$\rho $$ -Einstein soliton, then either $$M^{2n+1}$$ is Einstein or the potential function is pointwise collinear with the Reeb vector field $$\xi $$ on an open set $${\mathcal {O}}$$ of $$M^{2n+1}$$ . Moreover, we prove that if the metric of a $$(\kappa ,-2)'$$ -almost Kenmotsu manifold with $$h' e 0$$ admits a gradient $$\rho $$ -Einstein soliton, then the manifold is locally isometric to the Riemannian product $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$ and potential vector field is tangential to the Euclidean factor $${\mathbb {R}}^n$$ . We show that there does not exist gradient $$\rho $$ -Einstein soliton on generalized $$(\kappa ,\mu )$$ -almost Kenmotsu manifold of constant scalar curvature. Finally, we construct an example for gradient $$\rho $$ -Einstein soliton.

Abstract: In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$ -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$ -Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$ -almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$ -almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.

Abstract: The object of the present paper was to introduce the notion of hyper generalized φ-recurrent Sasakian manifold and quasi generalized φ-recurrent Sasakian manifold and study its various geometric properties. The existence of hyper generalized φ-recurrent Sasakian manifold and quasi generalized φ-recurrent Sasakian manifold is proved by giving a proper example.

Abstract: 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.

Abstract: The present research paper consists of the study of an η1-Einstein soliton in general relativistic space-time with a torse-forming potential vector field. Besides this, we try to evaluate the characterization of the metrics when the space-time with a semi-symmetric energy-momentum tensor admits an η1-Einstein soliton, whose potential vector field is torse-forming. In adition, certain curvature conditions on the space-time that admit an η1-Einstein soliton are explored and build up the importance of the Laplace equation on the space-time in terms of η1-Einstein soliton. Lastly, we have given some physical accomplishment with the connection of dust fluid, dark fluid and radiation era in general relativistic space-time admitting an η1-Einstein soliton.

Abstract: The aim of this paper is to find some important classes of Einstein manifolds using conformal [Formula: see text]-Ricci solitons and conformal [Formula: see text]-Ricci almost solitons. We prove that a Kenmotsu metric as conformal [Formula: see text]-Ricci soliton is Einstein if it is [Formula: see text]-Einstein or the potential vector field [Formula: see text] is infinitesimal contact transformation or collinear with the Reeb vector field [Formula: see text]. Next, we prove that a Kenmotsu metric as gradient conformal [Formula: see text]-Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariants. Finally, we construct some examples to illustrate the existence of conformal [Formula: see text]-Ricci soliton, gradient almost conformal [Formula: see text]-Ricci soliton on Kenmotsu manifold.