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

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

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.

/pdf/general-relativistic-space-time-with-e1-einstein-metrics-1e672zis.pdf

General Relativistic Space-Time with η1-Einstein Metrics

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

Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

In this paper geometrical aspects of perfect fluid spacetime with torse-forming vector field $$\xi $$
 are described and Ricci soliton in perfect fluid spacetime with torse-forming vector field $$\xi $$
 are determined. Conditions for the Ricci soliton to be expanding, steady or shrinking are also given.

Ricci soliton and geometrical structure in a perfect fluid spacetime with torse-forming vector field

In this paper, we study contact metric manifold whose metric is a Riemann soliton. First, we consider Riemann soliton (g; V ) with V as  contact vector eld on a Sasakian manifold (M; g) and in this case we prove that M is either of constant curvature +1 (and V is Killing) or D-homothetically xed -Einstein manifold (and V leaves the structure tensor φ invariant). Next, we prove that if a compact K-contact manifold whose metric g is a gradient almost Riemann soliton, then it is Sasakian and isometric to a unit sphere S2n+1. Further, we study H-contact  manifold admitting a Riemann soliton (g; V ) where V is pointwise collinear with .Key words: Contact metric manifold, Riemann soliton, gradient almost Riemann soliton.

Riemann Soliton within the framework of contact geometry

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 $$h'\ne 0$$
 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.

Gradient $$\rho $$ ρ -Einstein soliton on almost Kenmotsu manifolds

We consider almost 
$$*$$

-Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of 
$$\eta $$

-Einstein paraKenmotsu manifold is 
$$*$$

Ricci soliton, then M is Einstein. Next, we show that if 
$$\eta $$

-Einstein paraKenmotsu manifold admits a gradient almost 
$$*$$

-Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field 
$$\xi $$

. Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature 
$$-1$$

. An illustrative example is given to support the obtained results.

/pdf/almost-ricci-soliton-on-parakenmotsu-manifolds-2c5chu7qpa.pdf

H. Aruna Kumara

Papers

∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds

Ricci soliton and geometrical structure in a perfect fluid spacetime with torse-forming vector field

Riemann Soliton within the framework of contact geometry

Gradient $$\rho $$ ρ -Einstein soliton on almost Kenmotsu manifolds

Almost $$*$$∗ -Ricci soliton on paraKenmotsu manifolds