Integral formulas in Riemannian geometry

In this paper, we study para-Sasakian manifold (M,g) whose metric g is an η-Ricci soliton (g,V ) and almost η-Ricci soliton. We prove that, if g is an η-Ricci soliton, then either M is Einstein and...

η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds

Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

https://www.degruyter.com/document/doi/10.1515/math-2019-0056/pdf

*-Ricci soliton on (κ, μ)′-almost 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

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

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

If M is a 3-dimensional contact metric manifold such that Qφ = φQ which admits a Yamabe soliton (g,V ) with the flow vector field V pointwise collinear with the Reeb vector field ξ, then we show th...

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

The purpose of this paper is to study 3-dimensional paraSasakian manifold and conformally flat K-paracontact manifold. Moreover, we show that in a K-paracontact manifold the conditions Einstein, conformally flat, semi-symmetric and Ricci semi-symmetric are all equivalent. Finally, compact regular 3-dimensional paraSasakian and conformally flat K-paracontact manifolds are studied.

Certain results on K-paracontact and paraSasakian 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

Devaraja Mallesha Naik

Papers

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

η-Ricci solitons and almost η-Ricci solitons on para-Sasakian manifolds

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

Certain results on K-paracontact and paraSasakian manifolds

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