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

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

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 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

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

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

V. Venkatesha

Papers

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

Riemann Soliton within the framework of contact geometry

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

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

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