Yamabe solitons on three-dimensional normal almost paracontact metric manifolds

Abstract

The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that •If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature −1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold.•If the semi-Riemannian metric of a three-dimensional paracosymplectic manifold is a Yamabe soliton, then it has constant scalar curvature. Furthermore either manifold is η-Einstein, or Ricci flat.• If the semi-Riemannian metric on a three-dimensional para-Kenmotsu manifold is a Yamabe soliton, then the manifold is of constant sectional curvature −1, reduces to an Einstein manifold. Furthermore, Yamabe soliton is expanding with λ = −6 and the vector field V is Killing.Finally, we construct examples to illustrate the results obtained in previous sections.2010 Mathematics Subject Classification. 53C25, 53C21, 53C44, 53D15.

Certain results of conformal and *-conformal Ricci soliton on para-cosymplectic and para-Kenmotsu manifolds

“…which holds for arbitrary vector fields X and Y of M. Now replacing Y by ξ and using (9) and the relations L V ξ = 0 and L V η = 0 in the foregoing equation we obtain…”

“…Recently in 2019, Venkatesha, Kumara and Naik [27] considered the metric of η-Einstein para-Kenmotsu manifold as * -Ricci soliton and proved that the manifold is Einstein. Erken [9] in 2019 considered Yamabe solitons on 3-dimensional para-cosymplectic manifold and proved some vital results like the manifold is either η-Einstein or Ricci flat. Motivated by above mentioned works, in this paper, we consider conformal Ricci soliton and * -conformal Ricci soliton in the framework of para-Kenmotsu manifold and conformal Ricci soliton in the framework of 3-dimensional para-cosymplectic manifold.…”

Three-dimensional para-Kenmotsu manifolds admitting eta-Ricci solitons

“…where S and r are the Ricci tensor and scalar curvature respectively and Q is the Ricci operator defined by g(QX, Y ) = S(X, Y ). It is known that the Ricci tensor of a three-dimensional para-Kenmotsu manifold is given by [15] S(X, Y…”

∗-η-Ricci Soliton and Gradient Almost ∗-η-Ricci Soliton Within the Framework of Para-Kenmotsu Manifolds

“…Furthermore, in 2019, V. Venkatesha et al [23] considered the metric of an η-Einstein para-Kenmotsu manifold as a p -Ricci soliton and proved that the manifold is Einstein. In another study performed in 2019, I. K. Erken [24] considered Yamabe solitons on a 3-dimensional para-cosymplectic manifold and proved some vital results, including the fact that the manifold is either η-Einstein or Ricci flat. Several authors have also studied the η-Ricci soliton and its abstraction on paracontact metric manifolds; for instance, Dey et al [25] considered a paracontact metric as a conformal Ricci soliton and a p -conformal Ricci soliton, Deshmukh et al [26] studied certain results on Ricci almost solitons, Sarkar et al [27] examined a conformal η-Ricci soliton on a para-Sasakian manifold, and Naik et al [28] considered a para-Sasakian metric as an η-Ricci soliton.…”