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

Abstract

The goal of the paper is to deliberate conformal Ricci soliton and *-conformal Ricci soliton within the framework of paracontact geometry. Here we prove that if an ?-Einstein para-Kenmotsu manifold admits conformal Ricci soliton and *-conformal Ricci soliton, then it is Einstein. Further we have shown that 3-dimensional para-cosymplectic manifold is Ricci flat if the manifold satisfies conformal Ricci soliton where the soliton vector field is conformal. We have also constructed some examples of para-Kenmotsu manifold that admits conformal and *-conformal Ricci soliton and verify our results.

A study on conformal Ricci solitons and conformal Ricci almost solitons within the framework of almost contact geometry

“…Thus, from (24) we have ν = 0 and so ν + λ = 2n + 1 2 (p + 2 2n+1 ) (it follows from (23)). Hence we have from (25) that Ric = −2ng and therefore r = −2n(2n + 1), as required. Now, we discuss an example of Kenmotsu manifold that admits a conformal Ricci soliton.…”

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

“…Lemma 2.1. [33] On para-Kenmotsu manifold M 2n+1 (ϕ, ξ, η, g) the following formulas hold for any X, Y ∈ χ(M ),…”

“…Recently, Patra [23] consider Ricci soliton on para-Kenmotsu manifold and proved that a para-Kenmotsu metric as a Ricci soliton is Einstein if it is η-Einstein or the potential vector field V is infinitesimal paracontact transformation. Further, η-Ricci soliton and its generalizations have been studied on contact and paracontact geometry by (see [11,14,27,28,29,30,31,32,33,34]) Motivated by these results we consider a para-Kenmotsu metric as η-Ricci solitons and η-Ricci almost solitons. This paper is organized as follows.…”

Geometry of $\ast$-$k$-Ricci-Yamabe soliton and gradient $\ast$-$k$-Ricci-Yamabe soliton on Kenmotsu manifolds

“…In [48], authors have considered * -Ricci solitons and gradient almost * -Ricci solitons on Kenmotsu manifolds and obtained some beautiful results. Very recently, Dey et al [13][14][15][16][17]29,30,34,38,41] have studied * -Ricci solitons and their generalizations in the framework of almost contact geometry. Recently D. Dey [11] introduced the notion of * -Ricci-Yamabe soliton ( * -RYS) as follows :…”