Conformal Einstein soliton within the framework of para-K$ä$hler manifold

Abstract

The object of the present paper is to study some properties of para-Kähler manifold whose metric is conformal Einstein soliton. We have studied some certain curvature properties of para-Kähler manifold admitting conformal Einstein soliton.

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

“…By virtue of this, the soliton equation (1) reduces to 2Ric(X, Y) (24) and using ( 8), (23) we get ξν = 0. It follows from (24) that Xν = 0. Putting it into (24) provides…”

“…This shows that M is η-Einstein and therefore from Theorem 1 we conclude that M is Einstein. 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.…”

“…In [29], authors have considered * -Ricci solitons and gradient almost * -Ricci solitons on Kenmotsu manifolds and obtained some beautiful results. Many authors studied conformal Ricci solitons and their generalizations in the framework of almost contact and paracontact geometries, e.g., Kenmotsu manifold [3,7,24], Sasakian manifold [6,23], f -Kenmotsu manifold [18,21], para-Kähler manifold [7] and (κ, µ)-paracontact metric manifolds [27]. Ricci solitons and their generalizations have been enormously studied by many authors within the framework of contact and paracontact metric manifolds (see in details [6, 8-10, 12, 13, 19, 20, 25, 26]).…”

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

“…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.…”

A Kenmotsu metric as a conformal $η$-Einstein soliton

“…In [8], Roy, Dey and Bhattacharyya have defined conformal Einstein soliton, which can be written as:…”