Almost Kenmotsu 3-h-metric as a cotton soliton

Dibakar Dey,Pradip Majhi

doi:10.1108/ajms-10-2020-0103

Abstract

Purpose Cotton soliton is a newly introduced notion in the field of Riemannian manifolds. The object of this article is to study the properties of this soliton on certain contact metric manifolds. Design/methodology/approach The authors consider the notion of Cotton soliton on almost Kenmotsu 3-manifolds. The authors use a local basis of the manifold that helps to study this notion in terms of partial differential equations. Findings First the authors consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next the authors assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to. Originality/value The results of this paper are new and interesting. Also, the Proposition 3.2 will be helpful in further study of this space.

Highlights

An almost contact metric manifold is an odd dimensional differentiable manifold M2nþ1 together with a structure (w, ξ, η, g) satisfying ([1, 2])w2X 1⁄4 −X þ ηðX Þξ; ηðξÞ 1⁄4 1; (1.1)gðwX ; wY Þ 1⁄4 gðX ; Y Þ À ηðX ÞηðY Þ (1.2)for all vector fields X, Y on M2nþ1, where g is the Riemannian metric, w is a (1, 1)-tensor field, ξ is a unit vector field called the Reeb vector field and η is a 1-form defined by η(X) 5 g(X, ξ).Here fξ 5 0 and η 8 f 5 0; both can be derived from (1.1)
On a non-Kenmotsu almost Kenmotsu 3-h-manifold such that the Reeb vector field is an eigen vector of the Ricci operator, there exist no Cotton soliton with potential vector field pointwise collinear with the Reeb vector field
On a 3-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure, there exist no Cotton soliton with potential vector field pointwise collinear with the Reeb vector field

Summary

Introduction

An almost contact metric manifold is an odd dimensional differentiable manifold M2nþ1 together with a structure (w, ξ, η, g) satisfying ([1, 2])w2X 1⁄4 −X þ ηðX Þξ; ηðξÞ 1⁄4 1; (1.1)gðwX ; wY Þ 1⁄4 gðX ; Y Þ À ηðX ÞηðY Þ (1.2)for all vector fields X, Y on M2nþ1, where g is the Riemannian metric, w is a (1, 1)-tensor field, ξ is a unit vector field called the Reeb vector field and η is a 1-form defined by η(X) 5 g(X, ξ).Here fξ 5 0 and η 8 f 5 0; both can be derived from (1.1) . A Cotton soliton is a metric g defined on 3-dimensional smooth manifold M3 such that the following equation ðLV gÞðX ; Y Þ þ CðX ; Y Þ À σgðX ; Y Þ 1⁄4 0; (1.4)

Results

Conclusion