Almost Cosymplectic (k,mu )-metrics as eta-Ricci Solitons

Wenjie Wang

doi:10.1007/s44198-021-00019-4

Abstract

In this paper, we study eta-Ricci solitons on almost cosymplectic (k,mu )-manifolds. As an application, it is proved that if an almost cosymplectic (k,mu )-metric with k<0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by k<0. In addition, a concrete example is constructed to illustrate the above result.

Highlights

Let (M, g) be a Riemannian manifold, and Ric its Ricci tensor with respect to g
In this paper we prove that if the metric of a non-cosymplectic almost cosymplectic (k, )-manifold is a Ricci soliton, the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and = 0
Theorem 3.2 There exist no almost cosymplectic (k, )-manifolds with k < 0 whose metric represents non-trivial -Ricci solitons and the potential vector field is collinear with the Reeb vector field

Summary

Introduction

Let (M, g) be a Riemannian manifold, and Ric its Ricci tensor with respect to g. The study of Ricci solitons in almost contact Riemannian geometry was initiated by Sharma [20] who proved that if the metric of a complete K-contact manifold represents a gradient Ricci soliton, the soliton must be shrinking and the manifold is compact Einstein and Sasakian. Wang in [24] who proved that if the metric of a three-dimensional cosymplectic manifold is a Ricci soliton, either the manifold is locally flat or the potential vector field is an infinitesimal contact transformation. Wang [23] proved that there exist no gradient Ricci solitons on a non-cosymplectic almost cosymplectic (k, )-manifold Extending this result, in this paper we prove that if the metric of a non-cosymplectic almost cosymplectic (k, )-manifold is a Ricci soliton, the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and = 0.

Almost cosymplectic manifolds

Example

Conclusions