Ricci solitons in almost $f$-cosymplectic manifolds

TL;DR

This paper investigates Ricci solitons in almost $f$-cosymplectic manifolds, proving nonexistence in certain cases, characterizing three-dimensional cases, and classifying specific $ ext{eta}$-Einstein manifolds with Ricci solitons.

Contribution

It establishes nonexistence results, characterizes three-dimensional cases, and classifies $ ext{eta}$-Einstein almost $f$-cosymplectic manifolds admitting Ricci solitons.

Findings

No Ricci solitons exist on almost cosymplectic $(
,)$-manifolds.

Three-dimensional almost $f$-cosymplectic manifolds with Ricci solitons are cosymplectic.

Classification of three-dimensional $ ext{eta}$-Einstein almost $f$-cosymplectic manifolds with Ricci solitons.

Abstract

In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(\kappa,\mu)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimesional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $\eta$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton.