Almost Ricci-Yamabe soliton on Almost Kenmotsu Manifolds

TL;DR

This paper investigates almost Ricci-Yamabe solitons on almost Kenmotsu manifolds, establishing conditions for $ extit{ extbf{ ext{η}}}$-Einstein structures and characterizing when these solitons reduce to Ricci-Yamabe solitons, with specific geometric classifications.

Contribution

It provides new conditions for almost Kenmotsu manifolds with ARYS to be $ extit{ extbf{ ext{η}}}$-Einstein and classifies the geometric structure of such manifolds under certain conditions.

Findings

AKMs with ARYS are $ extit{ extbf{ ext{η}}}$-Einstein under specific conditions.

An ARYS on Kenmotsu manifolds becomes a Ricci-Yamabe soliton with restrictions.

$(2n+1)$-dimensional $( ext{ extkappa}, ext{ extmu})'$-AKMs with gradient ARYS are either locally isometric to $ ext{H}^{n+1}(-4) imes ext{R}^n$ or have codirectional Reeb and soliton vector fields.

Abstract

This manuscript examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci-Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $\eta$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes Ricci-Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional $(\kappa, \mu)'$-AKM equipped with a gradient ARYS is either locally isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of $3$-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.