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A note on compact almost Yamabe solitons

Ramesh Mete

Abstract

In this paper, we investigate almost Yamabe solitons on compact Riemannian manifolds without boundary of dimension greater than or equal to two. We provide some sufficient conditions for which the defining conformal vector field associated to a compact almost Yamabe soliton is a Killing vector field.

A note on compact almost Yamabe solitons

Abstract

In this paper, we investigate almost Yamabe solitons on compact Riemannian manifolds without boundary of dimension greater than or equal to two. We provide some sufficient conditions for which the defining conformal vector field associated to a compact almost Yamabe soliton is a Killing vector field.
Paper Structure (4 sections, 6 theorems, 32 equations)

This paper contains 4 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main results
  4. Some corrections in the article of Barbosa-Ribeiro Jr

Key Result

Theorem 1.1

Let $(M^n , g, X, \rho)$ be an almost Yamabe soliton. Then $X$ is a Killing vector field if one of the following conditions holds: Here and throughout of the paper, $\langle \cdot, \cdot \rangle := g (\cdot, \cdot)$ and $\mathrm{dvol}_{g}$ denotes the volume element for the Riemannian metric $g$.

Theorems & Definitions (10)

  • Theorem 1.1: Barbosa-Ribeiro Jr-2013
  • Remark 1.2
  • Theorem 1.3: Hwang-Yun-2025
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1: Petersen-Wylie-2009
  • Lemma 2.2: Barbosa-Ribeiro Jr-2013
  • proof : Proof of Theorem \ref{['thm:main-compact case']}
  • proof : Proof of Corollary \ref{['cor:corollary-1-compact-case']}
  • proof : Proof of Theorem \ref{['thm:Barbosa-Ribeiro Jr']}