Remarks on almost Riemann solitons with gradient or torse-forming vector field

TL;DR

This paper explores the properties of almost Riemann solitons in Riemannian manifolds, especially when the potential vector field is of gradient, solenoidal, or torse-forming type, and relates them to almost Ricci solitons.

Contribution

It explicitly expresses the function λ in terms of the gradient vector field for gradient almost Riemann solitons and analyzes special cases involving solenoidal and torse-forming vector fields.

Findings

Explicit formula for λ in gradient case using Bochner formula

Properties of solenoidal and torse-forming potential vector fields

Illustrative examples demonstrating theoretical results

Abstract

We consider almost Riemann solitons $(V,\lambda)$ in a Riemannian manifold and underline their relation to almost Ricci solitons. When $V$ is of gradient type, using Bochner formula, we explicitly express the function $\lambda$ by means of the gradient vector field $V$ and illustrate the result with suitable examples. Moreover, we deduce some properties for the particular cases when the potential vector field of the soliton is solenoidal or torse-forming, with a special view towards curvature.