Remarks on Einstein solitons with certain types of potential vector field

TL;DR

This paper investigates Einstein solitons with specific vector fields on Riemannian manifolds, deriving explicit formulas for the potential function and exploring geometric properties under symmetry conditions.

Contribution

It provides explicit expressions for the potential function in Einstein solitons with gradient, solenoidal, or concircular vector fields and analyzes geometric properties under Ricci tensor symmetries.

Findings

Explicit formula for λ in terms of V

Examples illustrating the results

Geometric properties under Ricci symmetry

Abstract

We consider almost Einstein solitons $(V,\lambda)$ in a Riemannian manifold when $V$ is a gradient, a solenoidal or a concircular vector field. 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 geometric properties when the Ricci curvature tensor of the manifold satisfies certain symmetry conditions.