On Ricci almost solitons arising from conformal vector fields

TL;DR

This paper explores how conformal vector fields on a semi-Riemannian manifold influence Ricci almost soliton structures on immersed hypersurfaces, providing a classification of such hypersurfaces under certain conditions.

Contribution

It establishes a natural link between conformal vector fields and Ricci almost solitons on totally umbilic hypersurfaces, including a classification result inspired by Tashiro's theorem.

Findings

Hypersurfaces inherit Ricci almost soliton structures from ambient conformal vector fields.

A classification of certain Riemannian hypersurfaces with Ricci almost solitons is provided.

Few explicit examples illustrate the theoretical results.

Abstract

Let $\overline{M}^{n+1}$ be a semi-Riemannian manifold of constant sectional curvature, and endowed with a conformal vector field . Consider a Riemannian manifold $M^n$, isometrically immersed into $\overline{M}^{n+1}$. With these hypotheses in mind, the ultimate goal of this paper is to investigate the intimate relationship between conformal vector fields on $\overline{M}^{n+1}$ and the Ricci almost soliton structure on $M^n$. Assuming that the latter manifold is connected and totally umbilic, we prove that it is naturally given a Ricci almost soliton structure by means of the tangential part of a conformal vector field on the ambient manifold. This result is rather general in nature, and few concrete examples are worked out to illustrate its true power. Furthermore, with Tashiro's theorem in mind, we reach the climax of this paper with a classification of a class of Riemannian hypersurfaces that naturally inherit their Ricci almost soliton structure by the existence of a conformal vector field on the ambient manifold.