Ricci solitons and concurrent vector fields

TL;DR

This paper classifies Ricci solitons with concurrent potential fields, explores conditions for submanifolds to be Ricci solitons in such manifolds, and specifically classifies shrinking Ricci solitons on Euclidean hypersurfaces, with various applications.

Contribution

It provides a complete classification of Ricci solitons with concurrent potential fields and characterizes submanifolds that are Ricci solitons in manifolds with concurrent vector fields.

Findings

Complete classification of Ricci solitons with concurrent potential fields.

Necessary and sufficient conditions for submanifolds to be Ricci solitons.

Classification of shrinking Ricci solitons on Euclidean hypersurfaces.

Abstract

A Ricci soliton $(M^n,g,v,\lambda)$ on a Riemannian manifold $(M^n,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. In the first part of this paper we completely classify Ricci solitons with concurrent potential fields. In the second part we derive a necessary and sufficient condition for a submanifold to be a Ricci soliton in a Riemannian manifold equipped with a concurrent vector field. In the last part, we classify shrinking Ricci solitons with $\lambda=1$ on Euclidean hypersurfaces. Several applications of our results are also presented.