Ricci solitons in manifolds with quasi-constant curvature

TL;DR

This paper explores Ricci solitons within manifolds of quasi-constant curvature, extending the Eisenhart problem to symmetric tensors and identifying conditions for expanding or shrinking solitons in specific geometric contexts.

Contribution

It reinterprets the Eisenhart problem for symmetric tensors in quasi-constant curvature manifolds and characterizes Ricci solitons in para-Sasakian spaces and quasi-umbilical hypersurfaces.

Findings

Ricci solitons occur in para-Sasakian spaces with constant scalar curvature.

Shrinking Ricci solitons are found in certain quasi-umbilical hypersurfaces.

The work links Ricci solitons to specific geometric structures in quasi-constant curvature manifolds.

Abstract

The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant scalar curvature and vanishing $D$-concircular tensor field and ii) shrinking on a class of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space form.