Some remarks on Yamabe solitons

TL;DR

This paper investigates properties of Yamabe solitons on compact Riemannian manifolds, deriving bounds on the soliton constant and characterizing conditions under which the soliton vector field is geodesic or produces Killing fields.

Contribution

It provides new insights into the geometric behavior of Yamabe solitons, including bounds on the soliton constant and conditions for the soliton vector field to be geodesic or Killing.

Findings

Bound on the soliton constant derived.

Commutator of soliton vector fields yields a Killing vector field.

Soliton vector field is geodesic iff the manifold has constant curvature.

Abstract

In this paper we have obtained evolution of some geometric quantities on a compact Riemannian manifold $M^n$ when the metric is a Yamabe soliton. Using these quantities we have obtained bound on the soliton constant. We have proved that the commutator of two soliton vector fields with the same metric in a given conformal class produces a Killing vector field. Also it is shown that the soliton vector field becomes a geodesic vector field if and only if the manifold is of constant curvature.