Some results on $\eta$-Ricci Soliton and gradient $\rho$-Einstein soliton in a complete Riemannian manifold

TL;DR

This paper investigates properties of $ ho$-Einstein and $ ho$-Ricci solitons on complete Riemannian manifolds, establishing conditions for compactness, scalar curvature positivity, and characterizations of the Euclidean sphere.

Contribution

It proves that certain gradient $ ho$-Einstein solitons with conformal potentials are isometric to spheres and provides conditions for compactness and scalar curvature positivity.

Findings

Manifolds with gradient $ ho$-Einstein solitons and conformal potentials are isometric to Euclidean spheres.

Convex Einstein potentials imply non-negative scalar curvature.

A sufficient condition for compactness of manifolds satisfying almost $ ho$-Ricci solitons.

Abstract

The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $\rho$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $\rho$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $\eta$-Ricci soliton (see Theorem 2).