Some characterizations of $\rho$-Einstein solitons

TL;DR

This paper investigates properties of $ ho$-Einstein and Ricci solitons, establishing conditions under which they are isometric to spheres, have constant scalar curvature, or are steady, contributing to geometric analysis.

Contribution

It provides new characterizations of $ ho$-Einstein and Ricci solitons under various geometric and boundedness conditions, including isometry to spheres and conditions for steadiness.

Findings

Gradient $ ho$-Einstein soliton with bounded vector field norm is isometric to the Euclidean sphere.

Complete gradient $ ho$-Einstein soliton with finite weighted Dirichlet integral has constant scalar curvature and is Ricci flat.

Non-shrinking or non-expanding gradient traceless Ricci soliton under certain conditions must be steady.

Abstract

In this article we have showed that a gradient $\rho$-Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient $\rho$-Einstein soliton with finite weighted Dirichlet integral and certain restriction on Ricci curvature must be of constant scalar curvature and steady Ricci flat. Finally, we have proved that a non-shrinking or non-expanding gradient traceless Ricci soliton possessing some conditions must be steady.