Compact gradient $\rho$-Einstein soliton is isometric to the Euclidean sphere

TL;DR

This paper proves that compact gradient $ ho$-Einstein Ricci solitons are isometric to Euclidean spheres and shows that under certain conditions, non-compact solitons have zero scalar curvature, advancing understanding of their geometric structure.

Contribution

It establishes the isometry of compact gradient $ ho$-Einstein Ricci solitons to spheres and characterizes non-compact cases with zero scalar curvature under integral conditions.

Findings

Compact gradient $ ho$-Einstein solitons are isometric to Euclidean spheres.

Non-compact solitons satisfying certain integral conditions have zero scalar curvature.

The scalar curvature becomes constant in the compact case.

Abstract

In this paper we have investigated some aspects of gradient $\rho$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $\rho$-Einstein soliton is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have showed that in a non-compact gradient $\rho$-Einstein soliton satisfying some integral condition, the scalar curvature vanishes.