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

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 satisfying some curvature conditions is isometric to the Euclidean sphere by showing that the scalar curvature becomes constant. Second, we have shown that in a non-compact gradient \(\rho\)-Einstein soliton satisfying an integral condition, the scalar curvature vanishes.