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

In this paper we have investigated some aspects of gradient $ρ$-Einstein Ricci soliton in a complete Riemannian manifold. First, we have proved that the compact gradient $ρ$-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 $ρ$-Einstein soliton satisfying some integral condition, the scalar curvature vanishes.

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