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 Some results in $\eta$-Ricci Soliton and gradient $\rho$-Einstein soliton in a complete Riemannian manifold Commun. Korean Math. Soc.Published online August 5, 2019 Chandan Kumar Mondal and Absos Ali Shaikh University of Burdwan 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). Keywords : Gradient $\rho$-Einstein soliton, almost $\eta$-Ricci soliton, Hodge-de Rham potential, Einstein potential, convex function, harmonic function, conformal vector field MSC numbers : 53C15, 53C21, 53C44, 58E20, 58J05 Full-Text :

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