Communications of the
Korean Mathematical Society CKMS

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.

Keywords: gradient $\rho$-Einstein soliton, almost $\eta$-Ricci soliton, Hodge-de Rham potential, Einstein potential, convex function, harmonic function, conformal vector field

MSC numbers: Primary 53C15, 53C21, 53C44, 58E20, 58J05

Supported by: The first author greatly acknowledges to The University Grants Commission, Government of India for the award of Junior Research Fellowship.