Rigidity Characterization of Compact Ricci Solitons
J. Korean Math. Soc.
Published online 2019 Feb 21
Fengjiang Li, and Jian Zhou
Department of Mathematics East China normal university, Department of Mathematics Yunnan normal university
Abstract : In this paper, we firstly define the Ricci mean value along the gradient vector field of the Ricci potential function and show that it is non-negative on a compact Ricci soliton. Furthermore a Ricci soliton is Einstein if and only if its Ricci mean value is vanishing. Finally, we obtain a compact Ricci soliton $(M^{n},g) (n \geq 3)$ is Einstein if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.
Keywords : Ricci soliton; Einstein manifold; Ricci mean value; Weyl conformal curvatue tensor
MSC numbers : 53C25; 53C44
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