Rigidity characterization of compact Ricci solitons
J. Korean Math. Soc. 2019 Vol. 56, No. 6, 1475-1488
https://doi.org/10.4134/JKMS.j180747
Published online November 1, 2019
Fengjiang Li, Jian Zhou
East China Normal University; 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 : Primary 53C25; Secondary 53C44
Supported by : This work was financially supported by the National Natural Science Foundations of China (No.11531012 and 61663049) and Provincial Natural Science Foundation of Yunnan (No.2018FB002).
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