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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.
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참고문헌 (24)
참고문헌 신청
Christoph Böhm / 2008 / Manifolds with positive curvature operators are space forms / Annals of Mathematics 167(3):1079~1097
H.-D. Cao / 1994 / Existence of gradient Kahler-Ricci solitons / Elliptic and parabolic methods in geometry :1~16
Huai-Dong Cao / 2012 / On locally conformally flat gradient steady Ricci solitons / Transactions of the American Mathematical Society 364(5):2377~2391
Xiaodong Cao / 2007 / Compact gradient shrinking Ricci solitons with positive curvature operator / Journal of Geometric Analysis 17(3):425~433
XIAODONG CAO / 2011 / ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS / Communications in Contemporary Mathematics 13(02):269~282
H.-D. Cao / 1994 / Existence of gradient Kahler-Ricci solitons / Elliptic and parabolic methods in geometry :1~16
Huai-Dong Cao / 2012 / On locally conformally flat gradient steady Ricci solitons / Transactions of the American Mathematical Society 364(5):2377~2391
Xiaodong Cao / 2007 / Compact gradient shrinking Ricci solitons with positive curvature operator / Journal of Geometric Analysis 17(3):425~433
XIAODONG CAO / 2011 / ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS / Communications in Contemporary Mathematics 13(02):269~282
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