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数学进展 - 2018, Vol. 47(2): 231-242
研究论文
Gauss-Bonnet-Chern曲率的变分公式
On Variational Formulas of Gauss-Bonnet-Chern Curvatures

郭希1,吴岚2
GUO Xi1,*, WU Lan2,**

1. 湖北大学数学与统计学学院, 应用数学湖北省重点实验室, 武汉, 湖北, 430062;
2. 中国人民大学信息学院数学系, 北京, 100872
1.Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, Hubei, 430062, P. R. China;
2. Department of Mathematics, School of Information, Renmin University of China, Beijing, 100872, P. R. China

收稿日期: 2016-10-16
出版日期: 2018-05-16
2018, Vol. 47(2): 231-242
DOI: 10.11845/sxjz.2016121b


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摘要 $p$阶Gauss-Bonnet-Chern曲率$L_p$是数量曲率的一种推广. 本文考虑了由此曲率定义的黎曼泛函$\widetilde{F}^p$.计算了$\widetilde{F}^p$的二阶变分公式. 应用该公式证明了球面上的标准度量和复射影空间上的Fubini-Study度量是$\widetilde{F}^p$的鞍点.
关键词 Gauss-Bonnet-Chern曲率黎曼泛函变分公式    
Abstract:The $p$-th Gauss-Bonnet-Chern curvature $L_p$ is a generalization of the scalar curvature. We consider the Riemannian functional $\widetilde{F}^p$ defined by this curvature. In this paper, we calculate the second variational formula of $\widetilde{F}^p$. As its application, we prove that the standard metric of the unit sphere and the Fubini-Study metric of the complex projective space are saddle points for $\widetilde{F}^p$.
Key wordsGauss-Bonnet-Chern curvature    Riemannian functional    variational formulas
PACS:  O186.12  
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