收稿日期: 2003-03-04
修回日期: 2003-06-02
网络出版日期: 2004-03-19
Several Inequalities in Differential Geometry and Their Generalizations
Received date: 2003-03-04
Revised date: 2003-06-02
Online published: 2004-03-19
马宏宾 , 孙振祖 . 微分几何中几个不等式及其推广[J]. 中国科学院大学学报, 2004 , 21(2) : 153 -163 . DOI: 10.7523/j.issn.2095-6134.2004.2.002
Several inequalities are discussed. (1) For Schur. s inequality on convex curves of plane, we give a newanalytic proof for it, which maybe is simpler or clearer than known ones; we make further discussions by means ofintegral geometry and get more results. Moreover several related inequalities are put forward and proved. We alsopropose a conjecture which is generalization of Schur. s inequality in case of spherical surface. (2) For Fáry. s inequalityon closed curves of Euclidean space E3, we generalize it into spherical surface (i. e. surface with positiveconstant Gauss curvature) using method of moving frame. (3) For Fáry. s inequality on closed surface of Euclideanspace E3 : 
, we enhance it to 
using method of moving frame. Moreover thisinequality has been also generalized into 4-dimension case: 
Furthermore, a conjectureon further generalization to higher dimension case or general Euclidean space is proposed which requires furtherstudy.
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