| 微分几何中几个不等式及其推广 |
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| 作者姓名： | 马宏宾 孙振祖 |
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| 作者单位： | 1. 中国科学院系统科学研究所, 北京 100080;
2. 郑州大学数学系, 郑州 450052 |
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| 摘 要： | 研究了微分几何中的几个不等式,提出了几个相关的不等式.(1)对平面上的Schur定理,给出了一种解析的证法,它比已知的一些 (几何的)证法显得简洁、明快,进而还用积分几何方法作了些讨论.(2)对欧氏空间中闭曲线的Fáry不等式,用活动标架法,将其推广到了球面 (正常高斯曲率曲面)中.(3)对三维欧氏空间中闭曲面的Fáry不等式,用活动标架法,将其中积分式前的常系数 4 π进一步改进为 1;此外,还将其推广到四维的欧氏空间中.这一不等式可能推广于更高维或一般的欧氏空间中,有待进一步研究.
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| 关 键 词： | Schur定理 Fá ry不等式 积分几何 活动标架法 高斯曲率 |
| 收稿时间： | 2003-03-04 |
| 修稿时间： | 2003-06-02 |
Several Inequalities in Differential Geometry and Their Generalizations |
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| Authors: | MA Hong-Bin SUN Zhen-Zu |
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| Institution: | 1. Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100080, China;
2. Department of Mathematics and Systems Science, Zhengzhou University, Zhengzhou 450052, China |
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| Abstract: | 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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| Keywords: | Schur’s inequality F?ry’s inequality integral geometry moving frame Gauss curvature |
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