同向单形到欧氏空间的等距嵌入及其应用

数学物理学报

同向单形到欧氏空间的等距嵌入及其应用

杨定华

(四川师范大学数学与软件科学学院成都 610066)

收稿日期:2006-01-08 修回日期:2008-03-15 出版日期:2008-10-25 发布日期:2008-10-25
通讯作者: 杨定华
基金资助:
国家973计划基金(2004CB318003)、四川省教育厅自然科学重点基金(07ZA087)资助

The Isometric Embedding of Synclastic Simplexes to Euclidean Space and Applications

Yang Dinghua

(College of Mathematics and Software Sciences, Sichuan Normal University, Chengdu 610066)

Received:2006-01-08 Revised:2008-03-15 Online:2008-10-25 Published:2008-10-25
Contact: Yang Dinghua

摘要/Abstract

摘要： 该文利用矩阵的方法, 获得了两个同向的 n 维单形同时等距嵌入 Eⁿ维欧氏空间的一个充分必要条件是: 对于预给(n+1)²个距离,满足一组具有行列式形式的不等式组det(△_k)<0, 由此可以得到两组等数量的有限点集合到 Eⁿ维欧氏空间中等长嵌入的一个充分必要条件. 然后利用杨路和张景中引进的代数方法, 应用广义等距嵌入定理, 提出了关于两组两个完全同向的 n 维单形“广义度量加”的概念, 并且证明了涉及“广义度量加”的一个几何不等式, 它推广了杨路和张景中关于Alexander猜想的结果. 同时我们将杨路和张景中关于Neuberg-Pedoe不等式的高维推广形式推广到两组两个完全同向的 n 维单形中, 获得了涉及四个单形的一类几何不等式, 它们蕴含近期诸多文献的主要结果.

关键词: 欧氏空间, 同向单形, 等距嵌入, 广义度量加, 几何不等式

Abstract: In this paper, by the matrix method, a necessary and sufficient condition for two n-dimensional synclastic simplexes to be able embedded in the n-dimensional Euclidean space is obtained, as well as for two group finite point sets. Then, by using the algebraic method which was introduced by Yang L. and Zhang J. Z., and the generalized isometric embedding theory, the generalized concept of metric addition is brought forward, a geometric inequality about the generalized metric addition is given, and the main result of Yang L. and Zhang J. Z. on Alexander conjecture is generalized, the higher dimensional Neuberg-Pedoe inequality generalized by Yang L. and Zhang J. Z. is generalized also. By these inequalities, a class of geometric inequalities about four simplexes is obtained, These imply some recent results in literature.

Key words: Euclidean space, Synclastic simplexes, Isometric embedding, Generalized metric addition, Geometry inequality

中图分类号:

51K05

杨定华. 同向单形到欧氏空间的等距嵌入及其应用[J]. 数学物理学报, 2008, 28(5): 914-922.

Yang Dinghua. The Isometric Embedding of Synclastic Simplexes to Euclidean Space and Applications[J]. Acta mathematica scientia,Series A, 2008, 28(5): 914-922.