Geometry of polyhedron from the view point of differential geometry

从微分几何的角度看多面体几何

基本信息

批准号：
12640079
负责人：
ITOH Jin-ichi
金额：
$ 2.37万
依托单位：
Kumamoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640079/
关键词：
polyhedron curvature total curvature cut locus tightness acute triangulation 最遠点 voronoi領域体積公式

项目摘要

The analogous results for polyhedron of Gauss's Theorema Egregium and Weyl's volume formula were proved and written in the 2-dimensional case. The Cohn-Vossen type inequality for 2-polyhedron, the total curvature of graphs and its tightness are written.There are the related new problems, for examples, the acute troangulations, the structure of essential cut locus, the length of cycles of cut locus, etc. All these problems are very exploratory and are expected to produce greate results by continuing studies.With respect to the acute triangulations, we proved that the cubed surface admitts an acute triangulations with 24 triangles, the icosahedral surface admitts an acute triangulations with 12 triangles, and these are the least numbers. 'The dodecahedral surface does not have any acute triangulations with triangles less than 11 and admits an acute triangulation with 14 triangles. Moreover we discussed several other cases and got some fundamental ideas how to treat the general convex surfaces.Withrespect to the essential cut locus, we define it in the case of a surface as the essential part of cut locus containing all critical points of distance function, and proved that the number of end points or the degree of vertices is related with several invariants of its inner metric. Moreover, we consider its structure in the case of convex polyhedron in general dimension.With respect to the length of cycles of cut locus, we proved that there is a point p on any torus with diameter 1 such taht the length of cycles in the cut locus of p is greater than 2. It is the best possible estimate and there is no upper bound.

在二维情况下，证明并写出了高斯理论egregium和Weyl的体积公式的多面体结果的类似结果。 Cohn-vossen类型不平等的不平等不平等，图形的总曲率及其紧密度是书面的。相关的新问题，例子，示例，急性转移，必不可少的切割基因座的结构，切割基因座的循环长度，临界的循环长度等。所有这些问题都非常易于探索，并有望通过cubs triand of trian the trian the trian。带有24个三角形的急性三角剖分，二十面体表面的急性三角剖分有12个三角形，而这些数量最少。 '十二面体表面没有任何三角形小于11的急性三角形，并且接受了14个三角形的急性三角剖分。此外，我们讨论了其他几个案例，并获得了一些基本的想法，如何处理一般凸表面。请注意，对于基本的切割基因座，我们将其定义为在表面的情况下，因为切割基因座的必不可少的部分包含距离距离的所有关键点，并证明了端点或顶点的程度与其内部内部的几个无敌量相关。此外，我们认为在凸多面体的情况下，我们认为其结构在一般维度上。对于切割基因座的循环的长度，我们证明了直径的任何圆环上有一个点p。1这样的taht taht taht tht tht tht tht cut cut cut locus t of p cut ocus cut locus the cut stoce of P的长度均大于2。这是最佳的估计，并且没有上限。