Geometry of polyhedron from the view point of differential geometry

在二维情况下，对多面体的高斯极限定理和Weyl体积公式的类似结果进行了证明和书写。给出了2-多面体的Cohn-Vossen型不等式、图的总曲率及其紧性。在此基础上还提出了一些新的问题，如急性三角、基本切迹的结构、切迹的周期长度等。所有这些问题都是非常探索性的，并有望通过继续研究产生巨大的成果。对于锐角三角形，我们证明了立方曲面上有24个锐角三角形，二十面体表面上有12个锐角三角形，这些是最小的数。十二面体表面不存在任何小于11个三角形的锐角三角形，但允许有14个三角形的锐角三角形。此外，我们还讨论了其他几种情况，得到了处理一般凸面的一些基本思路。对于本质切割轨迹，我们在曲面的情况下将其定义为包含距离函数的所有临界点的切割轨迹的本质部分，并证明了端点的个数或顶点的程度与其内度规的几个不变量有关。此外，在一般维数的凸多面体情况下，我们考虑了它的结构。关于切迹的环长度，证明了在任意直径为1的环面上存在点p，使得p的切迹的环长度大于2。这是最好的估计，没有上界。

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.