Geometry of polyhedron from the view point of differential geometry

在二维情形下，证明并写出了高斯多面体、埃氏定理和Weyl体积公式的类似结果。给出了2-多面体的Cohn-Vossen型不等式、图的全曲率及其紧性等新问题，如锐化三角剖分、本质割轨迹的结构、割轨迹圈的长度等，这些问题都是很有探索性的，经过不断的研究可望得到更好的结果。对于锐化三角剖分，证明了三次曲面允许24个三角形的锐化三角剖分，二十面体曲面允许12个三角形的锐化三角剖分，这些都是最少的。“十二面体表面没有任何小于11个三角形的锐角三角剖分，允许有14个三角形的锐角三角剖分。”对于本质割轨迹，我们将其定义为包含距离函数所有临界点的割轨迹的本质部分，并证明了端点数或顶点的度数与其内度量的几个不变量有关。此外，我们还考虑了一般维凸多面体的结构.关于割轨迹的圈的长度，我们证明了在任何直径为1的环面上存在一个点p，使得p的割轨迹的圈的长度大于2.这是最好的估计，并且不存在上界.

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.