The relation between Riemannian geometry and discrete geometry from the view point of minimulity

本文对其中的一些问题进行了总结，特别是锐角三角剖分、曲线的全曲率、椭球的截迹等问题，并对柏拉图立体曲面上的锐角三角剖分进行了研究。证明了二十面体曲面可以用8个非钝角三角形和12个锐角三角形进行三角剖分。我们还证明了这些数字是尽可能小的。在正十二面体曲面的情形下，我们证明了存在一个只有10个非钝角三角形的三角剖分，并且这是最好的可能，我们还证明了一个有14个锐角三角形的三角剖分的存在性，以及不存在少于12个三角形的三角剖分。我们看一看具有固定端点和端点方向的曲线集，看看这个集中的总绝对曲率的下确界如何取决于端点和端点方向。我们考虑了曲线长度固定和长度自由的两种情况，并比较了两种情况下的不同结果。在此基础上，我们进一步确定了S ^2中端点、端点方向和长度固定的非闭曲线集中使总曲率最小的曲线的形状，研究了正则3维、4维和5维单形可通过的小孔，证明了任意椭球上任意点的截轨线都是通过其对极点的曲率线上的弧。证明了共轭轨迹恰好有四个尖点，这是Jacobi的最后一个几何命题。在此基础上，我们进一步研究了一类Liouville曲面的情形，得到了类似的结果。

We studied various problems, here we summarize some of them, especially acut triangulation, total curvature of curves, cut locus of ellipsoids.We studied acute triangulations on the surfaces of Platonic solids. We proved that the surface of icosahedron can be triangulated with 8 non-obtuse and with 12 acute triangles. We also showed these numbers to be smallest possible. In the case of the regular dodecahedral surface, we proved that there exists a triangulation with only 10 non-obtuse triangles, and that this is best possible, we also proved the existence of a triangulation with 14 acute triangles, and the non-existence os such triangulations with less than 12 triangles.The total absolute curvature of non closed curves in S^2 is studied. We look at the set of curves with fixed end points and end-directions and see how the infimum of the total absolute curvature in this set depends on the endpoints and the end-directions. We consider both the case when the length od curves is fixed and the case when the length is free, and see the difference results between them. Furthermore, we determin the shape of the curve in S^2 which minimize the total curvature in the set of nonclosed curves with fixed end points, end-directions and length.We studied small holes through which regular 3-,4- and 5-dimensional simplices can pass through.We proved that the cut locus of any point on any ellipsoid is an arc on the curvature line through the antipodal point. Also, we proved that the conjugate locus has exactly four cusps, which is known as the last geometric statement of Jacobi. Furthermore we studied in the case of some kind of Liouville surfaces and get the similar results.