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

我们研究了各种问题，在这里我们总结了其中的一些问题，特别是切三角剖分，曲线的全曲率，椭球的割轨迹。我们研究了柏拉图实体表面上的锐化三角剖分。证明了二十面体的曲面可以用8个非钝三角形和12个锐三角形进行三角剖分。我们还表明这些数字是可能的最小值。在正十二面体曲面的情况下，我们证明了存在一个只有10个非钝三角形的三角剖分，这是最佳可能的，我们还证明了有14个锐化三角形的三角剖分的存在，以及少于12个三角形的三角剖分的不存在。研究了S^2中非闭合曲线的全绝对曲率。我们观察具有固定端点和端点方向的曲线集，并了解该集合中总绝对曲率的下确界如何依赖于端点和端点方向。我们既考虑了曲线长度固定时的情况，又考虑了长度为自由时的情况，并观察了它们之间的差异。在此基础上，我们确定了S[2]中以最小化全曲率为目标的曲线的形状，研究了具有固定端点、端点和长度的非闭合曲线集上的小孔，证明了任意椭球面上任意一点的割线都是曲率线上通过对极点的弧形.证明了共轭轨迹恰好有四个尖点，这是雅可比的最后一个几何命题。此外，我们还研究了一些Liouville曲面的情况，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.