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Geometry of Points at Infinity

无穷远点的几何

基本信息

批准号：
12640069
负责人：
SUGAHARA Kunio
金额：
$ 1.98万
依托单位：
Osaka Kyoiku 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 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640069/
关键词：
Riemannian manifold Liouville manifold ideal boundary Alexandrov空間 Busemann関数

项目摘要

For a Riemannian manifold there are several definitions of points at infinity. Gromov defined points at infinity using only the metric structure of the Riemannian manifold and named it ideal boundary.Although his definition is abstract and universal, the relation between the global Riemannian structure and the ideal boundary is not clear. And it is hard to determine the ideal boundary for each Riemannian manifold. In fact, in the global study of Hadamard manifolds, he did not essentially make use of the idea of the ideal boundary.In this research, in order to study the geometry of the ideal boundary, we tried to determine the ideal boundary for some Riemannian manifolds.The quadratic surfaces in Euclidean space are Liouville manifolds. Making use of their classical coordinates, we studied the differential equation of their geodesies and see that the points at infinity of elliptic paraboloids are determined by the limit of the distance between two singular sets as Liouville manifolds.For the quadratic surfaces in hyperbolic space, we can use the analogous method in low dimensions. Hyperboloids of two sheets has finite Maeda constant and the same geodesic structure with elliptic paraboloids in Euclidean spaces and their points at infinity are also determined by the limit of the distance between two singular sets as Liouville manifolds.

对于黎曼流形，无穷远处的点有几种定义。格罗莫夫只用黎曼流形的度量结构来定义无穷远处的点，并将其命名为理想边界。虽然他的定义是抽象和普遍的，但全局黎曼结构与理想边界之间的关系并不明确。对于每一个黎曼流形，理想边界是很难确定的。事实上，在阿达玛流形的全局研究中，他并没有从本质上利用理想边界的概念。在本研究中，为了研究理想边界的几何性质，我们尝试确定一些黎曼流形的理想边界。欧几里得空间中的二次曲面是刘维尔流形。利用它们的经典坐标，研究了它们的测地线微分方程，发现椭圆抛物面的无穷远点是由两个奇异集之间的距离极限决定的。对于双曲空间中的二次曲面，我们可以在低维空间中使用类似的方法。两片双曲面具有有限的前田常数，与欧几里得空间中的椭圆抛物面具有相同的测地线结构，它们在无穷远处的点也是由两个奇异集之间的距离极限决定的，如Liouville流形。