Geometry of Points at Infinity

对于黎曼流形，无穷远处的点有几种定义。Gromov仅利用黎曼流形的度量结构定义了无穷远处的点，并将其命名为理想边界，尽管他的定义是抽象的和普遍的，但整体黎曼结构和理想边界之间的关系并不清楚。而且很难确定每个黎曼流形的理想边界。事实上，在Hadamard流形的全局研究中，他并没有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，我们尝试确定了一些黎曼流形的理想边界。利用它们的经典坐标，研究了它们的测地线的微分方程式，发现椭圆抛物面在无穷远处的点是由两个奇异集之间的距离作为Liouville流形的极限所决定的，对于双曲空间中的二次曲面，我们可以在低维上使用类似的方法。两叶双曲面具有有限的前田常数，且在欧氏空间中与椭圆抛物面具有相同的测地结构，它们在无穷远处的点也由两个奇异集之间的距离极限决定为Liouville流形。

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.