Geometry of Points at Infinity

对于黎曼流形，无穷远点有几种定义。Gromov仅利用黎曼流形的度量结构定义了无穷远处的点，并称之为理想边界，虽然他的定义是抽象的和普遍的，但整体黎曼结构与理想边界之间的关系并不明确。并且很难确定每个黎曼流形的理想边界。实际上，在Hadamard流形的整体研究中，他并没有实质性地利用理想边界的思想。在本研究中，为了研究理想边界的几何，我们试图确定某些黎曼流形的理想边界。欧氏空间中的二次曲面是Liouville流形。利用它们的经典坐标，研究了它们的测地线的微分方程，发现椭圆抛物面的无穷远点是由两个奇异集之间的距离的极限作为Liouville流形决定的，对于双曲空间中的二次曲面，在低维情况下也可以采用类似的方法.双叶双曲面具有有限的Maeda常数，与欧氏空间中的椭圆抛物面具有相同的测地线结构，其无穷远点也由两个奇异集之间的距离的极限决定，如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.