Compactifications of Riemannian manifolds and points at infinity

比较了Riernannian流形的各种紧化，主要研究了Gromov利用Riemann度量导出的距离函数所得到的最自然的紧化。他呼吁一套点增加了他的紧化一个理想的边界。我们确定了以下空间的理想边界的结构。1.椭圆抛物面的紧化是一个2-球面。它的理想边界是一个端点为Busemann函数的区间.由两条抛物线围成的2-平面上的n-平片构成的锥的紧化是边为Busemann函数的n-边形.以旋转抛物面为界的域的二重的紧化是一个3-球面。它的理想边界是一个端点为Busemann函数的区间，上述结果表明Busemann函数在理想边界中起着关键作用。然后证明了Busernann函数决定理想边界的大小，无穷远点的截轨线应由Busemann函数定义，Busemann函数由该点的梯度流射线决定。

Comparing various compactifications of Riernannian manifolds, we mainly studied the most natural compactification due to Gromov by means of the distance function induced from the Riemannian metric. He called the set of points added in his compactification an ideal boundary. We determined the structure of the ideal boundary of the following spaces.1. The compactification of an elliptic paraboloid is a 2-sphere. Its ideal boundary is an interval whose ends are Busemann functions.2. The compactification of a cone consists of n-flat pieces in 2-plane bounded by two parabolas is an n-gon whose edges are Busemann functions.3. The compactification of the double of a domain bounded by a paraboloid of revolution is a 3-sphere. Its ideal boundary is an interval whose ends are Busemannfunctions.Results above suggested that Busemann functions play key role in the ideal boundary. Then we proved that Busernann functions determine the size of the ideal boundary.The cut locus of a point at infinity should be defined by means of Busemann functions determined by the rays which are gradient flows of the point.