Compactifications of Riemannian manifolds and points at infinity

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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.