Radial curvature and topology of Riemannian manifolds

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The angle comparison theorem plays an important role for the study of the radial curvature and topology of pointed complete Riemannian manifolds. We have proved that the angle comparison for all the angles of all the generic geodesic triangles is valid. We are led to a new idea by thinking of radially curved manifolds as certain complete real hypersurfaces on the compat rank one symmetric spaces. This corresponds to the maximal diameter theorem for compact manifolds with positive curvature. Thus the global characterization of certain complete real hyprsurfaces are obtained.