On a Distance between Directions in an Aleksandrov Space of Curvature ≤K

摘要

In this note we consider an analog of the notion of angle between two directions, possibly based at different points, for a space of curvature bounded above. The set of all unit tangent vectors of an n-dimensional Riemannian manifold M~n constitutes a sphere-bundle T_1(M~n) over M~n. In [13; 14], Sasaki introduced a Natural rimemannian metric on a sphere-bundle of a Riemannian space , Which now is known as the Sasaki metric. Let ξ and ξ be a pair of vectors in T_1 (M~n) at the nearby points x and x.