Singularity theory for Riemannian distance functions on non-positively curved surfaces

We construct Morse theory and singularity theory for Riemannian distance functions d(p) on a compact non-positively curved surface M-2. Stable critical points and their indexes are described due to the connection between Min-type functions and Riemannian distance functions on non-positively curved manifolds; Min-type functions are minima of a finite number of smooth functions. We prove that the set of non-positively curved metrics such that (for a given p is an element of M-2) the distance function d(p) is Morse, is open and everywhere dense in the set of non-positively curved Riemannian metrics. For a generic metric on M-2 distance functions d(q) are not Morse for some points q is an element of M-2; the description of the singularities of d(q) is based on Morse theory for non-positively curved Riemannian metrics p(x, y) which are Min-type functions on M-2 x M-2. We prove that a generic negatively curved metric is Morse.