The structure of cut locus and global Riemannian geometry

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Recently, it was showed that the total length of cut locus of Riemannian surface is finite in the study of the structure of cut locus and by using this Ambrose's problem of surface was answered affirmatively. The purposes of this research are to study the structure of cut locus of Riemannian manifold and to study global Riemannian geometry by using the above results.The most main result that the distance function to the cut locus is Lipshitz continuous is proved the 1'st year, and now printing (joint work with M.Tanaka). It follows that the distance is derived naturally on the cut locus. It happens the following interesting question "what is the natural geometric structure on the cut locus?''In the study of some kind of stratification of cut locus of CィイD1∞ィエD1 Riemannian manifold, at the beginning, we tried the problem "Is the cut locus locally a submanifold around any cut point except for any subset which is of local dimensional Hausdorff measure zero?" We proved this affirmatively by very complicated method (j.w. with M.Tanaka.). Now, we are searching simpler method.Also we studied the set of critical points of distance function which is closely related with the cut locus. We proved last year that the set of all critical values of the distance function for a submanifold of a 3-dimensional complete Riemannian manifold is of Lebesgue measure zero (Sard type theorem). Now, we extend this result that ィイD71(/)2ィエD7-dimensional Hausdorff measure is zero (j.w. with M.Tanaka). By using method we expect that in the 4-dimensional case it is of Lebesgue measure zero.We cannot answered Ambrose's problem of general dimension, but we get the evaluation of face of Voronoi domain in any Hadamard manifold and showed that the subset (essential cut locus) of the cut locus which is contained any critical point of distance function is not so complicated in the case of low dimensional convex polytope.