Comprehensive studies of cut locus

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We studied the cut loci and related several topics, for example, the farthest points or simple closed geodesies on convex surfaces and got many results.As the problem to determine the cut locus, we proved that some compact Liouville manifolds have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds. We discussed on an extension to general dimension of Jacobi's last theorem(conjugate loci on ellipsoids have exact four cusps), and we got some remarkable progress and are planning to continue the research. Moreover, as the related topics, we got a modern proof of thread constructions of general quadric(hyper)surfaces by using the first integral (jointed work with K. Kiyohara).As the problem to study structures of cut locus, under some non-degenerating assumption we proved that the cut locus admits a nice stratification, some cone structure locally. Under stronger assumptions we have simpler procedure of Morse theory by using of critical points of distance functions (jointed work with T. Sakai).We established, for general convex surfaces, inequalities involving the diameter, the area and the length of simple closed quasi-geodesics (jointed work with C. Vilcu).Using the above simple closed quasi-geodesics we proved that any polyhedra are unfolded to a planar simple polygon by some cutting (jointed work with J. O'Rourke, C. Vilcu).We discussed several other unfolding by using simple clodes quasi geodesics, also