The research of the behavior of geodesics

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1) The structure of the cut locus of a standard torus of revolution in Euclidean space was completely determined by J.Gravesen, S.Markvorsen, R.Sinclair and M.Tanaka (the head investigator). Furthermore the structure of a family of torus of revolution containing the standard tori.2) R.Sincalir and M.Tanaka had a conjecture of the upper bound of the number of endpoints of a cut locus on a surface of revolution experimentally. R.Sinclair checked it by making use of his computer programming.3) M.Tanaka proved that the cut locus of a surfase whith real analytic Riemannian metric has at least four cusps if the surface is homeomorphic to a sphere.4) T.Matsuyama and M.Tanaka treated the outside problem of the wave equation. We obtained the L^2 estimates of the solution and proved the nondecay of the energy of the solution. Furthermore M.Tanaka proved that the boundary of the convex hull of a smooth hypersurface in any dimensional Euclidean space is only continuously differentiable, but not smooth and gave a counterexample of it.