2005 Fiscal Year Final Research Report Summary
The research of the behavior of geodesics
| Project/Area Number |
14540091
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Tokai University |
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Principal Investigator |
TANAKA Minoru Tokai University, School of Science, Professor, 理学部, 教授 (10112773)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAGUCHI Masaru Tokai University, School of Science, Professor, 理学部, 教授 (10056252)
NOGUCHI Mitsunori Meijo University, School of Commerce, Professor, 商学部, 教授 (00208331)
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| Project Period (FY) |
2002 – 2005
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| Keywords | cut locus / geodesic / fractal set / Hausdorff dimension / surface of revolution / torus / ellipsoid / cusp |
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| Research Abstract |
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.
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