2001 Fiscal Year Final Research Report Summary
The theoretical approach of the behaviors of geodesics
| Project/Area Number |
11640094
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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 |
TANKAA Minoru Tokai University Department of Mathematics Professor, 理学部, 教授 (10112773)
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| Co-Investigator(Kenkyū-buntansha) |
NOGUCHI Mitsunori Meijyo University Department of Commerce Professor, 商学部, 教授 (00208331)
YAMAGUCHI Masaru Tokai University Department of Mathematics Professor, 理学部, 教授 (10056252)
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| Project Period (FY) |
1999 – 2001
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| Keywords | geodesic / sut locus / fractal set / Hausdorff dimension / entropy dimension |
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| Research Abstract |
The Behaviors of geodesics on a Riemannian manifold has been greately investigated even in the period of classical differential geometry. Because there was no idea of cut loci in that period, there is no research paper from the point of the cut locus view. In the early stage of 20^<th> century, Poincare introduced the idea of cut loci. In 1950's, Klingenberg clarified the importance of the idea by proving some theorems in Riemannian geometry. Between 1999 and 2001, we got the following two main results on the behaviors of geodesics from the point of the cut locus view.
Theorem 1 Let N be a closed submanifold of a complete Riemannian manifold M. The distance function to the cut locus of N on the unit normal bundle over N is locally Lipschitz.
The next theorem was proved by applying the above theorem. Theorem 2 Let N be a closed submanifold of a complete Riemannian manifold M. If the dimension of M does not exceed 4, then the set of all critical values of the distance function from N is of Lebesgue measure zero. In particular, for almost all t>0, the set of points whose distance is t, is a hypersurface of M.
After joint work with Prof. Sinclair, we proved that the cut locus of a point on an ellipsoid has no branch point.
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