2002 Fiscal Year Final Research Report Summary
Geometry of Points at Infinity
| Project/Area Number |
12640069
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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 | Osaka Kyoiku University |
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Principal Investigator |
SUGAHARA Kunio Osaka Kyoiku U., Faculty of Education, Prof., 教育学部, 教授 (20093255)
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| Co-Investigator(Kenkyū-buntansha) |
MACHIGASHIRA Yoshiroh Osaka Kyoiku U., Faculty of Education, Asso. Prof., 教育学部, 助教授 (00253584)
KOYAMA Akira Osaka Kyoiku U., Faculty of Education, Prof., 教育学部, 教授 (40116158)
KATAYAMA Yoshikazu Osaka Kyoiku U., Faculty of Education, Prof., 教育学部, 教授 (10093395)
ITOH Jin-ichi Kumamoto U., Faculty of Education, Prof., 教育学部, 教授 (20193493)
INNAMI Nobuhiro Niigata U., Faculty of Science, Prof., 理学部, 教授 (20160145)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Riemannian manifold / Liouville manifold / ideal boundary |
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| Research Abstract |
For a Riemannian manifold there are several definitions of points at infinity. Gromov defined points at infinity using only the metric structure of the Riemannian manifold and named it ideal boundary.
Although his definition is abstract and universal, the relation between the global Riemannian structure and the ideal boundary is not clear. And it is hard to determine the ideal boundary for each Riemannian manifold. In fact, in the global study of Hadamard manifolds, he did not essentially make use of the idea of the ideal boundary.
In this research, in order to study the geometry of the ideal boundary, we tried to determine the ideal boundary for some Riemannian manifolds.
The quadratic surfaces in Euclidean space are Liouville manifolds. Making use of their classical coordinates, we studied the differential equation of their geodesies and see that the points at infinity of elliptic paraboloids are determined by the limit of the distance between two singular sets as Liouville manifolds.
For the quadratic surfaces in hyperbolic space, we can use the analogous method in low dimensions. Hyperboloids of two sheets has finite Maeda constant and the same geodesic structure with elliptic paraboloids in Euclidean spaces and their points at infinity are also determined by the limit of the distance between two singular sets as Liouville manifolds.
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