1999 Fiscal Year Final Research Report Summary
Studies on Curvatures of Submanifolds
| Project/Area Number |
10640061
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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 | Utsunomiya University |
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Principal Investigator |
KITAGAWA Yoshihisa Utsunomiya Univ., Faculty of Education, Associate Professor, 教育学部, 助教授 (20144917)
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| Co-Investigator(Kenkyū-buntansha) |
SHIRASOU Takeo Utsunomiya Univ., Faculty of Education, Professor, 教育学部, 教授 (50007960)
OCHIAI Shoji Utsunomiya Univ., Faculty of Education, Professor, 教育学部, 教授 (30031545)
KIMURA Shigeru Utsunomiya Univ., Faculty of Education, Professor, 教育学部, 教授 (70007962)
FUJIHIRA Hideyuki Utsunomiya Univ., Faculty of Education, Associate Professor, 教育学部, 助教授 (70114171)
KIMURA Hiroshi Utsunomiya Univ., Faculty of Education, Professor, 教育学部, 教授 (70017953)
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| Project Period (FY) |
1998 – 1999
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| Keywords | submanifold / flat torus / isometric deformation |
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| Research Abstract |
In this research, using a method established by Y.Kitagawa, we studied flat tori in the unit 3-sphere SィイD13ィエD1, and obtained many interesting results. We summarize the results as follows.
(1) Isometric deformations of flat tori in SィイD13ィエD1 with nonconstant mean curvature : In this research, we studied isometric deformations of an isometric immersion f of a flat torus M into SィイD13ィエD1, and proved that if the mean curvature of f is not constant, then the immersion f admits a nontrivial isometric deformation preserving the total mean curvature.
(2) Isometric deformations of flat tori in SィイD13ィエD1 with constant mean curvature : It is easy to see that if f is an isometric immersion of a flat torus M into SィイD13ィエD1 with constant mean curvature, then f is a covering map onto a Clifford torus in SィイD13ィエD1. In this research, we classified the covering maps which admit no isometric deformation.
(3) Periodicity of the asymptotic curves of n-dimensional flat tori isometrically immersed in SィイD12n-1ィエD1 : In 1988, it was shown that if M is a 2-dimensional flat torus isometrically immersed in SィイD13ィエD1, then every asymptotic curve of M is periodic. In this research, we studied periodicity of the asymptotic curves of n-dimensional flat tori isometrically immersed in SィイD12n-1ィエD1, and proved that there exists a 3-dimensional flat torus isometrically embedded in SィイD15ィエD1 all of whose asymptotic curves have no period.
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