2003 Fiscal Year Final Research Report Summary
Group action and Grassmann Geometry
| Project/Area Number |
13640063
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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 | Tokyo University of Agriculture and Technology |
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Principal Investigator |
MASHIMO Katsuya Tokyo University of Agriculture and Technology, Faculty of Technology, Professor, 工学部, 教授 (50157187)
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| Co-Investigator(Kenkyū-buntansha) |
TOJO Koji Chiba Institute of Technology, Faculty of Technology, Associate Professor, 工学部, 助教授 (30296313)
TASAKI Hiroyuki University of Tsukuba, Intsitute of Mathematics, Associate Professor, 数学系, 助教授 (30179684)
HASHIMOTO Hideya Meijo University, Faculty of Science and Technology, Professor, 理工学部, 教授 (60218419)
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| Project Period (FY) |
2001 – 2003
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| Keywords | 6 dimensional sphere / G_2 / Grassmann geometry / CR submanifolds / J holomorphic curve / tube / CR submanifold / Chen's inequality |
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| Research Abstract |
We studied submanifolds of S^6 from the viewpoint of Grassmann geometry.
The compact simple Lie group of type G_2 acts on S^6. Extend the action to the action on the Grassmann bundle G^p (TS^6) of all p-dimensional subspaces of a tangent space of S^6 and decompose G^p (TS^6). Take a single orbit Σ of the action. We say that a p-dimensional submanifold N is a Σ-submanifokd if and only if the tangent space of N is contained in Σ. As a special case of Σsubmanifold, we have the class of totally real submanifolds and CR-submanifolds.
(1)Submanifolds which are obtained as a tube over another submanifolds are studied by many nathematicians. We classified all 3-dimensional submanifolds which are obtained as tubes in the direction of first or second normal space over a J-holomorphic curve. Concerning the problem, Hashimoto studied a method of construction of J-holomorphic curves from 2-tori to S^6 by a joint work with Taniguchi and Udagawa.
(2)We have a general inequality, proved by B.Y.Chen in 1991, for submanifolds of space forms. We studied minimal CR submanifolds of S^6 which attains the equality in Chen's inequality. Similar problem were studied by Dillen and Vrancken for 3-dimensional totally real submanidolds of S^6.
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Research Products
(8 results)
