2000 Fiscal Year Final Research Report Summary
Submanifolds of homogeneous spaces and Grassmann geometry
| Project/Area Number |
10640066
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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) |
KODA Takashi Toyama University, Faculty of Science, Associate Professor., 理学部, 助教授 (40215273)
HASHIMOTO Hideya Nippon Institute of Technology, Faculty of Technology, Associate Professor., 工学部, 助教授 (60218419)
TASAKI Hiroyuki University of Tsukuba, Department of Mathematics, Associate Professor., 数学系, 助教授 (30179684)
TOJO Koji Chiba Institute of Technology, Lecturer., 工学部, 講師 (30296313)
IKAWA Osamu Fukushima National College of Technology, Department of General Education, Associate Professor., 助教授 (60249745)
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| Project Period (FY) |
1998 – 2000
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| Keywords | 6-dimensional sphere / Grassmann Geometry / G_2 |
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| Research Abstract |
The exceptional simple Lie group G_2 acts naturally on the 6-dimensional sphere S^6.Consider the decomposition of the Grassmann bundle G_p(TS^6) of all p-dimensional subspaces of tangent space of S^6. For a G_2-orbit ν of G_p(TS^6), a submanifold N of S^6 is said to be a ν-submanifold if all of the tangent space of N is contained in ν. We investigated the properties of ν submanifolds.
1. Construction and existence :
(1) Case p=2, there exists a ν-submanifold for any G_2-orbit ν of G_p(TS^6).
(2) Case p=3, the orbit space of G_p(TS^6) is identified with the real projective plane. If a compact ν-submanifold exists ν is contained in a line of the real projective plane.
We studied if the tubes over a J-holomorphic curve in the direction of the first (or second) Normal bundle is a ν-submanifold.
(3) Case p=4, we constructed many 4-dimensional CR submanifolds. But for another orbit ν the existence of ν-submanifold is open.
2. G_2 rigidity of CR submanifols
We gave a condition that two CR submanifolds are G_2 congruent and as its application we gave a characterization of CR submanifolds given by K.Sekigawa.
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