1998 Fiscal Year Final Research Report Summary
Exceptional Lie Groups and Grassmann Geomtry
| Project/Area Number |
09640126
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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 | Nippon Institute of Technology |
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Principal Investigator |
HASHIMOTO Hideya Nippon Institute of Technology, Associated Professor, 工学部, 助教授 (60218419)
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| Co-Investigator(Kenkyū-buntansha) |
KODA Takashi Toyama University, Associated Professor, 理学部, 助教授 (40215273)
MASHIMO Katsuya Tokyo University of Agriculture and technology, Associated Professor, 工学部, 助教授 (50157187)
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| Project Period (FY) |
1997 – 1998
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| Keywords | octonions / the Lie group G_2 of automorphisms of the octonions O / the 6-dimensional unit sphere S^6=G_2 / SU (3) / grassmann subbundles / J-holomorphic curves / CR-submanifolds / totally real submanifolds |
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| Research Abstract |
Let S^6 be the 6-dimensional unit sphere centered at the origin in a 7-dimensional Eu-cliclean spacc. We identified 7-dimensional Euclidean space with purely imaginary octo-nions IrnO (or Cayley algebra). Taking account of algebraic properties of octonions we can define the homogeneous almost Herinitian structure on S^6 We denote by G_2 the Lie group of automorphisms of the octonions 0. Then we have S^6 - G_2/SU(3).
This almost complex structure satisfy the nearly Kah1er condition ((*xJ)X=0) where * is the Levi-Civita connection of S^6, and X is any vector field of S^6
First, we gave the representaion of homogeneous grassmann subbundles corresponding to the invariant submanifolds (J-holomorphic curves, totally real and CR-submani[olds) of S^6=( S^6, J,<, >).
Next, we obtained some constructions of invariant submanifolds of S^6=(S^6, J, < , >).
(1). We gave many examples of 3-dimensional homogeneous CR-submanifolds or S^6 and extension cC the 3-dimensional CR-submanifold which is obtained by K.Sekigawa.
(2). We obtained some rigidity theorem of 3-dimensional CR-submanifolds up to the action or G2 and determine G2 invariants.
(3). We construct 3-dimensional totally real submanifolds and 3-dimensional CR-submanifolcls as a tube of the first or second normal bundle of J-holomorphic curves.
(4). We give some examples of 4-dimensional CRsubmanifolds and studied the obstructions of the existence of such submanifolds.
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