1999 Fiscal Year Final Research Report Summary
Research on transformation groups from the topological viewpoint
| Project/Area Number |
10640058
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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 | Yamagata University |
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Principal Investigator |
UCHIDA Fuichi Faculty of Science, Yamagata Univ. Professor, 理学部, 教授 (90028126)
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| Co-Investigator(Kenkyū-buntansha) |
ASOH Tohl Graduate School of Information Sciences, Tohoku Univ. Associate Professor, 情報科学研究科, 助教授 (00111352)
YASUI Tsutomu Faculty of Education, Kagoshima Univ. Professor, 教育学部, 教授 (60033891)
II Kiyotaka Faculty of Science, Yamagata Univ. Associate Professor, 理学部, 助教授 (10007180)
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| Project Period (FY) |
1998 – 1999
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| Keywords | transformation groups / non-compact Lie group / maximal compact subgroup / complex projective space |
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| Research Abstract |
In a previous project, F. Uchida have studied smooth SOィイD20ィエD2(p,q)-actions on SィイD1p+q-1ィエD1, each of which is an extension on the standard SO(p) x SO(q) action on SィイD1p+q-1ィエD1.
In this project, as a main theme, we study smooth Sp(p,q)-action on SィイD14p+4q-1ィエD1, each of which is an extension of the standard Sp(p) x Sp(q) action on SィイD14p+4q-1ィエD1. This standard action has condimension-one principal orbits with Sp(p-1)x Sp(q-1) as the principal isotropy subgroup. Furthermore, the fixed point set of the restricted Sp(p-1) x Sp(q-1) action is diffeomorphic to the seven-sphere SィイD17ィエD1.
We can show such SOィイD20ィエD2(p,q)-action on SィイD1p+q-1ィエD1 is characterized by a pair (φ,∫) satisfying certain conditions, whereφ is a smooth Sp(1,1)-action on SィイD17ィエD1, and ∫: SィイD17ィエD1 → PィイD21ィエD2(H) is a smooth function.
The pair(φ,∫) was introduced by T. Asoh to consider smooth SL(2,C)-actions on the 3-sphere, and was improved by F. Uchida.
As related topics, T. Yasui has a result on embeddings of n-manifolds into complex projective n-space, T. Asoh studies smooth actions of SL(2,C) on the 4-sphere, each of which is an extension of an orthogonal action of SU(2) on the 4-sphere, and K. Ii has given a method of construction and a characterization of complex structures on tangent bundles of complex projective spaces and quaternion projective spaces.
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