| Project/Area Number |
10640056
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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 | Tohoku University |
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Principal Investigator |
ASOH Toru Graduate School of Information Science, Tohoku University, Asso. Professor, 大学院・情報科学研究科, 助教授 (00111352)
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| Co-Investigator(Kenkyū-buntansha) |
OKADA Masami Graduate School of Information Science, Tohoku University, Professor, 大学院・情報科学研究科, 教授 (00152314)
UAKAWA Hajime Graduate School of Information Science, Tohoku University, Professor, 大学院・情報科学研究科, 教授 (50022679)
YASUI Tsutomu Faculty of Education, Kagoshima University, Professor, 教育学部, 教授 (60033891)
SHIMOKAWA Koya Graduate School of Information Science, Tohoku University, Assistant, 大学院・情報科学研究科, 助手 (60312633)
TAYA Hisao Graduate School of Information Science, Tohoku University, Assistant, 大学院・情報科学研究科, 助手 (40257241)
内田 興二 東北大学, 大学院情報科学研究科, 教授 (20004294)
金子 誠 東北大学, 大学院情報科学研究科, 教授 (10007172)
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| Project Period (FY) |
1998 – 1999
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| Project Status |
Completed (Fiscal Year 1999)
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| Budget Amount *help |
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1998: ¥2,100,000 (Direct Cost: ¥2,100,000)
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| Keywords | Lie transformation group / non-compact group / fixed point / four dimensional sphere / complex special linear group / 2次特殊復素線形群 / グラフ理論 |
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| Research Abstract |
Let G be a Lie group, and M be a compact smooth manifold. Assume that G acts smoothly on 4M4, and the fixed point set F(G,M) is not empty.
If G is compact, then there exists an open neighborhood of a fixed point such that the restricted G-action is linear by the differential slice theorem. But if G is not compact, then there does not, in general, exists such a neighborhood around the fixed point.
The aim of our research is to study the behavior of the non-compact Lie group G action around a fixed point. In this research, we consider the case that G is the complex special linear group SL(2,C) and M is the four dimensional sphere SィイD14ィエD1. SL(2,C) has the maximal compact subgroup SU(2). We also assume that the restricted SU(2) action on SィイD14ィエD1 is the linear action induced from four dimensional irreducible real representation. Under these assumptions, we have the following results.
1. The fixed point sets of SL(2,C) and SU(2) coincide with each other, and hence the fixed point set F(SL(2,C), SィイD14ィエD1) consists of two points.
2. SL(2,C) acts smoothly on SィイD14ィエD1 - F(SL(2,C),SィイD14ィエD1), which is diffeomorphic to SィイD13ィエD1 × R. Under a certain condition, this action is characterized by the affine group Aff(1) action on R. Here Aff(1) is the two dimensional closed subgroup of SL(C,2).
Consider our problem in the topological aspect, we have the following several results. T.Yasui has studied the isotopy classes of embeddings homotopic to the given embedding f : M → N. He studied the case that N is the complex projective space CPィイD1nィエD1. K.Shimokawa s studied one dimensional submanifolds in compact oriented three dimensional manifolds.
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