2004 Fiscal Year Final Research Report Summary
Invariants On the Geometric Manifolds with Group Actions
| Project/Area Number |
14340026
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 Metropolitan University |
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Principal Investigator |
KAMISHIMA Yoshinobu Tokyo Metropolitan University., Science, Professor, 理学研究科, 教授 (10125304)
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| Co-Investigator(Kenkyū-buntansha) |
IMAI Jun Tokyo Metropoiltan University, Science, Associate Professor, 理学研究科, 助教授 (70221132)
KAMIYA Shigeyasu Okayama Science University, Thchnology, Professor, 工学部, 教授 (80122381)
SOMA Teruhiko Tokyo Denki University, Science and Thchnology, Pofessor, 理工学部, 教授 (50154688)
OHSHIKA Kenichi Osaka University, Science, Professor, 大学院・理学研究科, 教授 (70183225)
FUJIWARA Kouji Tohoku University, Science, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Pseudo-conformal auaternionic structure / Gemetric structure / Quaternionic CRstructure / Pseudo-conformal auaternionic CR structure / Uniformization / Conformal structure / Vanishing / Obstruction |
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| Research Abstract |
We have studied a geometric structure on a (4n+3)-dimensional smooth manifold M which is an integrable, nondegenerate codimension 3 subbundle D on M whose fiber supports the structure of 4n-dimensional quaternionic vector space. We call it a psesuo-conformal quterninonic structure. This structure has a refinement which is said to be a psesuo-conformal quterninonic CR structure. The structure is thought of as a generalization of the quaternionic CR structure. In order to study this geometric structure on M, we single out an sp(1)-valued 1-form ω locally on a neighborhood U of M such that Ker ω = D|U. We shall construct the invariants on the pair (M, ω) whose vanishing implies that M is uniformized with respect to a finite dimensional flat quaternionic CR geometry. In fact we have proved the standard psesuo-conformal quterninonic structure on the spahere S^<4n+3> coincides with the standard pseudo-quaternionic CR structure on S^<4n+3> The invariants obtained on a (4n+3)-manifold M have the same formula as the curvature tensor of quaternionic (indefinite) Kaehler manifolds. From this viewpoint, we shall exhibit a quaternionic analogue of Chern-Moser's CR structure. As to the global existence of the 1-form ω on a (4n+3)-manifold M is related to the Pontrjagin classes. We have shown the relation that 2p_1(M)=(n+2)p_1(L). In particular, if 2p_1(M)=0, then there exists a global 1-form co on M which represents a pseudo-conformal quaternionic structure D. As a consequence, there exists a hyperoomplex structure {I, J, K} on D.
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