| Project/Area Number |
17340019
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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, Graduate School of Science and Technology, Professor (10125304)
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| Co-Investigator(Kenkyū-buntansha) |
KAMIYA Sigeyasu Okayama Science University, Engineering, Professor (80122381)
SOMA Teruhiko Tokyo Metropolitan University, Graduate School of Science and Technology, Professor (50154688)
OHSHIKA Ken'ichi Osaka University, Department of Mathematics, Professor (70183225)
FUJIWARA Koji Tohoku University, Department of Information Science, Professor (60229078)
GUEST Martin Toyko Metropolitan University, Graduate School of Science and Technology, Professor (10295470)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥6,650,000 (Direct Cost: ¥6,200,000、Indirect Cost: ¥450,000)
Fiscal Year 2007: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2006: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2005: ¥1,900,000 (Direct Cost: ¥1,900,000)
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| Keywords | Quaterninbic CR structure / Pseudo-conformal quaterninic structure / Uniformization / Cusp- cross section / Heisenberg nilmanifold / Complex hyperbolic manifold / Homogenous space / Seifert Rigidity / 一意化 / 展開写像 / ホロノミー群 / 非球形空間 / ローレンツ空間 / コボルディズム / 冪零多様体 / 複素双曲多様体 / Seifert Fiber space / 可解多様体 / 剛性 / 表現 / 擬共形4元数構造 / 普遍モデル / 擬共形4元数CR幾何構造 / Chern-Moser曲率形式 / Spherical CR幾何 / パラボリック幾何 / 擬共形4元数CR曲率 / 標準平坦モデル |
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| Research Abstract |
(1) We have studied an integrable, nondegenerate codimension 3 -subbundle D on a 4n+3- manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an sp (1)-valued 1-form ω loally on a neighborhood U such that Null ω= DIU and construct the curvature invariant on (M,ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic Kaehler 4n-manifolds. From this viewpoint, we have exhibited a quaternionic analogue of Chern-Moser's CR structure.
(2) Long and Reid have shown that the diffeomorphism class of every Riemannian flat manifold of dimension n>2 arises as some cusp cross-section of a complete finite volume real hyperbolic orbifold. For the complex hyperbolic case, D. B. McReynolds proved that every 3-dimensional infranilmanifold is diffeomorphic to a cusp cross-section of a complete finite volume complex hyperbolic 2-orbifold. We study this realization problem by using Seifert fibration. Let π be an n-dimensional crystallographic group. Then there is a faithful representation B: π Z^n×GL (n, Z). In particular, every compact Riemannian flat orbifold R^n/π can be realized as a cusp cross-section of a complete finite volume real hyperbolic orbifold.
(3) We have proved that every compact aspherical homogeneous manifold is the total space of a fibration with solv-geometry on the fibers over a base which is a locally symmetric orbifold of non-positive curvature. We construct an iterated injective Seifert fibered structure on such fibrations, and this allows to prove that every homotopy equivalence between such manifolds is induced by a diffeomorphism. In particular, two compact homogeneous aspherical manifolds are diffeomorphic if and only if their fundamental groups are isomorphic.
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