2007 Fiscal Year Final Research Report Summary
Research on Geometric invariant on Manifolds and Lie transformation groups
| 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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| Keywords | Quaterninbic CR structure / Pseudo-conformal quaterninic structure / Uniformization / Cusp- cross section / Heisenberg nilmanifold / Complex hyperbolic manifold / Homogenous space / Seifert Rigidity |
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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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