1991 Fiscal Year Final Research Report Summary
Foliations and Geometric Structures
| Project/Area Number |
02640015
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
代数学・幾何学
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| Research Institution | Chiba Universtiy |
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Principal Investigator |
INABA Takashi Chiba University, College of Arts and Sciences, Assistant Professor, 教養部, 助教授 (40125901)
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| Co-Investigator(Kenkyū-buntansha) |
YASUDA Masami Chiba University, College of Arts and Sciences, Professor, 教養部, 教授 (00041244)
HINO Toshiyuki Chiba University, College of Arts and Sciences, Professor, 教養部, 教授 (70004405)
ANDO Tetsuya Chiba University, College of Arts and Sciences, Assistant Profeesor, 教養部, 助教授 (20184319)
NOZAWA Sohei Chiba University, College of Arts and Sciences, Assistant Profeesor, 教養部, 助教授 (20092083)
KUGA Ken'ichi Chiba University, College of Arts and Sciences, Assistant Professor, 教養部, 助教授 (30186374)
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| Project Period (FY) |
1990 – 1991
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| Keywords | Foliation / Geometric Structure / Global Holonomy Group |
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| Research Abstract |
There are two natural ways to introduce geometric structures into foliations. One is to introduce them in the direction transverse to the leaves, and the other is to do so in the direction tangent to the leaves. The former way has already been studied very much by many people, but it seems that works about the latter way are not so many in the literature.
In this research, we investigated these both types of geometric structures on foliations, mainly from the viewpoint of differential topology. Firstly, as for the transverse geometric structures, we studied co-dimension one foliations with transverse projective structure. We clarified the relation between the global holonomy group and the topological properties of the foliations. Furthermore, we showed that a transversely projective foliation cannot have exceptional leaves if the ambient manifold has amenable fundamental group. Secondly, as for the tangential geometric structures, we studied(1)tangentially affine foliations and(2)tangentially holomorphic foliations : (1)Note that tangentially affine foliations appear as Lagrangian foliations on symplectic manifolds. We determined the space of all leafwise affine, functions on a tangentially affine foliation on the torus. We also proved that the three dimensional sphere does not admit any co-dimension one tangentially affine foliation. (2)A Levi-flat real hypersurface in a complex surface has a tangentially holomorphic foliation, which is usually called the Levi foliation. We obtained some topological properties of compact Levi-flat hypersurfaces by investigating the holonomy of total leaves in their Levi foliations. In particular, we showed that the three dimensional sphere cannot be embedded as a Levi-flat hypersurface. This result is applied to two dimensional complex dynamical systems.
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