1998 Fiscal Year Final Research Report Summary
STUDY ON THE GROUPS OF DIFFEOMORPHISMS
| Project/Area Number |
09440028
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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 | THE UNIVERSITY OF TOKYO |
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Principal Investigator |
TSUBOI Takashi University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40114566)
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| Co-Investigator(Kenkyū-buntansha) |
MINAKAWA Hiroyuki Hokkaido University, Graduate School of Sciences, Assistant, 大学院・理学研究科, 助手 (30241300)
KAMISHIMA Yoshinobu Kumamoto University, Faculty of Science, Associate Professor, 理学部, 助教授 (10125304)
OHSHIKA Ken'ichi University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (70183225)
SHISHIKURA Mitsuhiro University of Tokyo, Graduate School of Mathematical Sciences, Associate Profess, 大学院・数理科学研究科, 助教授 (70192606)
MORITA Shigeyuki University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
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| Project Period (FY) |
1997 – 1998
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| Keywords | diffeomorphisms / volume preserving / Lipschitz / boundary / contact structure / Anosov / foliation |
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| Research Abstract |
We studied the groups of diffeomorphisms of the circle and the disk. In particular, we found an important relationship between the group of diffeomorphisms of the circle and the group of the area preserving diffeomorphisms of the disk. Namely using tha fact that any diffeomorphism of the boundary circle of the disk has an extention to an area preserving diffeomorphism of the disk, we wrote down the relationship between the Euler class for the group of diffeomorphisms of the circle and the Calabi invariant for the group of the area preserving diffeomorphisms of the disk. Moreover we showed the sane result for the groups of Lipschitz homeomorphisms.
We studied similar relationship between the group of diffeomorphisms of 2-sphere and that of 3-ball, between the group of diffeomorphisms of 3-sphere and that of 4-ball, or more generally, the group of diffeonorphisms of the boundary of a compact manifold and that of the manifold. We also studied the group of Lipschitz homeomorphisms and we obtained a new result on the perfectness of such groups.
Relating to the contact structure on the 3-manifolds, we investigated the generalization of the notion of the projectively Anosov flows in higher dimensions. We studied the diffeomorphism classes of such objects. We look at the algebraic models in detail. We also studied Anosov flows and found a classification for regular projectively Anosov flows without compact leaves. We also studied complex vector fields and complex contact structures.
We studied the characteristic classes for foliations and the SC^-*S algebra for the foliations. In particular we investigated the case where the foliation has transversely piecewise linear structure. We also looked at the finitely presented simple groups.
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