Project/Area Number  09440028 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
Geometry

Research Institution  THE UNIVERSITY OF TOKYO 
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)

CoInvestigator(Kenkyūbuntansha) 
KAMISHIMA Yoshinobu Kumamoto University, Faculty of Science, Associate Professor, 理学部, 助教授 (10125304)
水谷 忠良 埼玉大学, 理学部, 教授 (20080492)
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)
矢野 公一 東京大学, 大学院・数理科学研究科, 助教授 (60114691)
MORITA Shigeyuki University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
MINAKAWA Hiroyuki Hokkaido University, Graduate School of Sciences, Assistant, 大学院・理学研究科, 助手 (30241300)
中山 裕道 広島大学, 総合科学部, 講師 (30227970)
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)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥7,900,000 (Direct Cost : ¥7,900,000)
Fiscal Year 1998 : ¥4,100,000 (Direct Cost : ¥4,100,000)
Fiscal Year 1997 : ¥3,800,000 (Direct Cost : ¥3,800,000)

Keywords  diffeomorphisms / volume preserving / Lipschitz / boundary / contact structure / Anosov / foliation / 微分同相 / 保積変換 / リプシッツ / 境界 / 接触構造 / アノソフ / 葉層 / 微分同相群 / リプシッツ同相 / 力学系 / 区分線形同相 / 特性類 / 多様体 / 葉層構造 
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 2sphere and that of 3ball, between the group of diffeomorphisms of 3sphere and that of 4ball, 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 3manifolds, 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.
