2003 Fiscal Year Final Research Report Summary
Synthetic studies of foliations and discrete group actions
| Project/Area Number |
13304005
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (A)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Nihon University |
|---|
Principal Investigator |
MATSUMOTO Shigenori Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (80060143)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TSUBOI Takashi University of Tokyo, Graduate school of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40114566)
MORITA Shigeyuki University of Tokyo, Graduate school of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
INABA Takashi Chiba University, Graduate school of Natural Sciences, Professor, 大学院・自然科学研究科, 教授 (40125901)
KANAI Masahiko Nagoya University, Graduate school of Mathematics, Professor, 大学院・多元数理研究科, 教授 (70183035)
NAKAYAMA Hiromichi Hiroshima University, Faculty of Integate Sciences, Assistant Professor, 総合科学部, 助教授 (30227970)
|
|---|
| Project Period (FY) |
2001 – 2003
|
| Keywords | foliations / locally free lie group actions / foliated cohomology / parameter rigidity / horocycle flow / Ruelle invariant / unique ergodicity / minimal set |
|---|
| Research Abstract |
The purpose of this project is to study the geometic and dynamical properties of foliations, or more generally, of discrete group actions. Throughout the project, we have obtained the following results.
1.When a solvable Lie group G acts on a closed manifold M, it determines the orbit foliation. Assume another G action on M with the same orbit foliation is given. We consider the problem whether or not the two actions are C^∞ conjugate up to an isomorphism of G. We interpreted this problem in terms of the foliated cohomology of the orbit foliations, and gave some examples of actions for which this problem has a positive answer.
2.Some groups of the sense preserving diffeomorphisms of the closed interval are shown to be perfect. The examples are, the group of Lipschitz homeomorphism, the group of C^∞ diffeomorphisms which are C^∞ tangent to the identity at the end points, and the group of C^1 diffeomorphisms C^1 tangent to the identity at the end points.
3.Suppose two codimension one foliations on a closed 3-manifold intersect transversely. Can it possible to isotope one foliation so that it is also tangent to the other, but in a different way? We considered this problem and have shown that the stable and unstable foliations of the suspension flow of hyperbolic toral automorphisms have unique intersection property, but those of geodesic flows of hyperbolic surfaces have not.
|
