| Project/Area Number |
09640082
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | HOKKAIDO UNIVERSITY |
|---|
Principal Investigator |
KIYOHARA Kazuyoshi Grad.School of Sci., Hokkaido Univ., Assoc.Prof., 大学院・理学研究科, 助教授 (80153245)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
IGARASHI Masayuki Sci.Univ.of Tokyo, Fac.Ind.Sci.of Tech., Lect., 基礎工学部, 講師 (60256675)
MINAKAWA Hiroyuki Grad.School of Sci., Hokkaido Univ., Assistant, 大学院・理学研究科, 助手 (30241300)
KAWAZUMI Nariya Grad.School of Sci., Hokkaido Univ., Assoc.Prof., 大学院・理学研究科, 助教授 (30214646)
ISHIKAWA Goo Grad.School of Sci., Hokkaido Univ., Assoc.Prof., 大学院・理学研究科, 助教授 (50176161)
YAMAGUCHI Keizo Grad.School of Sci., Hokkaido Univ., Prof., 大学院・理学研究科, 教授 (00113639)
森吉 仁志 北海道大学, 大学院・理学研究科, 助教授 (00239708)
泉屋 周一 北海道大学, 大学院・理学研究科, 教授 (80127422)
|
|---|
| Project Period (FY) |
1997 – 1998
|
| Project Status |
Completed (Fiscal Year 1998)
|
| Budget Amount *help |
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,700,000 (Direct Cost: ¥1,700,000)
|
|---|
| Keywords | Integrable geodesic flow / Integrable system / Hamiltonian mechanics / Symplectic geometry / Riemannian geometry / Liouville manifolds / Geodesic flow / Semiclassical approximation / クウヴィル多様体 |
|---|
| Research Abstract |
For "classical theory" we have obtained two major results. One of them may be stated as follows : Let M be a riemannian manifold diffeomorphic to 2-sphere, and let F be a first integral of its geodesic flow that is a polynomial of degree kappa on each fiber. Such a pair (M, F) is well-understood if kappa = 1,2. When kappa <greater than or equal> 3, however, no nontrivial example has been known except one-parameter families for kappa 3 and 4. In this research we constructed families of such (M, F) (parametrized by functions in one variable) for every kappa <greater than or equal> 3. Moreover, we proved that they are C_<2pi>-manifolds, i.e., every geodesic is closed and has the common length 2pi. The other result is a construction of "Hermite-Liouville structure" on Hopf surfaces. The idea is analogous to Kahler-Liouville manifold, which is a complexifled version of Liouville manifold established by the head investigator. Despite the significance of the Kahler condition in the whole theory of Kahler-Liouville manifolds, this result seems to suggest the existence of another complexification scheme for Liouvil
For "quantization" we studied spectra of the laplacian on Liouville surfaces diffeomorphic to 2-sphere. We decomposed the defining equation of the eigenfunctions into a pair of ordinary differential equations on circles, and applied semiclassical approximation to each of them. As a result, we found that this method gives a nice approximation when the corresponding invariant tori are sufficiently close to a critical one, as well as the case where the tori are located far from the critical ones. We think this result will be more refined.
|
