2000 Fiscal Year Final Research Report Summary
Integrable geodesic flows and semi-classical approximations
| Project/Area Number |
11640053
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | HOKKAIDO UNIVERSITY |
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Principal Investigator |
KIYOHARA Kazuyoshi Hokkaido Univ., Grad.School of Sci., Assoc.Prof., 大学院・理学研究科, 助教授 (80153245)
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| Co-Investigator(Kenkyū-buntansha) |
IGARASHI Masayuki Sci.Univ.of Tokyo, Fac.Ind.Sci.of Tech., Lect., 基礎工学部, 講師 (60256675)
ISHIKAWA Goo Hokkaido Univ., Grad.School of Sci., Assoc.Prof., 大学院・理学研究科, 助教授 (50176161)
IZUMIYA Shuichi Hokkaido Univ., Grad.School of Sci., Prof., 大学院・理学研究科, 教授 (80127422)
TSUKAMOTO Chiaki Kyoto Kougei-Sen-i Univ., Fac.of Textile, Assoc.Prof., 纎維学部, 助教授 (80155340)
SUGAHARA Kunio Osaka Kyouiku Univ., Fac.of Edu., Prof., 教育学部, 教授 (20093255)
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| Project Period (FY) |
1999 – 2000
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| Keywords | Integrable geodesic flow / Integrable system / Hamiltonian mechanics / Symplectic geometry / Riemannian geometry / Liouville manifolds / Geodesic flow / Semiclassical approximation |
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| Research Abstract |
We investigated the structures of Kahler-Liouville manifolds which are not necessarily of type (A). As a consequence, we showed that every compact, proper Kahler-Liouville manifold has a bundle structure such that the fiber is a Kahler-Liouville manifold whose geodesic flow is integrable, and the base is (locally) a product of one-dimensional Kahler manifolds. The fiber naturally possesses the structure of toric variety. Also, we obtain another class, called of type (B), of Kahler-Liouville manifolds whose geodesic flows are integrable.
Also, we obtained many examples of "Hermite-Liouville" manifolds whose geodesic flows are integrable. The first examples are those defined over Hopf surfaces. Others are those defined over complex projective spaces. In the latter case, the idea of construction is as follows. We use two structures of real Liouville manifolds on the real projective space. One of them is used to prepare the "frame of complexification", while the other is used to produce a comlexified metric and first integrals. If those two Liouville structures are the same, then the result becomes Kahler-Liouville manifold.
Moreover, 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 them. As a result, we found that this method gives new approximations when the corresponding invariant tori tend to a critical one.
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