2005 Fiscal Year Final Research Report Summary
Integrable geodesic flows and related problems
| Project/Area Number |
16540069
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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 | Okayama University |
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Principal Investigator |
KIYOHARA Kazuyoshi Okayama Univ., Grad School of Natural Sci., Prof., 大学院・自然科学研究科, 教授 (80153245)
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| Co-Investigator(Kenkyū-buntansha) |
KATSUDA Atsushi Okayama Univ., Grad School of Natural Sci., Assoc.Prof., 大学院・自然科学研究科, 助教授 (60183779)
IKEDA Akira Okayama Univ., Fac.Edu., Prof., 教育学部, 教授 (30093363)
SAKAI Takashi Okayama Univ.Sci., Fac.Sci., Prof., 理学部, 教授 (70005809)
ITOH Jin-ichi Kumamoto Univ., Fac.Edu., Prof., 教育学部, 教授 (20193493)
IGARASHI Masayuki Sci.Univ.Tokyo, Fac.Ind.Sci.of Tech., Assoc.Prof., 基礎工学部, 助教授 (60256675)
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| Project Period (FY) |
2004 – 2005
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| Keywords | ellipsoid / cut locus / conjugate locus / Liouville manifold / Jacobi / Integrable geodesic flow / tri-axial ellipsoid / quadratic surface |
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| Research Abstract |
We made a series of researches on "cut locus". First, we showed that the cut locus of any point on two-dimensional ellipsoids which is not an umbilic point is a segment of the curvature line passing through the antipodal point. Moreover, we proved that the conjugate locus of that point has exactly four cusps, and they appear on the two curvature lines passing through the antipodal point. The latter result was stated in Jacobi's "Lectures on dynamical systems" in the case of rotational ellipsoids, and had remained unproved.
Secondly, we showed that on certain Liouville surfaces including the ellipsoids the cut locus of a general point is "simple", i.e., a curve segment in compact case, and either empty or a curve segment or two curve segments in noncompact case. In particular, in the case of (a connected component of) two-sheeted hyperboloids, it was proved that there are two cases : In one case all of the above three types of cut loci appear ; and in the other case only connected cut loci appear. Thirdly, we proved that for a certain class of Liouville manifold diffeomorphic to the sphere, the cut locus of a general point is diffeomorphic to the closed disk of codimension one.. In particular, this class contains the ellipsoids whose principal axes have distinct length.
Also, we studied "Hermite-Liouville manifolds", which are not necessarily Kaehler-Liouville manifolds, and completely determined their local structures. Among them are involved the cases where the infintesimal automorphisms are not associated. Moreover, we constructed Hermite-Liouville manifolds over the complex projective space as a complexification of real Liouville manifolds over the real projective space. Our construction involves the parameters which almost meets the local possibility.
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