2007 Fiscal Year Final Research Report Summary
Problems related to integrable geodesic flows
| Project/Area Number |
18540087
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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 University, Grad. School of Nat. Sci. Tech., Professor (80153245)
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| Co-Investigator(Kenkyū-buntansha) |
ITOH Jin-ichi Kumamoto Univ., Fac. Edu., Prof (20193493)
IGARASHI Masayuki Sci. Univ. of Tokyo, Fac. Ind. Sci. Tech., Prof (60256675)
SAKAI Takashi Sci. Univ. Okayama, Fac. Sci, Prof (70005809)
KATSUDA Atsushi Okayama Univ., Grad. School of Nat. Sci. Tech, Assoc. Prof (60183779)
IKEDA Akira Okayama Univ., Fac Edu., Prof (30093363)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Ellipsoid / Integrable geodesic flow / Cut locus / Conjugate locus / Cusp / Kahler-Liouville / Hermite-Liouville / Liouville manifold |
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| Research Abstract |
It is well known that the geodesic flow of ellipsoid is completely integrable. We studied in this research much finer properties of it. One of the results we obtained is the determination of the cut loci for any points; they are closed balls of codimension one for general points and those of codimension two for special points. The other result is the clarification of the structure of the first conjugate loci for general points. In particular, we showed that the set of singularities of conjugate locus of a general point consists of three connected component and each component (an open and dense part) is a cuspidal edge. This is a higher dimensional version of the so-called Jacobi's last geometric statement: "The conjugate locus of any non-umbilic point on two-dimensional ellipsoid has exactly four cusps". The main ingradient in the proofs of those results is the detailed investigation of Jacobi fields and their zeros. Moreover, we showed that the above results equally hold for some Liouville manifolds.
Also, we investigated local structures of Hermite-Liouville manifolds and clarified them completely, even when they do not have the action of infinitesimal automor-phisms as for the case of Kahler-Liouville manifolds. Moreover, we illustrated a way of local construction of Hermite-Liouville manifolds in the case of having infinitesimal automorphisms, which almost corresponds to a global construction of Hermite-Liouville manifolds on complex projective spaces. In this construction, one can easily check which one is Kahler-Liouville and which one is not.
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