Geometric structures on submanifolds in space forms and differential equations related to their structures.
| Project/Area Number |
16340020
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Fukuoka University |
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Principal Investigator |
SUYAMA Yoshihiko Fukuoka University, Fac. Sci., Professor (70028223)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOHAMA Katsuhiro Fukuoka University, Fac. Sci., Part-time Lecturer (20016059)
KUROSE Takashi Fukuoka University, Fac. Sci., Professor (30215107)
HAMADA Tatsuyoshi Fukuoka University, Fac. Sci., Assistant Professor (90299537)
KAWAKUBO Satoshi Fukuoka University, Fac. Sci., Assistant Professor (80360303)
MATSUURA Nozomu Fukuoka University, Fac. Sci., Assistant Professor (00389339)
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| Project Period (FY) |
2004 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥9,690,000 (Direct Cost: ¥9,000,000、Indirect Cost: ¥690,000)
Fiscal Year 2007: ¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2006: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2004: ¥2,300,000 (Direct Cost: ¥2,300,000)
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| Keywords | conformally flat hypersurface / Mobius Geometry / warped Product manifold / Affine Geometry / Kirchhoff elastic rod / integrable systems / Guichard net / variational methods / 共形不変量 / アフィン超球面 / 統計多様体 / キルヒホッフ弾性捧 / 離散可積分系 |
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| Research Abstract |
(1) Suyama obtained remarkable results in the research on generic conformally flat hypersurfaces. The subject is an open problem since the work by E. Cartan. Suyama and Hertrich-Jeromin (Bath Univ.) constructed all conformally flat hypersurfaces with cyclic Guichard net and gave a complete classification of them by conformal equivalence. This work gives new development in the research on higher dimensional submanifolds from the integrable system view-points.
(2) Shiohama studied the relation between radial curvature and topology of warped product manifolds. He gave Alexandrov-Toponogov comparison theorem for various geodesic triangles, and applied it for a classification of warped product manifolds.
(3) Kurose studied the method of constructing improper affine hyperspheres and gave a new representation formula and a characterization of the class of improper affine hypersurfaces obtained by this formula.
(4) Kurose and Fujioka gave a representation as a Hamiltonian system to the motions of curves in a complex hyperbola associated with the integrable equations of the Burgers hierarchy.
(5) Kawakubo studied Kirchhoff elastic rods in three-dimensional space forms, and obtained the explicit formulas for them. Also, he proved that the energy functional for Kirchhoff elastic rods satisfies the Palais-Smale condition. The result gives an effective method of studying the stability of closed Kirchhoff elastic rods.
(6) Matsuura studied discretizations of curves and surfaces in connection with the theory of discrete integrable systems. He made a geometric interpretation of the discrete KdV equation, so that he successfully proposed a non-autonomous version of it.
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Report
(5 results)
Research Products
(80 results)
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[Presentation] q-正規分布とエスコート分布2007
Author(s)
田中 勝
Organizer
福岡大学微分幾何学研究会
Place of Presentation
福岡大学セミナーハウス
Year and Date
2007-02-02
Description
「研究成果報告書概要(和文)」より
Related Report
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[Presentation] 正則統計多様体とアフィン幾何2005
Author(s)
黒瀬 俊
Organizer
福岡大学微分幾何学研究会
Place of Presentation
福岡大学セミナーハウス
Year and Date
2005-10-08
Description
「研究成果報告書概要(和文)」より
Related Report
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[Presentation] 離散KdV方程式2004
Author(s)
松浦 望
Organizer
第51回幾何学シンポジウム
Place of Presentation
東京都立大学
Year and Date
2004-08-08
Description
「研究成果報告書概要(和文)」より
Related Report
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