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1999 Fiscal Year Final Research Report Summary

Study of geometric structures on projective submanifolds in view of differential equations

Research Project

Project/Area Number 09304010
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKobe University

Principal Investigator

SAKAI Takeshi  Kobe Univ, Fac of Sci, Prof, 理学部, 教授 (00022682)

Co-Investigator(Kenkyū-buntansha) KANEYUKI Souji  Sophia Univ, Fac of Sci, Prof, 理工学部, 教授 (40022553)
TAKANO Kyoichi  Kobe Univ, Fac of Sci, Prof, 理学部, 教授 (10011678)
TAKAYAMA Nobuki  Kobe Univ, Fac of Sci, Prof, 理学部, 教授 (30188099)
YAMAGUCHI Keizo  Hokkaido Univ, Grad. School of Sci, Prof, 大学院・理学研究科, 教授 (00113639)
YOSHIDA Masaaki  Kyushu Univ, Grad. School of Math, Prof, 大学院・数理学研究科, 教授 (30030787)
Project Period (FY) 1997 – 1999
Keywordsprojective submanifold / hypergeometric system / contact geometry / projective differential geometry / projective invariants / affine differential geometry / integrable system / line congruence
Research Abstract

The result of the project is summarized as follows :
1.Publishing of notes on the projective differential geometry of curves, scurfaces, and hypersurfaces, classification of projectively homogeneous surfaces, and a study of correspondences between line congruence and Laplace transforms of projective surfaces.
2.Study of hypergeometric differential equations from several point of views ; say, a study of projective surfaces defined by Appell's systems FィイD22ィエD2 and FィイD24ィエD2, determination of the uniformizing equation associated with the moduli space of cubic surfaces, a new formulation of intersection theory, explicit presentation of fundamental solutions of the system E(3,6), a combinatorial study of A-hypergeometric systems, and algorithmic study of D-modules.
3.Investigation of the relation holding between Painlevequation and Backlund transformation, characterization of Painleve equation in terms of Hamiltonian structure, and description of timelike Bonnet surfaces by Paileve equation.
4.Contact geometric study of 3rd order ordinary differential equations by introducing differential invariants and formulation of Schwarzian derivatives related with contactomorphisms.
5.Study of various geometric structures ; e.g., projective flat structures on homogeneous spaces, Weyl-Yang-Mills connection para-kahler structures on affine symmetric spaces, Hesse structures, and so on.
6.Study of 3-dimensional manifolds using DS-diagram and harmonic analysis on graphs.
7.Study of special surfaces such as minimal surfaces, surfaces with constant mean curvature and, further, a study of nilpotent manifolds and knot theory.
Along with the studies above, new computer systems for algebraic computation was developed, by which mathematically rigorous computer support was available for our study.

  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] T. Sasaki: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc. Japan Acad., Ser A. 75. 129-133 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] B. Strumfels: "Grobner Bases and Applications"London Math Soc Lecture Notes. 251. 246-258 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Matano: "On some Hamiltonian structure of Painleve systems II"J. Math. Soc. Japan. 51. 843-866 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] W. Rossman: "A New Flux for Mean Curvature 1 Surfaces in Hyperbolic 3-space, and Applications"Proc. Amer. Math. Soc.. 12. 2147-2154 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M. Yoshida: "A hyperbolic structure on the real locus of the moduli space of marked cubic surfaces"Topology. (in press).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] H. Shima: "Homogeneous spaces with invariant projectively flat affine connections"Trans. American Math. Soc.. 351. 4713-4726 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M. Saito: "Grobner Deformations of Hypergeometric Differential Equations"Springer Verlag. 254 (199)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T. Sasaki: "Projective Differential Geometry and Linear Homogeneous Differential Equations"Rokko Lectures in Math., vol. 5. 115 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Sasaki and M.Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc. Japan Acad., Ser A. 75. 129-133 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Sasaki: "Projective Differential Geometry and Linear Homogeneous Differential Equations"Rokko Lectures in Math. vol.5. 115 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] B.Strumfels and N.Takayama: "Grobner Bases and Applications"London Math Soc. Lecture Notes. 251. 246-258 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Matano, A.Matumiya and K.Takano: "On some Hamiltonian structures of Painleve systems II"J. Math Soc. Japan. 51. 843-866 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] W.Rossman, M.Umehara, and K.Yamada: "A New Flux for Mean Curvature 1 Surfaces in Hyperbolic 3-space, and Applications"Proc. Amer. Math. Soc.. 12. 2147-2154 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Yoshida: "A hyperbolic structure on the real locus of the moduli space of marked cubic surfaces"Topology. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Shima: "Homogeneous spaces with invariant projectively flat affine connections"Trans. American Math. Soc.. 351. 4713-4726 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] H.Urakawa: "On invariant projectively flat affine connections"Hokkaido Math. J.. 28. 333-356 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ichiyama, H.Huruhata and H.Urakawa: "A conformal gauge invariant functional for Weyl structures and the first variation formula"Tsukuba J. Math.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Gindikin and S.Kaneyuki: "On the automorphism group of the generalized conformal structure of a symmetric R-space"Differential Geom. Appl.. 8. 21-33 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Sakane: "Homogeneous Einstein metrics on flag manifolds"Lobachevskii J. Math.. 4. 71-87 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ozawa and H.Sato: "Linearizations of ordinary differential equations by area preserving maps"Nagoya. Math. J.. 156. 109-122 (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2001-10-23  

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