Project/Area Number  09304010 
Research Category 
GrantinAid for Scientific Research (A).

Section  一般 
Research Field 
Geometry

Research Institution  Kobe University 
Principal Investigator 
SAKAI Takeshi Kobe Univ, Fac of Sci, Prof, 理学部, 教授 (00022682)

CoInvestigator(Kenkyūbuntansha) 
YAMAGUCHI Keizo Hokkaido Univ, Grad. School of Sci, Prof, 大学院・理学研究科, 教授 (00113639)
志磨 裕彦 山口大学, 理学部, 教授 (70028182)
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)
YOSHIDA Masaaki Kyushu Univ, Grad. School of Math, Prof, 大学院・数理学研究科, 教授 (30030787)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥19,900,000 (Direct Cost : ¥19,900,000)
Fiscal Year 1999 : ¥4,600,000 (Direct Cost : ¥4,600,000)
Fiscal Year 1998 : ¥5,800,000 (Direct Cost : ¥5,800,000)
Fiscal Year 1997 : ¥9,500,000 (Direct Cost : ¥9,500,000)

Keywords  projective submanifold / hypergeometric system / contact geometry / projective differential geometry / projective invariants / affine differential geometry / integrable system / line congruence / 射影部分多様体 / 超幾何系 / 接触幾何学 / 射影微分幾何学 / 射影不変量 / アファイン幾何学 / 可積分系 / 線叢 / projective invariant / Hesse geometry 
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 Ahypergeometric systems, and algorithmic study of Dmodules. 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, WeylYangMills connection parakahler structures on affine symmetric spaces, Hesse structures, and so on. 6.Study of 3dimensional manifolds using DSdiagram 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.
