Project/Area Number  01302002 
Research Category 
GrantinAid for Cooperative Research (A).

Research Field 
代数学・幾何学

Research Institution  Okayama University 
Principal Investigator 
SAKAI Takashi Okayama U., Fac. Sci., Professor, 理学部, 教授 (70005809)

CoInvestigator(Kenkyūbuntansha) 
KASUE Atushi Osaka U., Fac. Sci., Associate Prof., 理学部, 助教授 (40152657)
NISHIKAWA Seiki Tohoku U., Fac. Sci., Associate Prof., 理学部, 助教授 (60004488)
市田 良輔 横浜市立大学, 文理学部, 助教授 (10094294)
SATO Hajime Nagoya U., Col. of Gen. Ed., Professor, 教養部, 教授 (30011612)
TANNO Shukichi Tokyo Inst. Tech., Fac. Sci., Professor, 理学部, 教授 (10004293)
OHNITA Yoshihiro Tokyo Metro. U., Fac. Sci., Assistant, 理学部, 助手 (90183764)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1990 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1989 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  Geometric Structure / Manifold Structures / Curvature and Topology / 幾何構造 / 多様体構造 / 曲率と位相 
Research Abstract 
The purpose of the present research project is to investigate collectively various properties of geometric structures on manifolds and their relations to the manifold structures, in the cooperation of many geometers in Japan. As for the metric structures, T.Sakai considered the inequalities which hold between metric invariants with respect to measure and get results on the isosystolic inequality and isodiametric inequality for compact surfaces. Now series of Gromov's outstanding works are having a great influence upon the study on the relation between the metric structures and the manifold structures. A. Kasue gave a proof of Gromov's convergence theorem and applied it to the study of the structure of manifolds of asymptotically nonnegative curvature. He also constructed the coordinates at infinity of ALEmenifolds under some geometric condition with S.Bando and H. Nakajima. K. Fukaya and T.Yamaguchi get decisive results on the structure of almost nonpositive and almost nonnegative c
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urvature using excellent techniques extending Gromov's ideas. As for the other geometric structures, the study of contact metric structures was initiated and has been done mainly in Japan. S. Tanno gave a characterization of the canonical contact metric structures associated with the contact form from the variational view point. H.Sato proposed to study Lie's sphere geometry from the view point of Gstructure and applied Tanaka's theory with K. Yamaguchi. On the other hand, one of the main trends of the recent progress on the study of geometric structures, for examples the study of minimal surfaces and harmonic mappings, has been made by applying new techniques from the analysis. S. Nishiwaka applied the heat equation method to the study of geometric foliations on the manifolds. As for harmonic mappings we have the researches by A. Kasue, H. Naito and others from the analytic view point and the researches by Y. Ohnita and others from the geometric view point. Now under the present Grantinaid for scientific research, we organized the following symposiums: "Surveys in Geometry  Minimal surfaces" (Univ. of Tokyo), "Variational problems appearing in geometry" (International Center of Osaka City Univ.), "Geometric structures and manifold structures" (Okayama niv.). We distributed printed research materials to the participants and had active and fruitful discussions. Less
