Geometric Structures and Manifold Structures

点击翻译按钮获取中文摘要

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 non-negative curvature. He also constructed the coordinates at infinity of ALE-menifolds under some geometric condition with S.Bando and H. Nakajima. K. Fukaya and T.Yamaguchi get decisive results on the structure of almost non-positive and almost non-negative c … More 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 G-structure 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 Grant-in-aid 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