Geometric Structures and Manifold Structures

本研究项目的目的是在日本许多几何学家的合作下，共同研究流形上几何结构的各种性质及其与流形结构的关系。对于度量结构，T.Sakai考虑了度量不变量之间关于度量的不变项，得到了紧致曲面上的等晶不等式和等径不等式的结果。目前，格罗莫夫的一系列优秀著作对度量结构和流形结构之间的关系的研究产生了重大影响。A.Kasue给出了Gromov收敛定理的一个证明，并将其应用于渐近非负曲率流形的结构研究。在一定的几何条件下，他还与S.Bando和H.Nakajima一起构造了A-M流形在无穷远处的坐标。K.Fukaya和T.Yamaguchi在几乎非正和几乎非负c…的结构上得到了决定性的结果对于其他几何结构，接触度规结构的研究起步较早，主要在日本进行。S.Tanno从变分的角度刻画了与接触形式相关的正则接触度量结构。H.Sato提出从G-结构的角度研究Lie的球面几何，并与K.Yamaguchi一起应用田中的理论。另一方面，应用分析中的新技术，取得了几何结构研究的主要趋势之一，例如极小曲面和调和映射的研究。S.Nishiwaka将热方程法应用于流形上的几何叶理的研究。对于调和映射，我们从解析的角度介绍了A.Kasue、H.Naito等人的研究成果，从几何的角度介绍了Y.Ohnita等人的研究成果。东京大学)、“几何中出现的变分问题”(大阪城市大学国际中心)、“几何结构和流形结构”(Okayama Niv.)。我们向与会者分发了印刷的研究材料，进行了积极和富有成果的讨论。较少

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