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测度的不等式，得到了紧致曲面的等收缩不等式和等径不等式的结果。格罗莫夫的一系列杰出作品对研究度量结构与流形结构之间的关系产生了很大的影响。a . Kasue给出了Gromov收敛定理的证明，并将其应用于渐近非负曲率流形结构的研究。他还与S.Bando和H. Nakajima在某些几何条件下构造了ale - menfolfold的无穷远坐标。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…更多的不确定性使用了扩展Gromov思想的优秀技术。至于其他几何结构，接触度量结构的研究是最早的，主要是在日本进行的。S. Tanno从变分的角度给出了与接触形式相关的标准接触度量结构的表征。佐藤提出从g结构的角度研究李氏球几何，并与山口将田中的理论应用于李氏球几何。另一方面，近年来几何结构研究的主要进展之一，例如最小曲面和调和映射的研究，是通过应用分析中的新技术而取得的。Nishiwaka将热方程方法应用于流形几何叶状的研究。对于调和映射，我们有A. Kasue、H. Naito等人从解析的角度进行的研究，也有Y. Ohnita等人从几何的角度进行的研究。在目前的科学研究资助下，我们组织了以下研讨会：“几何调查-最小曲面”（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”（冈山niv.）。我们向与会者分发了印刷的研究资料，并进行了积极而富有成果的讨论。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