Geometry of Planar 3-webs

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04838
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Ochanomizu University
• Principal Investigator: Nakai Isao お茶の水女子大学, 基幹研究院, 教授 (90207704)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: WEB 幾何学 / 曲率形式 / 位相剛性

Outline of Final Research Achievements

A simple elementary method of computation of the curvature form of singular planar 3-webs has been established. A planar web is a configuration of excessively many, more than or equal to 3, foliations of the plane, one parameter families of curves filling up the plane. This structure is seen ubiquitously in nature, for instance in configurations of waves. The canonical model for this structure is represented by 3 families of parallel lines, which is called a hexagonal 3-web. In general almost all webs differ topologically from this canonical model. The difference is measured by quantified by the so-called web curvature form. The curvature form was given by Blaschke et al. in 1930's for nonsingular webs. By this research project, the curvature was computed for arbitrarily singular webs.

Free Research Field

幾何学、微分位相幾何学、複素力学

Academic Significance and Societal Importance of the Research Achievements

Web 構造は波の重ね合わせの構造である。このような構造は自然界だけでなく、数理的考察の場面でいたるところに現れる。本研究課題の3-web の曲率形式は,WEB幾何学の研究の基礎をなしている。特異3-web に対するその計算方法の確立は、今後の一般のWEB 構造の研究への応用が期待でき、自然科学への将来の応用が期待できる。