Geometric structures which admits singular points and the realization problem

2018 Fiscal Year Final Research Report

• Project/Area Number: 16K17605
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Yokohama National University (2017-2018); Miyakonojo National College of Technology (2016)
• Principal Investigator: HONDA Atsufumi 横浜国立大学, 大学院工学研究院, 准教授 (90708611)
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: 曲面 / 特異点 / 波面 / カスプ辺 / ツバメの尾 / Kossowski計量 / 連接接束 / 等長実現

Outline of Final Research Achievements

When we study surfaces under some natural curvature conditions, such as surfaces of constant Gaussian/mean curvature, singular points frequently appear on such surfaces. Recently, based on such a differential geometric study of surfaces with singular points, the theory of intrinsic geometry with singularities is progressing rapidly. Although the isometric realization problem of positive semi-definite metrics as surfaces with singular points is one of the fundamental problem, the problem was solved only in a specified case. In this study, we proved every real analytic Kossowski metric can be isometrically realized locally, where Kossowski metrics are positive semi-definite metric modeled on induced metrics of wave fronts. Including such a theorem, we obtained several fundamental results on surfaces with singular points, and intrinsic geometry with singularities.

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究では，Kossowski計量という半正定値計量は実解析的な場合に局所等長実現可能であることを示すなど，特異点を持つ曲面の幾何学や特異点付きの内在的幾何学の理論における中心的な結果を得た．このような結果は，特異点を持つ曲面の理論を内在的な半正定値計量の幾何学に一般化しているため今後の応用が期待される．すでにいくつかの結果はローレンツ多様体の混合型曲面の型変化の理論の構築に応用されており，本研究で得られた結果は，理論物理等を含む広範な分野へも応用されることが期待できると思われる．