| Project/Area Number |
20K03617
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Tokyo Denki University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2025-03-31
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| Project Status |
Completed (Fiscal Year 2024)
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| Budget Amount *help |
¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2022: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
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| Keywords | 微分幾何 / 広義の正則性 / 解析的拡張 / 特異点 / 曲率 / 平均曲率 / 平坦 / 波面 / 対称性 / 正則性 / 微分幾何学 / 曲面 |
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| Outline of Research at the Start |
曲面に関する数学の研究は長い歴史を持ち，正則曲面（滑らかな曲面）について非常にたくさんの研究があります．その研究対象を少し広い範囲に拡げて，研究に取り組むことが目的です．現在，広義正則曲面（得点を持つ曲面，異質な性質の混在する曲面）については何人かの研究者により様々な興味深い結果が導かれているところです．それらの本質的な部分に着目して，より体系的な理論構築への寄与を目指します．
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| Outline of Final Research Achievements |
In addition to ordinary smooth surfaces, we focused on surfaces containing singularities and studied them geometrically and analytically as regular surfaces in a wider sense.In particular, for a catenoid of mean curvature 1 in 3-dimensional de Sitter space, we showed the existence of analytic extensions beyond the constraints of space and that they are analytically maximal.
In hyperbolic spaces, flat wavefronts with regular polyhedral symmetry were constructed, and five specific examples and their geometric features were identified. Moreover, new concepts such as analytic completeness and double cone manifolds were introduced to deepen our understanding of the structure of surfaces with singularities. We also discovered surfaces for which Gauss curvature contours are concentric circles. These results were made public through papers and conference presentations.
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| Academic Significance and Societal Importance of the Research Achievements |
本研究では、特異点を許容する広義の正則曲面という新たな視点を導入し、従来の滑らかな曲面理論を拡張した。特に、de Sitter 空間内のカテノイドの解析的極大性の証明や、双曲空間における対称性をもつ波面の構成は、曲面の微分幾何に新たな知見を加えるものである。また、「解析的完備性」や「二重錐多様体」といった概念の提案は、複素解析や位相幾何との接点を広げ、今後の理論展開の足がかりとなる。さらに、Gauss 曲率の等高線構造に注目した研究は、曲面分類への新たな手がかりを提供している。
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