| Project/Area Number |
17K05238
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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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| Research Field |
Geometry
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| Research Institution | Muroran Institute of Technology |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
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| Keywords | 特異点論 / 微分幾何学 / 微分方程式 / 包絡面 / 枠付き曲線 / 枠付き曲面 / ルジャンドル特異点論 / ラグランジュ特異点論 / 双対性 / クレロー型微分方程式 / アファイン曲線 / 不変量 / 曲面論 / 曲線論 / 曲率 / 幾何学 |
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| Outline of Final Research Achievements |
As study on surfaces with singular points and singlarity theory, we have studied framed surfaces and one-parameter families of framed curves. We gave relations between framed surfaces and one-parameter families of framed curves. Moreover, we investigated envelopes of one-parameter families of framed curves, one and two-parameter families of framed surfaces and families of Legendre surfaces.
As applications, we gave conditions that singular solutions of completely integrable first order ordinary and partial differential equations are envelopes.
Moreover, we gave a generic classification of bifurcations of Lagrangian submanifold germs by using a relation between equivelence relations of families of Lagrangian submanifold germs and graphlike Legendrian unfoldings.
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| Academic Significance and Societal Importance of the Research Achievements |
特異点を許容する曲線、曲面論の研究は、現在進行形で発展しつつあります。それは、既存の正則では現れない現象や理論の拡張を考察する必要性と様々な不変量や特異点の型の判定等の道具が揃ってきたことによります。本研究課題では特異点を許容する曲線、曲面として枠付き曲線、枠付き曲面と捉えることによりユークリッド変換の意味で完全不変量を得ることができましたので、微分幾何学的研究を行いました。また、特異点の型や微分方程式の特異解が包絡面となる条件を求めました。
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