Development of analysis and discretization in differential geometry
| Project/Area Number |
20K03585
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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 | Kobe University |
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Principal Investigator |
Rossman W.F 神戸大学, 理学研究科, 教授 (50284485)
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| Co-Investigator(Kenkyū-buntansha) |
安本 真士 徳島大学, 大学院社会産業理工学研究部(理工学域), 講師 (70770543)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Project Status |
Discontinued (Fiscal Year 2022)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2023: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2022: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 離散的微分幾何学 / 離散曲面 / 離散曲線 / 特異点 / Darboux変換 / differential geometry / surface theory / discretization / integrable systems / analysis |
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| Outline of Research at the Start |
本研究課題は，現代微分幾何の根幹をなす曲面の微分幾何学を，多角的なアプローチを用いて離散化し，離散化された曲面を解析する手法を確立することを目的とする．微分幾何的対象の離散化は，純粋数学だけでなく，CGや材料工学などの関連諸分野からも高い注目を集めている．一方，上記の研究分野を研究する際には，従来の数学研究をそのまま適用するだけでは不十分であり，これまでの微分幾何を，離散的な土台のもとで再整備・再構築することが求められている．
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| Outline of Annual Research Achievements |
This is research on discrete curves and surfaces established by the use of transformation theory, a theory that can be viewed as a discrete construction within the smooth category that allows for natural notions of discretization. Previously understood discrete isothermic surfaces have been extended here to the wider class of discrete Omega surfaces, together with a transformation theory for this wider class that preserves underlying structures in the corresponding smooth case. In particular, Darboux transformations for the full class have been developed. Additionally, discretization of the potential mKdV equation has been seen from the perspective of Darboux transformations of curves. Also, in a related topic, singularities and signature changes of smooth surfaces in de Sitter 3-space have been explored, with particular application to various catenoids in that space.
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Report
(3 results)
Research Products
(42 results)
