2013 Fiscal Year Final Research Report
Applications to differential geometry and singularity theory of differential equations
| Project/Area Number |
23740041
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Muroran Institute of Technology |
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Principal Investigator |
TAKAHASHI MASATOMO 室蘭工業大学, 工学(系)研究科(研究院), 准教授 (80431302)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 特異点 / 微分方程式 / 微分幾何学 / 完全解 / ルジャンドル曲線 / 双対性 / 縮閉線 / 伸開線 |
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| Research Abstract |
As applications of singularity theory, we have studied a qualitative theory for implicit ordinary differential equations. We gave a generic classification of semi-local first order ordinary differential equations of Clairaut type. Moreover, we defined complete solutions for singular first order ordinary differential equations and gave its an existence condition. We also gave types of completely integrable implicit second order ordinary differential equations.
As applications to differential geometry, we have studied a theory of Legendre curves. We gave the existence and the uniqueness for Legendre curves via Legendre curvatures. By using the Legendre curvature, we investigated the evolutes and the involutes. Moreover, we gave a generic classifications of tangent surfaces associated with G2 and D4 geometries.
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