| Project/Area Number |
14540097
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Numazu National College of Technology |
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Principal Investigator |
MACHIDA Yoshinori Numazu National College of Technology, Liberal arts, assistant professor, 教養科, 助教授 (90141895)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Hajime Nagoya university, Graduate school of mathematics, professor, 多元数理科学研究科, 教授 (30011612)
石川 剛郎 北海道大学, 大学院・理学研究科, 助教授 (50176161)
藤井 一幸 横浜市立大学, 理学部, 教授 (00128084)
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| Project Period (FY) |
2002 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2002: ¥700,000 (Direct Cost: ¥700,000)
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| Keywords | twistor theory / Monge-Ampere equation / Goursat equation / Clairaut equation / instanton / Legendre singularity theory / モンジュ・アンペール方程式 / 2階常微分方程式 / ルジャンドル測地線 / 非ホロノーム分布 / モンジュ・アンペール系 / ラグランジェ部分空間対 / 3次コーン構造 / カルタン接続 / ルジャンドル観測線 / 双対性 / 超幾何方程式 / ラプラス方程式 / 半平坦部分接続 / 双曲構造 / 非可換性 |
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| Research Abstract |
We regard the twister theory as the correspondence via a double fibering, which is the duality between geometric structures on different spaces. From this point of view, we applied to the study of the essence and the construction of equations themselves and solutions for various differential equations and integrable field theory.
1.Extending Monge-Ampere equations which are Hessian=const. and Gaussian curvature=const., we intrinsically defined and studied Monge-Ampere systems with Lagrangian pair on contact manifolds. In the case of 5, 7 dimensins, we saw that generic geometric solutions have four and eleven kinds of singularities respevtively via Legendre duality.
2.A Goursat equation is a second order PDE whichi is of parabolic type and the Monge characteristic of which is completely integrable. It has the interpretation of twistor theory. We constructed equations themselves by Lagrange-Grassmann duality and solutions by Cartan-Legendre duality.
3.We saw that Clairaut equations are nothing but the essence of twistor theory. We considered the various extention by the method of twistor theory.
4.Nondegenerate type type (4,7) distributions are of finite type unlike contact structures. We constructed the normal Cartan connections and they have two curvature invariants. Hyperbolic type distributions have relation to Legendre geodesies via twistor theory.
5.We define SU(3) type U(1) instantons on 6-dimensional manifolds with SU(3) structure.
In particular, in the case of nearly Kahler structure, we constructed SU(3) type U(1) instantons on Hopf bundles of CP^3 and F_{12} which are the twistor space of S^4 and CP^2 respectively.
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