2002 Fiscal Year Final Research Report Summary
Research of Systems of Partial Differential Equations from the view point of Contact Geometry
| Project/Area Number |
11304002
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | HOKKAIDO UNIVERSITY |
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Principal Investigator |
YAMAGUCHI Keizo Hokkaido Univ. Grad School of Sci., Prof., 大学院・理学研究科, 教授 (00113639)
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| Co-Investigator(Kenkyū-buntansha) |
ISHIKAWA Goo Hokkaido Univ. Grad School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (50176161)
KIYOHARA Kazuyoshi Hokkaido Univ. Grad School of Sci., Asso. Prof., 大学院・理学研究科, 助教授 (80153245)
IZUMIYA Shuichi Hokkaido Univ. Grad School of Sci., Prof., 大学院・理学研究科, 教授 (80127422)
SASAKI Takeshi Kobe Univ., Fac. of Schi., Prof., 理学部, 教授 (00022682)
SATO Hajime Nagoya Univ. Grad. School of Poly Math., Prof., 大学院・多元数理科学研究科, 教授 (30011612)
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| Project Period (FY) |
1999 – 2002
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| Keywords | Contact transformations / Monge-Ampere equations / Systems of higher order partial differential equations of finite type / Harmonic mappings and Integrable systems / Geodesic flows and Integrade systems / Gauss-Schwarz theory |
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| Research Abstract |
The purpose of this project is to study systems of partial differential equations as geometric objects, i.e., as submanifolds of Jet spaces, from the view points of differential geometry and singularities theory, the central theme of which is the contact equivalence problems of systems of differential equations.
For the last year of the project, we did our research to summarize the following our original 6 projects :
(1) Contact equivalence problem of systems of partial differential equations of second order for one unknown function. Especially the research of G2type partial differential equations of second order d'apres E. Cartan.
(2) Formation of shock wave solutions and singularities of solutions of Monge-Ampere equations.
(3) The research of graded Lie algebras induced from symbols of partial differential equations and the contact equivalence of systems of higher order partial differential equations of finite type.
(4) Application of the equivalence problem for linear partial differential equations of finite type to projective submanifolds theory and Gauss-Schwarz theory.
(5) Characterization of the notion of genre in exterior differential systems in terms of Web geometry.
(6) The research of riemannian monifolds whose geodesic flows are completely integrable.
The head investigator summarized the content of (1) in "G2-geometry of overdetermined systems of second order". Izumiya summarized the content of (2) in the journal "Sugaku Exposition". The contents of (3) was summarized by the head investigator and Yatsui in "Geometry of higher order differential equations of finite type associated with Symmetric spaces". As for the content of (4), Sato and Ozawa contributed to construct "Schwarzian derivative" in case of contact diffeomorphisms. The content of (6) was summarized by Kiyohara in "On Kahler-Liouville manifolds".
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