1998 Fiscal Year Final Research Report Summary
On the correspondences and relations betweendifferent geometric structares by twistor theory
| Project/Area Number |
09640141
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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 College of Technology |
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Principal Investigator |
MACHIDA Yoshinon Numazu College of Techuotagy, Liberal Arts, Assoc.Professor, 一般科目, 助教授 (90141895)
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| Co-Investigator(Kenkyū-buntansha) |
KAWADA Hiroyuki Liberal Arts, Assis.Professor, 一般科目, 講師 (00249799)
AIHARA Yoshinori Liberal Arts, Assoc.Professor, 一般科目, 助教授 (60175718)
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| Project Period (FY) |
1997 – 1998
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| Keywords | twistor theory / geometric structure / Grassmannian structure / self-duality / Monge-Ampere equation |
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| Research Abstract |
An aspect of the twistor theory is to know the relations and correspondences between different geometric structures defined by a double fibration.
1. Grassmannian structures : It is interesting to study the correspondences between Grassmannian structures of type (n, 2) and projective structures. We showed that. after we defined a tautological distribution on the null plane bundle, the distribution is completely integrable if and only if the structure is half-f lat. And then we showed the lifting theorem, the reduction theorem and the twistor theorem on the twistor theory of Grassmannian structures.
2. Self-dual octonian structures : On 8-dimensional manifolds with Spin (7) structures, the notion of self-duality is defined. We can construct 12-dimensional twistor spaces with fiber S*4. We showed that the structure is self-dual if and only if the twistor space has a semi-integrable quaternion structure.
3. The fundamental solutions of Laplace equations : We studied the twistor integral representation of the fundamental solution of the (complex) Laplace equation on the flat complex space-time. It is represented by integrating some closed differential form associated with the notion of tree in graph theory.
4. Monge-Ampere equations : We defined a remarkable class called decomposable Mange-Ampere equations in more than three independent variables. We showed that we can associate to the class the characteristic systems.
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