2003 Fiscal Year Final Research Report Summary
Characterization of curvatures by differential equations
| Project/Area Number |
13640074
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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 | Hiroshima University |
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Principal Investigator |
AGAOKA Toshio Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (50192894)
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| Co-Investigator(Kenkyū-buntansha) |
KANNO Hitoshi Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (00291477)
NAKAYAMA Hiromichi Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (30227970)
USAMI Hiroyuki Hiroshima University, Faculty of Integrated Arts and Sciences, Associate Professor, 総合科学部, 助教授 (90192509)
YAMAGUCHI Keizo Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00113639)
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| Project Period (FY) |
2001 – 2003
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| Keywords | curvature / principal bundle / characteristic class / Weil algebra / differential equation / protective codimension / plethysm / decomposition formula |
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| Research Abstract |
Curvatures defined on principal fiber bundles satisfy several types of differential equations. A well know example is the closedness of differential forms which express classical characteristic classes. But in general there exist other types of differential equations on the curvature. To obtain such equations, we introduce the non-commutative version of the Weil algebra, and by using this formulations we show that all differential equation of curvatures are "basic", i.e., they can be pulled down to the base manifolds.
In order to characterize curvatures by differential equations, we must consider higher order equations. It is known that in the case where the structure group is semi-simple, second order equations are enough to characterize the curvature. In this serearch, we consider the second order differential equations in case the structure group is 2-step nilpotent, and give a method to obtain such equations as the image of certain linear map defined by the curvature. An we explicity give the 'number of independent second order equations for several types of 2-step nilpotent Lie groups.
We also study several related topis on this subject. We determine the value of projective codimension of some Lie groups, clarify their relation to centro-affine immersions. And we also give some decomposition formulas of plethysms, and the tensor product of two irreducible representations (the Littlewood-Richardson rule) appearing in representation theory in terms of generating functions, which are needed in developing our subject.
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