2012 Fiscal Year Final Research Report
The geometry of symmetric triad and Hermann actions
| Project/Area Number |
22540108
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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 | Kyoto Institute of Technology (2012)
Fukushima National College of Technology (2010-2011) |
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Principal Investigator |
IKAWA Osamu 京都工芸繊維大学, 工芸科学研究科, 教授 (60249745)
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| Project Period (FY) |
2010 – 2012
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| Keywords | 対称三対 / Hermann 作用 |
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| Research Abstract |
We introduced the notion of symmetric triad, which is a generalization of the notion of irreducible root system, and studied its fundamental properties. We defined a symmetric triad with multiplicities and showed that everysymmetric triad is obtainedfrom a commutative Hermann action. Applying these results,we studied the orbit spaces of Hermannactions on compact symmetric spaces.In particular we determined the structure of the orbit spaces of Hermann actions, and classified austere orbits. We gave a new formula which write down the moment map for holomorphic isometric action on a complete Kaehler manifold using the motion of a charged particle when the action is Hamiltonian.Further we gave a necessary and sufficient condition that a holomorphic isometric action on a complete Kaehler manifold to be Hamiltonian when the action has a fixed point. We also showed that if the first de-Rham cohomolgy group of a complete Kaehler manifold vanishes, then a holomorphic isometric action is Hamiltonian.
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