2007 Fiscal Year Final Research Report Summary
Research of submanifolds in a symmetric space by using infinite dimensional geometry
| Project/Area Number |
18540099
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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 | Tokyo University of Science |
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Principal Investigator |
KOIKE Naoyuki Tokyo University of Science, Faculty of Science, Associate Professor (00281410)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIOKA Akira Tokyo University of Science, Faculty of Science, Professor (40200935)
TAMARU Hiroshi Hiroshima University, Faculty of Science, Associate Professor (50306982)
SAKAI Takashi Osaka City University, Faculty of Science, Special Duty Assistant (30381445)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Differential Geoimetry |
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| Research Abstract |
As the first study result of 2006, we obtained a Chevalley type restriction theorem for a proper complex equifocal submanifold in a symmetric space of non-compact type. The proof was performed by investigating the lift of its complexification to some infinite dimensional anti-Kaehlerian space. Here we note that principal orbits of Hermann type actions on the symmetric Name are proper complex equifocal submanifolds. This research was performed by the head investigator. As the second study result, we almost classified complex hyperpolar actions with total geodesic orbit on a symmetric space of non-compact type. Here we note that principal orbits of complex hyperpolar actions are complex equifocal submanifolds and that conversely homogeneous complex equifocal submanifolds war as principal orbits of complex hyperpolar actions. Also, we note that Hermann type actions are complex hyperpolar actions. This research was performed by the head investigator. As the third study result, we classified cohomogeneity one actions on rankone symmetric spaces of non-compact type. This research was performed by Professor Hiroshi Tamaru of the investigator and Professor Jurgen Berndt.
As the first study result of 2007, we completed almost the proof of the homogeneity theorem for irreducible proper complex equifocal submanifolds of codimension greater than one in a symmetric space of non-compact type. The proof was performed by investigating the lift of its complexification to some infinite dimensional anti-Kaehlerian space. This research was performed by the head investigator. As the second study result, we completed almost the proof of the non-existence theorem of equifocal submanifolds with non-Bat section in an irreducible symmetric space of rank greater than one, which is an open problem. This research was performed by the head investigator.
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