Research of submanifolds in a symmetric space by using infinite dimensional geometry

作为2006年的第一个研究结果，我们得到了非紧型对称空间中真复等焦子流形的Chevalley型限制定理。证明是通过研究它的复化到某个无穷维反凯勒空间的提升来进行的。这里我们注意到对称Name上的Hermann型作用的主轨道是真复等焦子流形。本研究由首席研究员进行。作为第二个研究结果，我们对非紧型对称空间上具有全测地轨道的复超极作用几乎进行了分类。这里我们注意到复超极作用的主轨道是复等焦子流形，反之齐次复等焦子流形也是复超极作用的主轨道。同时，我们注意到Hermann型作用量是复超极作用量.本研究由首席研究员进行。作为第三个研究结果，我们对非紧型秩对称空间上的上齐性作用进行了分类。本研究由研究者田丸浩教授和Jurgen Berndt教授共同完成，作为2007年的第一个研究成果，我们几乎完成了非紧型对称空间中余维大于1的不可约真复等域子流形的齐性定理的证明。证明是通过研究它的复化到某个无穷维反凯勒空间的提升来进行的。本研究由首席研究员进行。作为第二个研究结果，我们完成了秩大于1的不可约对称空间中具有非Bat截面的等焦子流形不存在性定理的基本证明，这是一个公开问题。本研究由首席研究员进行。

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.