Differential equations and theory of submanifolds

微分方程和子流形理论

基本信息

批准号：
14540090
负责人：
MIYAOKA Reiko
金额：
$ 1.79万
依托单位：
Kyushu University (2003)Sophia University (2002)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2003
项目状态：
已结题

项目摘要

I proved the homogeneity of isoparametric hypersurfaces with six principal curvatures with multiplicity two, which I had been tackling for several years. I also got a new proof of Dorfmeister-Neher's theorem which treats the multiplicity one case, in a unified manner.Investigating the resulted homogeneous hypersurfaces, I got the following As was known in the case of multiplicity one, the hypersurfaces with 6 principal curvatures are given as a fibration over those with 3 principal curvature, where the fibers aret otally geodesic spheres. In the case of multiplicity two, the fiber dimension is six, while in the case of multiplicity one, this is three. Discovery of the fibration structure is an extension of our former results on the degenerate Gauss mapping which was done with G. Ishikawa and M. Kimura.Moreover, using the fact that the family of isoparametric hypersurfaces fill the ambient space, we get an interesting relation between 13-dimensional sphere and 7-dimensional sphere. Furthermore, using that these hypersurfaces are given as orbits of the exceptional group G_2, we can show that there exists a metric on S^7-CP^2 of which holonomy group is G_2. From this, a real open version of Calabi conjecture will be considered, i.e., when a compact Riemannian manifolds with positive Ricci curvature from which a certain part removed, admits a metric with G_2 holonomy? In this way, hypersurfaces obtained as G_2 orbits suggest us very important and interesting problems.

我证明了等六个主曲面的同质性，具有六个主要曲线，具有多重性，我已经处理了几年。 I also got a new proof of Dorfmeister-Neher's theorem which treats the multiplicity one case, in a unified manner.Investigating the resulted homogeneous hypersurfaces, I got the following As was known in the case of multiplicity one, the hypersurfaces with 6 principal curvatures are given as a fibration over those with 3 principal curvature, where the fibers aret otally geodesic球。在多重性二的情况下，纤维尺寸为六个，而在多重性的情况下，这是三个。纤维化结构的发现是我们以前的结果对使用G. iShikawa和M. kimura进行的退化高斯映射的扩展。此外，使用等异端术的家庭填充环境空间的事实，我们得到了13维领域和7级级别和7级阶段的有趣关系。此外，使用这些超曲面作为特殊组G_2的轨道，我们可以证明在S^7-CP^2上存在一个度量，其中自动构图是G_2。由此，将考虑一个真正的开放版本的卡拉比猜想，即，当一个带有正ricci曲率的紧凑型riemannian歧管上移除某个部分时，请承认具有G_2 holomony的指标？通过这种方式，作为G_2轨道获得的高空曲面表明我们非常重要且有趣的问题。