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.

我证明了具有六个主曲率、重数为二的等参超曲面的同质性，多年来我一直在解决这个问题。我还得到了 Dorfmeister-Neher 定理的一个新证明，该定理以统一的方式处理重数一的情况。研究所得的齐次超曲面，我得到以下结果正如在重数一的情况下已知的那样，具有 6 个主曲率的超曲面是给出了具有 3 个主曲率的纤维的纤维化，其中纤维是完全测地线球体。在多重性为二的情况下，纤维尺寸为六，而在多重性为一的情况下，纤维尺寸为三。纤维化结构的发现是我们之前与 G. Ishikawa 和 M. Kimura 合作完成的简并高斯映射结果的延伸。此外，利用等参超曲面族填充周围空间的事实，我们得到了一个有趣的关系介于13维球体和7维球体之间。此外，利用这些超曲面作为例外群G_2的轨道，我们可以证明S^7-CP^2上存在一个度量，其完整群为G_2。由此，将考虑卡拉比猜想的真正开放版本，即，当一个具有正 Ricci 曲率的紧致黎曼流形从中删除了某个部分时，承认具有 G_2 完整性的度量？这样，作为G_2轨道获得的超曲面向我们提出了非常重要且有趣的问题。