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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且重数为二的等参超曲面的齐性，这是我多年来一直在解决的问题。对所得到的齐次超曲面进行了研究，得到了如下结果：在重数为1的情况下，已知6个主曲率的超曲面是3个主曲率的超曲面上的纤维化，其中纤维不全是测地球面。在多重数为2的情况下，纤维尺寸为6，而在多重数为1的情况下，纤维尺寸为3。纤维化结构的发现是G.石川和M.此外，利用等参超曲面族充满周围空间的事实，我们得到了13维球面与7维球面之间的一个有趣的关系。进一步，利用这些超曲面作为例外群G_2的轨道给出，我们可以证明在S^7-CP^2上存在一个度量，其完整群是G_2。由此，将考虑卡拉比猜想的一个真实的开放版本，即，当一个具有正Ricci曲率的紧致黎曼流形去掉某一部分后，允许一个具有G_2完整性的度量？这样，作为G_2轨道的超曲面就给我们提供了非常重要和有趣的问题。