Differential equations and theory of submanifolds

我证明了具有6个多重度为2的主曲率的等参超曲面的同质性，这个问题我已经研究了好几年了。我还得到了Dorfmeister-Neher定理的一个新的证明，它以统一的方式处理多重性。研究得到的齐次超曲面，我得到了下面的结论在多重性1的情况下，具有6个主曲率的超曲面被给出为具有3个主曲率的超曲面的纤维，其中纤维完全是测地球体。在多重性2的情况下，光纤维度是6，而在多重性1的情况下，这是3。纤维结构的发现是我们先前与G. Ishikawa和M. Kimura一起完成的简并高斯映射结果的扩展。此外，利用等参超曲面族填充环境空间的事实，我们得到了13维球面与7维球面之间的有趣关系。此外，利用这些超曲面作为例外群G_2的轨道，我们可以证明在S^7-CP^2上存在一个度规，其完整群是G_2。由此，我们将考虑Calabi猜想的一个实开放版本，即当一个具有正Ricci曲率的紧致黎曼流形被移除了某一部分时，允许一个具有G_2完整度规？这样，由G_2轨道得到的超曲面给我们提出了非常重要和有趣的问题。

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.