Research on conformally flat hypersurfaces

我们在四维一般共形平坦超曲面的研究方面取得了很大的进展。最重要的结果是，我们基金之间的这些超曲面的几种变换。这意味着我们对这些超曲面的研究向高维超曲面的积分系统理论迈进了一步。变换的存在性对于构造新的超曲面和研究这些超曲面的完全分类是有用的：一个超曲面有一个单参数的超曲面族。每个超曲面都有一个对偶超曲面。我们给出了对偶的积分公式。一般情况下，对偶对的成员不是共形等价的，超曲面的变换和它的共形变换也不是。对偶超曲面对与Guichard网的Ribaucour对之间存在一一对应关系。给出了Guichard网构造Ribaucour网伙伴的代数方法。

We had a great progress in the study on generic conformally flat hypersurfaces of dimension 4. The most important result is that we fund several kinds of transformations between these hypersurfaces. This means that our study on these hypersurfaces made progress to the theory of integral systems for higher dimensional hypersurfaces. Existence of transformations is useful for the construction of new hypersurfaces and for the study toward complete classification of these hypersurfaces: A hypersurface has a one parameter associated family of hypersurfaces. Each hypersurface has a dual hypersurface. We gave an integral formula for the dual. Generally, the members of dual pair are not conformally equivalent, and both duals of a hypersurface and its change by a conformal transformation are also not. There is a one to one correspondence between dual pairs of hypersurfaces and Ribaucour pairs of Guichard nets. We gave an algebraic method for a Guichard net to construct its Ribaucour partner.