Problems in Lie Sphere Geometry

本文的目的是证明任意四个主曲率为常数的不可约真Dupin超曲面通过李球变换等价于等参超曲面。这些方法将建立在三主曲率情况下这一问题的成功解的基础上。在这种情况下，第一步将是证明主曲率的多重性满足等参超曲面所要求的关系。当有三个以上的主曲率时，一个新的特征是任意四个主曲率的交比。下一步是证明如果这个交比是恒定的，那么它一定等于它在等参时的值。给出关于重数和交比的正确条件，最后一步是证明不可约意味着超曲面通过李球变换到等参超曲面是等价的。在这两种情况下，我们都将使用移动标架的方法来研究李球几何中Dupin超曲面的局部微分几何。如果一个普通空间中的曲面的主曲率为常数，则称之为等参曲面。这个条件是如此严格，以至于唯一的例子是平面、球体和圆柱体。如果假设每个主曲率仅沿其每条曲率线为常数，则该曲面称为Dupin圆。机械工程师之所以使用圆柱体，是因为这样的曲面可以沿着它们的曲率圆连接在一起。非等参圆环称为旋转圆环，它是普通甜甜圈的曲面。任何具有两个不同主曲率的圆环都可以通过刚性运动、球面内的反转或平行平移(点与曲面固定距离的轨迹)从单个环面获得。所有这样的变换都产生了李球变换群。在更多的变量中，球体中有许多值得注意的等参超曲面，但它们只能有1、2、3、4或6个不同的主曲率。Dupin的圆周推广到球面上的Dupin超曲面的思想。其中包括等参超曲面。有一种局部结构，它从较小的维度构建一个新的Dupin超曲面。如果Dupin超曲面不是由这些构造中的一种构造而成的，则称它为不可约曲面。对于具有四个不同主曲率的等参超曲面，主曲率的交比是常数。本文的目的是证明：如果一个不可约的Dupin超曲面有四个截然不同的交比恒定的主曲率，则它是一个等参超曲面的Lie球变换。

The goal of this proposal is to prove that any irreducible proper Dupin hypersurface with four principal curvatures whose cross ratio is constant is equivalent by a Lie sphere transformation to an isoparametric hypersurface. The methods will build on the successful solution of this problem in the case of three principal curvatures. As in that case, the first step will be to prove that the multiplicities of the principal curvatures satisfy the relations required by an isoparametric hypersurface. A new feature when there are more than three principal curvatures is the cross ratio of any four principal curvatures. The next step is to prove that if this cross ratio is constant, then it must equal its value in the isoparametric case. Given the correct conditions on the multiplicities and the cross ratio, the final step is to prove that irreducibility implies that the hypersurface is equivalent by Lie sphere transformations to an isoparametric hypersurface. In each case, the method of moving frames will be used to study the local differential geometry of Dupin hypersurfaces in Lie sphere geometry.A surface in ordinary space is called isoparametric if its principal curvatures are constant. This condition is so restrictive that the only examples are planes, spheres, and circular cylinders. If each principal curvature is assumed constant only along each of its lines of curvature, then the surface is called a cyclide of Dupin. Cyclides are used by mechanical engineers because of the way such surfaces can be fit together along their circles of curvature. A cyclide that is not isoparametric is the torus of revolution, which is the surface of an ordinary doughnut. Any cyclide with two distinct principal curvatures can be obtained from a single torus by rigid motions, inversions in spheres, or parallel translations (the locus of points a fixed distance from the surface). All such transformations generate the group of Lie sphere transformations. In more variables, there are many remarkable isoparametric hypersurfaces in spheres, but they can have only 1, 2, 3, 4, or 6 distinct principal curvatures. Dupin's cyclides generalize to the idea of Dupin hypersurfaces in spheres. These include the isoparametric hypersurfaces. There is a local construction that builds a new Dupin hypersurface from one of smaller dimension. A Dupin hypersurface is called irreducible if it is not built by one of these constructions. For an isoparametric hypersurface with four distinct principal curvatures, the cross ratio of the principal curvatures is constant. The goal of this proposal is to prove that, if an irreducible Dupin hypersurface has four distinct principal curvatures with constant cross ratio, then it is a Lie sphere transformation of an isoparametric hypersurface.