Problems in Lie Sphere Geometry

这个建议的目的是证明任何不可约的正常Dupin超曲面的四个主曲率，其交比是常数是等价的李球变换等参超曲面。 该方法将建立在成功解决这个问题的情况下，三个主曲率。 在这种情况下，第一步是证明主曲率的重数满足等参超曲面所要求的关系。 当存在多于三个主曲率时的新特征是任何四个主曲率的交比。 下一步是证明，如果这个交比是常数，那么它必须等于它在等参情况下的值。 给定重数和交比的正确条件，最后一步是证明不可约性意味着超曲面通过李球变换等价于等参超曲面。 在每一种情况下，我们都将使用活动标架的方法来研究Lie球几何中Dupin超曲面的局部微分几何。 这个条件是如此的严格，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沿着为常数，则曲面称为杜宾圆线。 机械工程师使用Cyclides是因为这样的曲面可以沿着它们的曲率圆沿着在一起。 非等参的圆线是旋转环面，它是普通甜甜圈的曲面。 任何具有两个不同主曲率的圆线都可以从一个环面通过刚体运动、球面反演或平行平移（点的轨迹距曲面固定距离）得到。 所有这样的变换都生成李群变换。 在更多的变量，有许多显着的等参超曲面的领域，但他们只能有1，2，3，4，或6个不同的主曲率。 Dupin的圆线推广到球面中的Dupin超曲面的概念。 这些包括等参超曲面。 有一个局部构造，从一个较小的维度建立一个新的Dupin超曲面。 一个Dupin超曲面被称为不可约的，如果它不是由这些构造之一。 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。 这个建议的目的是证明，如果一个不可约的Dupin超曲面有四个不同的主曲率与常数的交叉比，那么它是一个等参超曲面的李球变换。

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.