Problems in Lie Sphere Geometry

李球几何问题

• 批准号: 0604236
• 负责人: Gary Jensen
• 金额: $ 10.12万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604236&HistoricalAwards=false
• 关键词: Problems; Lie; Sphere; Geometry

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.

该提案的目标是证明任何具有四个主曲率且交叉比恒定的不可约真杜宾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个主曲率情况下该问题的成功解决的基础上。 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需的关系。 当主曲率超过三个时，一个新特征是任意四个主曲率的交叉比。 下一步是证明如果这个交叉比是常数，那么它必须等于等参情况下的值。 给定多重性和交叉比的正确条件，最后一步是证明不可约性意味着超曲面通过李球变换等效于等参超曲面。 在每种情况下，都将使用移动框架的方法来研究李球几何中杜宾超曲面的局部微分几何。普通空间中的曲面如果其主曲率恒定，则称为等参曲面。 该条件的限制性很大，因此唯一的例子是平面、球体和圆柱体。 如果假设每个主曲率仅沿着其每条曲率线恒定，则该表面称为杜宾循环面。 机械工程师使用循环装置，因为这种表面可以沿着曲率圆装配在一起。 非等参的环线是旋转环面，它是普通甜甜圈的表面。 任何具有两个不同主曲率的循环线都可以通过刚性运动、球体反转或平行平移（距表面固定距离的点的轨迹）从单个环面获得。 所有此类变换都会生成李球变换组。 在更多变量中，球体中存在许多显着的等参超曲面，但它们只能具有 1、2、3、4 或 6 个不同的主曲率。 杜宾环线推广到球体杜宾超曲面的思想。 其中包括等参超曲面。 有一种本地构造可以从较小尺寸的超曲面构建新的杜宾超曲面。 如果杜宾超曲面不是由这些构造之一构建的，则称为不可约超曲面。 对于具有四个不同主曲率的等参超曲面，主曲率交叉比是恒定的。 该提议的目标是证明，如果一个不可约的杜宾超曲面具有四个不同的主曲率且交叉比恒定，那么它是等参超曲面的李球变换。