RUI: Differential Geometry of Submanifolds

摘要：项目负责人：Thomas E. cecil，主要研究欧几里得空间和单位球中具有特殊曲率性质的子流形。特别有趣的是等参超曲面，它具有恒定的主曲率，以及dupin超曲面，它具有沿其每个曲率曲面的每个主曲率恒定的性质。尽管这些重要的超曲面类别自19世纪以来就被研究过，但许多自然问题至今仍未解决。首席研究员和他的合作者，来自华盛顿大学的Chi国信（guo - shin Chi）和Gary Jensen，将在李球几何中的legendresub流形上使用移动帧的方法来研究这些问题。这种方法适用于欧氏空间中任何子流形的研究，而不仅仅是这里提到的那些，并且它已经被首席研究员和他的合作者在以前的研究中成功地使用过。自1822年杜宾循环被引入以来，杜宾超曲面得到了广泛的研究，在过去的25年里，杜宾超曲面的分类取得了很大的进展。杜宾超曲面在许多数学理论中都扮演着重要的角色，如紧嵌入理论、流体动力型哈密顿系统的研究以及高维拉普拉斯不变量理论。杜平循环也出现在最近的计算机辅助几何设计论文中。球体等参超曲面的研究是由著名的法国数学家Elie Cartan在20世纪30年代发起的，许多数学家对这一美丽的理论做出了重大贡献。在等参超曲面及其焦点集中包含了许多著名的几何对象，如Veronese曲面和clifford - stiefel流形，它们已经被研究数学家从几个不同的角度进行了研究。在本科院校研究（RUI）活动和该提案的更广泛影响方面，首席研究员计划继续他成功的新课程开发和个别学生指导计划。在过去的15年里，这导致了13篇荣誉论文和总共25名几何课程的学生，他们已经参加或计划参加数学或相关领域的研究生院。

AbstractAward: DMS-0405529Principal Investigator: Thomas E. CecilThe goal of this proposal is to study submanifolds of Euclideanspace and the unit sphere which have special curvatureproperties. Of particular interest are isoparametrichypersurfaces, which have constant principal curvatures, andDupin hypersurfaces, which have the property that each principalcurvature is constant along each of its curvature surfaces.Although these important classes of hypersurfaces have beenstudied since the nineteenth century, many natural problemsremain open at this time. The principal investigator and hiscollaborators, Quo-Shin Chi and Gary Jensen of WashingtonUniversity, will employ the method of moving frames on Legendresubmanifolds in Lie sphere geometry in their research on theseproblems. This method is applicable to the study of anysubmanifold in Euclidean space, not just to those mentioned here,and it has been used successfully by the principal investigatorand his collaborators in previous research.Dupin hypersurfaces have been studied extensively since theintroduction of the cyclides of Dupin in 1822, and great progresshas been made over the past 25 years in their classification.Dupin hypersurfaces have played a major role in variousmathematical theories, such as the theory of taut embeddings, thestudy of Hamiltonian systems of hydrodynamic type, and the theoryof higher-dimensional Laplace invariants. The cyclides of Dupinhave also apperared in recent papers on computer aided geometricdesign. The study of isoparametric hypersurfaces in spheres wasinitiated by the renowned French mathematician, Elie Cartan, inthe 1930's, and many mathematicians have made significantcontributions to this beautiful theory. Included in the class ofisoparametric hypersurfaces and their focal sets are many famousgeometric objects, such as the Veronese surface and theClifford-Stiefel manifolds, which have been studied by researchmathematicians from several different points of view. In termsof Research in Undergraduate Institutions (RUI) activities andthe broader impact of the proposal, the principal investigatorplans to continue his successful program of new coursedevelopment and mentoring of individual students. Over the pastfifteen years, this has resulted in 13 honors theses and a totalof 25 students in geometry courses, who have attended or plan toattend graduate school in mathematics or related fields.