RUI: Differential Geometry of Submanifolds

摘要奖：DMS-0405529主要研究者：托马斯E. Cecil本文的目标是研究欧氏空间和单位球面中具有特殊曲率性质的子流形。 特别令人感兴趣的是等参超曲面，它具有恒定的主曲率，和Dupin超曲面，它具有每个主曲率沿其每个曲率曲面都是恒定的沿着的性质。虽然这些重要的超曲面类自19世纪以来一直在研究，许多自然问题仍然开放在这个时候。 主要研究者和他的合作者，华盛顿大学的Quo-Shin Chi和加里詹森，将在他们对这些问题的研究中使用李球几何中Legendre子流形上的移动标架方法。 这种方法不仅适用于这里所提到的那些子流形，而且也适用于欧氏空间中任何子流形的研究，并且它已经被主要的研究者和他的合作者成功地用于以前的研究中。自从1822年Dupin的圆线被引入以来，Dupin超曲面得到了广泛的研究，Dupin超曲面在各种数学理论中发挥了重要作用，如紧嵌入理论，流体动力学型哈密顿系统的研究和高维拉普拉斯不变量理论。Dupin圆线也出现在最近的计算机辅助几何设计论文中。 球面等参超曲面的研究是由法国著名数学家Elie Cartan于20世纪30年代提出的，许多数学家对这一美丽的理论做出了重要贡献。 在非参数超曲面及其焦集类中，有许多著名的几何对象，如Veronese曲面和Clifford-Stiefel流形，它们已经被数学家从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，校长计划继续他成功的新课程开发和个别学生指导计划。在过去的十五年里，这导致了13个荣誉论文和25名学生在几何课程，谁已经或计划参加研究生院在数学或相关领域。

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.