Mathematical Sciences: Curvature and Topology

数学科学：曲率和拓扑

• 批准号: 9208073
• 负责人: Stephan Stolz
• 金额: $ 6.86万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1996-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9208073&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Curvature; Topology

The investigator will continue to study the relationship between the curvature and the topology of manifolds. In particular, he will continue to study the existence and concordance classification of positive scalar curvature metrics. He has recently shown that if a closed manifold M admits a positive scalar curvature metric, then the concordance classes of such metrics are in bijective correspondence to a certain bordism group which depends only on the fundamental group and the first two Stiefel- Whitney classes of M. Also, M admits a positive scalar curvature metric if and only if M represents zero in this bordism group. If the universal cover of M is spin, this bordism group maps to the real K-theory of the 'twisted' group C*-algebra of the fundamental group of M, the twist being determined by the first two Stiefel- Whitney classes of M. An optimistic hope is that this is in fact an isomorphism. The investigator intends to continue his study of the relations between positive Ricci curvature, elliptic genera and elliptic homology; in particular, to pursue his conjecture that the Witten genus of a positive Ricci curvature manifold M vanishes provided M is spin and its first Pontrjagin class is zero. In a different direction, he hopes to show that all the possible multiplicities of a Dupin hypersurface with four distinct eigenvalues are realized by the known examples, i.e. the homogeneous examples and the Clifford examples. Topology treats those properties of geometric objects which are not dependent upon distances or angles, which are so fundamental that they persist after stretching and bending an object short of tearing it. In a topological sense, a phonograph record and a wedding ring are the same, for no topological properties distinguish one from the other. In fact, each is topologically the same as a coffee cup. Their shapes are wildly different, and yet there are some geometric properties that all will have in common as a result of their common topology. The famous Gauss-Bonnet theorem, well-known in electromagnetic theory as well as differential geometry, requires that the total (Gaussian) curvature of any of their surfaces will be zero. Now curvature is a very non-topological property, and locally it can have any value whatsoever on one of these surfaces, but the values cannot be totally unrelated as one moves from point to point, for the theorem says they will all sum to zero if one weights them by elements of area. Prominent among the results this project seeks are modern variants of this wonderful theorem for manifolds of other dimensions and for other kinds of curvature.

研究人员将继续研究 流形的曲率和拓扑之间的关系。 在 特别是，他将继续研究的存在和一致性， 正标量曲率度量的分类。 他 最近证明，如果闭流形M允许一个正标量 曲率度量，则此类度量的一致性类为 与某个协边群双射对应， 只依赖于基本群和前两个Stiefel- Whitney类的M. 而且，M容许正的标量曲率 度量当且仅当M在这个协边群中表示零。 如果 M的泛覆盖是自旋，这个协边群映射到 基本“扭”群C*-代数的真实的K-理论 M组，捻由前两个Stiefel确定。 Whitney类的M. 乐观的希望是， 同构 研究者打算继续研究 正Ricci曲率、椭圆亏格和 椭圆的同源性;特别是，追求他的猜想， 正Ricci曲率流形M的维滕亏格为零 假设M是自旋，并且它的第一个Pontrjagin类为零。 中 不同的方向，他希望表明，所有可能的 Dupin超曲面的重数 特征值由已知的示例实现，即， 齐次例子和Clifford例子。 拓扑处理几何对象的那些属性， 并不依赖于距离或角度， 最基本的是，它们在伸展和弯曲后仍然存在 在拓扑学的意义上，一台留声机 记录和结婚戒指是一样的，没有拓扑 属性将一个属性与另一个属性区分开。 事实上，每一个 拓扑上和咖啡杯一样 它们的形状 不同，但有一些几何性质，所有 由于它们的共同拓扑结构而具有共同点。 的 著名的Gauss-Bonnet定理，在电磁理论中很有名 以及微分几何，要求总的 任何曲面的（高斯）曲率都将为零。 现在 曲率是一种非常非拓扑的性质，局部地它可以 在这些表面上有任何价值，但是这些价值 当一个人从一个点移动到另一个点时，不可能完全无关，因为 该定理表明，如果对它们进行加权， 地区的元素。 在本项目寻求的结果中， 是这个奇妙定理的现代变体， 其他尺寸和其他类型的曲率。