Kurvature, Kohomology and K-Theory

曲率、上同调和 K 理论

• 批准号: 395901807
• 负责人: Privatdozent Dr. Manuel Amann
• 金额: --
• 依托单位: Lehrstuhl für Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/395901807?language=en
• 关键词: Kurvature; Kohomology; Theory

An important subject in Riemannian geometry is to understand topological implications of geometric structures. This becomes particularly appealing when one wonders how globally defined topological objects obstruct locally defined properties - in our case for example non-negative sectional curvature.In this project we want to investigate concrete questions of this kind. On the one hand we aim to understand which vector bundles over homogeneous spaces and their many generalisations permit metrics of non-negative sectional curvature - leading to various questions on moduli spaces of non-negatively curved metrics on bundles.On the other hand we are interested in questions on several singular spaces like orbifolds, Alexandrov spaces, etc., which generalise known results on manifolds. The characterisation of closed geodesics, the construction of metrics satisfying certain Ricci curvature conditions and contrasting orbifolds and manifolds from a certain homotopy theoretic viewpoint, are concrete problems in this direction.From the point of view of algebraic topology we want to tackle these questions mainly by a combination of (equivariant) K-theory and cohomology, as well as rational homotopy theory - thereby hoping for synergies both in tools and applications.

Riemannian几何形状的一个重要主题是了解几何结构的拓扑意义。当人们想知道全球定义的拓扑对象如何阻碍本地定义的特性时，这变得尤其有吸引力 - 例如，在我们的情况下进行非负分段曲率。在这个项目中，我们想研究此类具体问题。一方面，我们旨在了解哪些均匀空间及其许多概括允许对非负截面曲率的指标进行指标 - 导致了关于捆绑中非模态度量的模量空间的各种问题。另一方面，我们对诸如Orbifolds，Alexandrovs，Genter Saces等的几个奇异空间上的问题感兴趣，均为alexandrovs，等等，等等。封闭的大地测量学的表征，满足某些RICCI曲率条件的指标的构建以及从某些同义理论的观点来对比的孔隙和歧管，这是朝这个方向朝着这个方向朝着的具体问题。从代数的角度来看，我们希望通过（Homiagient and Equhoriant and Equhorizat and Equhory）来解决这些问题 - 善于地解决这些问题 - 希望在工具和应用程序中建立协同效应。