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.

黎曼几何的一个重要课题是理解几何结构的拓扑含义。当人们想知道全局定义的拓扑对象如何阻碍局部定义的属性时（在我们的例子中例如非负截面曲率），这变得特别有吸引力。在这个项目中，我们想要研究这类具体问题。一方面，我们的目标是了解齐次空间上的哪些向量丛及其许多推广允许非负截面曲率的度量 - 导致关于丛上非负弯曲度量的模空间的各种问题。另一方面，我们感兴趣关于几个奇异空间（如轨道折叠、亚历山德罗夫空间等）的问题，这些空间概括了流形上的已知结果。闭测地线的刻画、满足一定里奇曲率条件的度量的构造以及从某种同伦理论的角度对比轨道折叠和流形，都是这个方向的具体问题。从代数拓扑的角度来看，我们主要通过以下方式解决这些问题： （等变）K 理论和上同调以及有理同伦理论的结合 - 从而希望在工具和应用程序中实现协同作用。