Cohomogeneity, Curvature, Cohomology

同齐性、曲率、上同调

• 批准号: 441900967
• 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: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441900967?language=en
• 关键词: Cohomogeneity; Curvature; Cohomology

It remains a central task in Riemannian Geometry to understand global implications of locally defined concepts like curvature. Especially the interactions of the local geometries and the topological properties of the underlying manifolds are a worthwhile field of study. This extends to synthetic notions of curvature and singular spaces (as constituted by Alexandrov Geometry).This project is based on three pillars, which vary such questions (in particular, with a view towards sectional curvature and its generalisations): on the one hand we shall investigate Alexandrov spaces (and orbifolds, etc.) which admit actions of compact Lie groups of low cohomogeneity and their cohomological properties; on the other hand different approaches to equivariant K-theory will be used to equip vector bundles over suitable manifolds (like biquotients) with metrics of non-negative sectional curvature up to stabilisation. Finally, tame homotopy theory, in particular, will be applied in order to extend different results and techniques obtained via rational invariants to the setting of finite characteristic.Beside the discussion of curvature properties, further interdependencies of these questions can be found in generalisations and applications of concepts from equivariant cohomology and rational homotopy theory.

在Riemannian几何形状中，了解曲率等本地定义的概念的全球含义仍然是一项核心任务。尤其是局部几何形状的相互作用和基础歧管的拓扑特性是一个值得的研究领域。 This extends to synthetic notions of curvature and singular spaces (as constituted by Alexandrov Geometry).This project is based on three pillars, which vary such questions (in particular, with a view towards sectional curvature and its generalisations): on the one hand we shall investigate Alexandrov spaces (and orbifolds, etc.) which admit actions of compact Lie groups of low cohomogeneity and their cohomological 特性;另一方面，将使用Equivariant K理论的不同方法将用于在适当的歧管（例如生物品质）上配备具有非负分段曲率的指标，以稳定稳定。最后，尤其将应用驯服同义理论，以扩展通过合理不变的不同结果和技术扩展到有限特征的设置。Beside曲率特性的讨论，这些问题的进一步相互依赖性可以在概括和等同于等同的同源既有同源理论的概念中发现。