It remains a central task in Riemannian Geometry to understand global implications of locally defined concepts like curvature. Especially the interactions of the local geometric settings 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): mainly, 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, may be applied in order to extend different results and techniques obtained via rational invariants to the setting of finite characteristic.
Beside the focus on applications in the large field of curvature, the used techniques may furthermore be combined via generalising and applying concepts from equivariant cohomology and rational homotopy theory.
We characterize cohomogeneity one manifolds and homogeneous spaces with a compact Lie group action admitting an invariant metric with positive scalar curvature.
36Cohomogeneity, curvature, cohomology