Topological and equivariant rigidity in the presence of lower curvature bounds

• Applicants: Dr. Fernando Galaz-García; Dr. Martin Kerin
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 339994903

The theory of compact groups acting on manifolds has a long tradition and is quite well understood nowadays. The more general case of groups acting on metric and/or singular spaces is today mostly pursued in the context of non-positively curved metric spaces. However, fundamental to other parts of modern geometry are Alexandrov spaces (of curvature bounded below) equipped with an isometric action of a compact Lie group G. These metric spaces include, for example, Riemannian manifolds with a lower sectional curvature bound and an isometric G-action. Of particular interest to Riemannian geometers have been Riemannian manifolds with positive or non-negative curvature equipped with isometric compact Lie group actions. This interest is in great part propelled by the so-called Grove program, which proposes to classify Riemannian manifolds of positive or non-negative sectional curvature with a large isometry group.The simplest groups that one may consider acting on a manifold are compact abelian Lie groups of positive dimension, i.e., tori. By now the theory of smooth torus actions is well developed. In the context of Riemannian geometry and, in particular, of the Grove program, positively curved Riemannian manifolds with torus actions have been extensively studied. Thanks to the work of several authors, most notably Grove and Searle, Fang and Rong, and Wilking, fairly complete classification results are available, provided that either the manifold, or the torus acting upon it, has sufficiently large dimension. For the action of a circle and a 2-torus in dimensions 5 and 6, respectively, the usual methods have so far failed to yield topological and equivariant classification results.Taking as departure point the Grove program and the well-developed theory of cohomological methods for smooth torus actions on smooth manifolds, the present proposal aims, on the one hand, at applying and developing equivariant topological methods in the context of Riemannian manifolds with a lower sectional curvature bound and, on the other hand, at studying closed Alexandrov spaces of cohomogeneity one. The primary goals are, respectively, to obtain a topological and equivariant classification of closed, simply connected 6-manifolds with an effective, isometric 2-torus action, and to classify closed, positively curved Alexandrov spaces of cohomogeneity one. The solution of these problems entails solving many self-contained problems which are of interest in their own right, including the consideration of singular spaces (e.g. orbifolds and Alexandrov spaces) with torus actions.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity
• International Connection: China, United Kingdom, USA
• Cooperation Partners: Professor Dr. Lee Kennard; Professor Dr. Burkhard Wilking; Dr. Masoumeh Zarei