Cohomogeneity Two Manifolds of Positive Sectional Curvature (22frg800)


(Wichita State University)


The Banff International Research Station will host the "Cohomogeneity Two Manifolds of Positive Sectional Curvature" workshop in Banff from June 12 - June 25, 2022.

Our group split our time at BIRS between two projects: the first was the cohomogeneity two project we described in our proposal, and the second was on estimating the lengths of geodesics.

The broad goal here is to classify such cohomogeneity two actions: that is, to find all possible $M$, up to diffeomorphism, and $G$, and to describe the action of $G$ on $M$ up to (equivariant) diffeomorphism. Coming into our stay at BIRS, our goal was to prove a classification theorem in low dimensions for closed, simply-connected, positively curved manifolds admitting an isometric, non-polar, cohomogeneity two action.

A useful tool for such actions is that of the $G$-manifold reduction, see Grove and Searle and Skjelbred and Straume. The idea is to reduce the $G$-action on $M$ to the case of a core group, $_cG$, acting on a core manifold, $_cM$. The core group $_cG$ is defined to be $N_G(H)/H$, where $N_G(H)$ denotes the normalizer of $H$ in $G$, the set of $g$ such that $gHg^{-1} = H$. The core manifold $_cM$ is defined to be the closure of the set $M^H$ of points in $M$ fixed by (a particular copy of) $H$. The quotient $_cM / _cG$ is isometric to $M/G$, and the principal isotropy of the action of $_cG$ on $_cM$ is the identity only. Thus, the only properties of the original $G$-action that are not preserved in the reduction are that $_cM$ might not be simply connected and $_cG$ might not be connected. Note that the original action is polar if and only if the core group $_cG$ is finite.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).