Tits Geometry and Positive Curvature

TL;DR

This paper uncovers a new geometric structure related to polar actions on positively curved manifolds, linking chamber systems to spherical buildings, and classifies such actions up to equivariant diffeomorphism.

Contribution

It introduces a novel chamber system framework for polar actions on positively curved manifolds and proves that its universal Tits cover is often a spherical building, enabling classification.

Findings

Universal Tits cover of chamber system is a spherical building in most cases.

Classifies all polar actions on simply connected positively curved manifolds of cohomogeneity ≥ 2.

Establishes a new connection between polar actions, chamber systems, and spherical buildings.

Abstract

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.