Tits geometry and positive curvature

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 $3$ and irreducible topological spherical buildings of rank at least $3$.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type $\mathsf{M}$ associated with any polar action of cohomogeneity at least $2$ on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type $\mathsf{M}$, its universal Tits cover indeed is a building in all but two exceptional cases. 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 $2$.

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The first author is supported in part by an NSFC grant and he is grateful to the University of Notre Dame for its hospitality. The second author is supported in part by an NSF grant, a research chair at the Hausdorff Center at the University of Bonn and by a Humboldt research award. The third author is grateful to the University of Notre Dame and the Capital Normal University in Beijing for their hospitality.

Received 30 March 2015

Accepted 10 July 2016

Published 13 September 2017