Fat Flats in Rank One Manifolds

Abstract

We study closed nonpositively curved Riemannian manifolds M that admit “fat k-flats”; that is, the universal cover M˜ contains a positive-radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank 1 nonpositively curved manifolds with a fat 1-flat that corresponds to a twisted cylindrical neighborhood of a geodesic on M. As a result, M contains an embedded closed geodesic with a flat neighborhood, but M nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite volume hyperbolic manifolds. Our second main result is a proof of a closing theorem for fat flats, which implies that a manifold M with a fat k-flat contains an immersed, totally geodesic k-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when k≥2. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.