The Michigan Mathematical Journal

Fat Flats in Rank One Manifolds

D. Constantine, J.-F. Lafont, D. B. McReynolds, and D. J. Thompson

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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 k2. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.

Article information

Michigan Math. J., Volume 68, Issue 2 (2019), 251-275.

Received: 17 April 2017
Revised: 9 March 2018
First available in Project Euclid: 9 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx] 37D35: Thermodynamic formalism, variational principles, equilibrium states


Constantine, D.; Lafont, J.-F.; McReynolds, D. B.; Thompson, D. J. Fat Flats in Rank One Manifolds. Michigan Math. J. 68 (2019), no. 2, 251--275. doi:10.1307/mmj/1549681300.

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