Fat flats in rank one manifolds
Abstract
We study closed non-positively curved Riemannian manifolds $M$ which admit `fat $k$-flats': that is, the universal cover $\tilde M$ 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 $1$ non-positively curved manifolds with a fat $1$-flat which 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 to prove 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 $k \geq 2$. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2017
- DOI:
- 10.48550/arXiv.1704.00857
- arXiv:
- arXiv:1704.00857
- Bibcode:
- 2017arXiv170400857C
- Keywords:
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- Mathematics - Dynamical Systems;
- Mathematics - Differential Geometry;
- Mathematics - Geometric Topology
- E-Print:
- v3: 22 pages, 1 figure. Fixed some typos. To appear in the Michigan Mathematical Journal