Fat flats in rank one manifolds
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866910718930976768 |
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| author | Constantine, D. Lafont, J. -F. McReynolds, D. B. Thompson, D. J. |
| author_facet | Constantine, D. Lafont, J. -F. McReynolds, D. B. Thompson, D. J. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1704_00857 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Fat flats in rank one manifolds Constantine, D. Lafont, J. -F. McReynolds, D. B. Thompson, D. J. Dynamical Systems Differential Geometry Geometric Topology 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. |
| title | Fat flats in rank one manifolds |
| topic | Dynamical Systems Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/1704.00857 |