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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.
In this article, we consider a closed rank one Riemannian manifold $M$ without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on $M$ with length at most $t$, and $# P(t)$ its cardinality. We obtain the follo
Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $mathbb{R}^m$ triangulates the convex hull of $P$, provided that $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be de
The paper was withdrawn due to a gap in the proof of Lemma 3.
We analyze two different fibrations of a link complement M constructed by McMullen-Taubes, and studied further by Vidussi. These examples lead to inequivalent symplectic forms on a 4-manifold X = S x M, which can be distinguished by the dimension of
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.