Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fat flats in rank one manifolds   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.

7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
16
The categories of this paper are: stat
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom