  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.
