  We define and study the foliated centralizer: the group of $C^\infty$
centralizer elements of the lift of an Anosov system on a non-compact manifold
which additionally preserve the stable and unstable foliations. When the Anosov
system is the geodesic flow of a closed Riemannian manifold with pinched
negative sectional curvatures, we prove some rigidity properties for the
foliated centralizer of the lift of the flow to the universal cover: it is a
finite-dimensional Lie group, which is moreover discrete (modulo the action of
the flow itself) unless the metric is homothetic to some real hyperbolic
metric. This result is inspired by the study of isometries of universal covers
that appeared originally in the work of Eberlein, and later in Farb and
Weinberger, as well as centralizer rigidity results in dynamics.
