  In this paper, we study bijections on strictly convex sets of $\mathbf R
\mathbf P^n$ for $n \geq 2$ and closed convex projective surfaces equipped with
the Hilbert metric that map complete geodesics to complete geodesics as sets.
Hyperbolic $n$-space with its standard metric is a special example of the
spaces we consider, and it is known that these bijections in this context are
precisely the isometries. We first prove that this result generalizes to an
arbitrary strictly convex set. For the surfaces setting, we prove the
equivalence of mapping simple closed geodesics to simple closed geodesics and
mapping closed geodesics to closed geodesics. We also outline some future
directions and questions to further explore these topics.
