  In this article, we construct a new para-K\"ahler structure $({\mathcal
G},{\mathcal J},\Omega)$ in the space of oriented geodesics ${\mathbb L}(M)$ in
a non-flat, real space form $M$. We first show that the para-K\"ahler metric
${\mathcal G}$ is scalar flat and when $M$ is a 3-dimensional real space form,
${\mathcal G}$ is locally conformally flat. Furthermore, we prove that the
space of oriented geodesics in hyperbolic $n$-space, equipped with the
constructed metric ${\mathcal G}$, is minimally isometric embedded in the
tangent bundle of the hyperbolic $n$-space. We then study the submanifold
theory, and we show that ${\mathcal G}$-geodesics correspond to minimal ruled
surfaces in the real space form. Lagrangian submanifolds (with respect to the
canonical symplectic structure $\Omega$) play an important role in the geometry
of the space of oriented geodesics as they are the Gauss map of hypersurfaces
in the corresponding space form. We demonstrate that the Gauss map of a
non-flat hypersurface of constant Gauss curvature is a minimal Lagrangian
submanifold. Finally, we show that a Hamiltonian minimal submanifold is locally
the Gauss map of a hypersurface $\Sigma$ that is a critical point of the
functional $\mathcal{F}(\Sigma)=\int_{\Sigma}\sqrt{|K|}\,dV$, where $K$ denotes
the Gaussian curvature of $\Sigma$.
