A para-Kähler structure in the space of oriented geodesics in a real space form

Abstract

In this article, we construct a new para-Kahler structure $$({{\mathcal {G}}},{{\mathcal {J}}},\Omega )$$ on the space of oriented geodesics of a n-dimensional (non-flat) real space form. We first show that the para-Kahler metric $${{\mathcal {G}}}$$ is scalar flat and when $$n=3$$ , it is locally conformally flat. Furthermore, we prove that the space of oriented geodesics of hyperbolic n-space equipped with the constructed metric $${{\mathcal {G}}}$$ is minimally isometrically embedded in the tangent bundle of hyperbolic n-space. We then study submanifold theory, and show that $${{\mathcal {G}}}$$ -geodesics correspond to minimal ruled surfaces in the real space form. Lagrangian submanifolds (with respect to the symplectic structure $$\Omega $$ ) play an important role in the geometry of the space of oriented geodesics as they come from 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 $$\Sigma $$ , which is a critical point of the functional $$\mathcal {F}(\Sigma )=\int _{\Sigma }\sqrt{|K|}\,dV$$ , K denoting the Gaussian curvature of $$\Sigma $$ .