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

TL;DR

This paper constructs a new para-K"ahler structure on the space of oriented geodesics in a non-flat real space form, revealing geometric properties and minimal submanifold characterizations related to hypersurfaces.

Contribution

It introduces a novel para-K"ahler structure on the space of geodesics, analyzes its curvature properties, and links geodesic submanifolds to minimal surfaces and Gauss maps of hypersurfaces.

Findings

The para-K"ahler metric is scalar flat.

In 3D, the metric is locally conformally flat.

Geodesics correspond to minimal ruled surfaces.

Abstract

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 $\Sigma$.