On the Geometry of Spaces of Oriented Geodesics

TL;DR

This paper explores the geometric structures of the space of oriented geodesics in certain symmetric spaces, classifying invariant symplectic, complex, and metric structures on these spaces.

Contribution

It provides a comprehensive classification of invariant geometric structures on the space of oriented geodesics in specific symmetric spaces.

Findings

Classification of invariant symplectic structures on L(M)

Description of invariant (para)complex and (para)Kaehler structures

Analysis of pseudo-Riemannian metrics on L(M)

Abstract

Let M be either a simply connected pseudo-Riemannian space of constant curvature or a rank one Riemannian symmetric space (other than the octonion hyperbolic plane), and consider the space L(M) of oriented geodesics of M. The space L(M) is a smooth homogeneous manifold and in this paper we describe all invariant symplectic structures, (para)complex structures, pseudo-Riemannian metrics and (para)Kaehler structure on L(M).