Glasgow Mathematical Journal

Research Article

ON THE GEOMETRY OF THE SPACE OF ORIENTED LINES OF THE HYPERBOLIC SPACE

MARCOS SALVAIa1*

a1 FAMAF – CIEM, Ciudad Universitaria, 5000 Córdoba, Argentina e-mail: salvai@mate.uncor.edu

Abstract

Let H be the n-dimensional hyperbolic space of constant sectional curvature –1 and let G be the identity component of the isometry group of H. We find all the G-invariant pseudo-Riemannian metrics on the space $\mathcal{G}_{n}$ of oriented geodesics of H (modulo orientation preserving reparametrizations). We characterize the null, time- and space-like curves, providing a relationship between the geometries of $ \mathcal{G}_{n}$ and H. Moreover, we show that $\mathcal{G}_{3}$ is Kähler and find an orthogonal almost complex structure on $\mathcal{G} _{7}$.

(Received November 03 2006)

(Revised March 13 2007)

(Accepted March 19 2007)

Key Words:

  • 53A55;
  • 53C22;
  • 53C35;
  • 53C50;
  • 53D25

Footnotes

* Partially supported by foncyt, Antorchas, ciem (conicet) and secyt (unc).