On the geometry of the space of oriented lines of the hyperbolic space
Abstract
Description
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 OG_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 OG_n and H. Moreover, we show that OG_3 is Kähler and find an orthogonal almost complex structure on OG_7.
Communicated at ICM 2006. Submitted to Glasgow Math. J. on November 3, 2006
Communicated at ICM 2006. Submitted to Glasgow Math. J. on November 3, 2006