On the geometry of the space of oriented lines of the hyperbolic space

2007/02/13 by Marcos Salvai, Salvai, Marcos
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.math/0702365

Communicated at ICM 2006. Submitted to Glasgow Math. J. on November 3, 2006

arxiv created 2007/02/13 · arxiv updated 2009/12/01

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 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 OG₃ is Kähler and find an orthogonal almost complex structure on OG₇.

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