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

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

2007-02-13

arxiv.org/abs/math/0702365v1

Mathematics

Differential Geometry

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

Scientific paper

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 K\"{a}hler and find an orthogonal almost complex structure on OG_7.

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