On the space of oriented geodesics of hyperbolic 3-space

Nikos Georgiou, Brendan Guilfoyle

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20 Citations (Scopus)


We construct a Kähler structure (J, Ω, G) on the space L(H 3) of oriented geodesies of hyperbolic 3-space H3 and investigate its properties. We prove that (L(H3), J) is biholomorphic to P1 x P1-Δ̄, where Δ̄ is the reflected diagonal, and that the Kähler metric G is of neutral signature, conformally flat and scalar flat. We establish that the identity component of the isometry group of the metric G on L(H3) is isomorphic to the identity component of the hyperbolic isometry group. Finally, we show that the geodesies of G correspond to ruled minimal surfaces in H3, which are totally geodesic if and only if the geodesies are null.

Original languageEnglish
Pages (from-to)1183-1220
Number of pages38
JournalRocky Mountain Journal of Mathematics
Issue number4
Publication statusPublished - 2010
Externally publishedYes


  • Hyperbolic 3-space
  • Isometry group
  • Kaehler structure


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