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Geometriae Dedicata

, Volume 161, Issue 1, pp 129–155 | Cite as

The analytic continuation of hyperbolic space

  • Yunhi Cho
  • Hyuk Kim
Original Paper

Abstract

We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction inspires a new concrete insight for the study of the hyperbolic geometry and Lorentzian geometry as a unified object. We also discuss the advantages of this new geometric model as well as some of its applications.

Keywords

Hyperbolic space Analytic continuation Complex volume 

Mathematics Subject Classification (2000)

51M10 51M15 51M25 53B30 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of SeoulSeoulKorea
  2. 2.Department of MathematicsSeoul National UniversitySeoulKorea

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