A characterization of Weingarten surfaces in hyperbolic 3-space



We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ℍ3 orthogonal to the geodesics of Σ.

We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ℍ3 are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ℍ3.


Kähler structure Hyperbolic 3-space Weingarten surfaces 

Mathematics Subject Classification

51M09 51M30 


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© Mathematisches Seminar der Universität Hamburg and Springer 2010

Authors and Affiliations

  1. 1.Department of Computing and MathematicsInstitute of Technology, TraleeClash TraleeIreland

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