Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg

, Volume 80, Issue 2, pp 233–253

A characterization of Weingarten surfaces in hyperbolic 3-space

Article

DOI: 10.1007/s12188-010-0039-7

Cite this article as:
Georgiou, N. & Guilfoyle, B. Abh. Math. Semin. Univ. Hambg. (2010) 80: 233. doi:10.1007/s12188-010-0039-7

Abstract

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.

Keywords

Kähler structureHyperbolic 3-spaceWeingarten surfaces

Mathematics Subject Classification

51M0951M30

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2010

Authors and Affiliations

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