A Characterization of Weingarten Surfaces in Hyperbolic 3-Space

Nikos Georgiou, Department of Computing and Mathematics, Institute of Technology, Tralee, Clash Tralee, Co. Kerry, Ireland
Brendan Guilfoyle, Department of Computing and Mathematics, Institute of Technology, Tralee, Clash Tralee, Co. Kerry, Ireland

Abhandlung aus dem Mathematischen Seminar der Universität Hamburg, 80. © Springer 2010

Abstract

We study 2-dimensional submanifolds of the space L(H3) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian if there exists a surface in H3 orthogonal to the geodesics of Σ. We prove that the induced metric on a Lagrangian surface in L(H3) has zero Gauss curvature if 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 L(H3) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H3