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 ?³ 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 ?³ 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 ?³.