Mathematics | Physical Sciences and Mathematics
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.
Georgiou, N., Guilfoyle, B. A characterization of Weingarten surfaces in hyperbolic 3-space. Abh. Math. Semin. Univ. Hambg. 80, 233–253 (2010). https://doi.org/10.1007/s12188-010-0039-7