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 iff 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 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 L(ℍ3) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H3.
|Number of pages||21|
|Journal||Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg|
|Publication status||Published - 2010|
- Hyperbolic 3-space
- Kähler structure
- Weingarten surfaces