Flat surfaces in hyperbolic space as normal surfaces to a congruence of geodesics

We first present an alternative derivation of a local Weierstrass representation for flat surfaces in the real hyperbolic three-space, H-3, using as a starting point an old result due to Luigi Bianchi. We then prove the following: let M subset of H-3 be a flat compact connected smooth surface with partial derivative M not equal circle divide, transversal to a foliation of H-3 by horospheres. If, along partial derivative M, M makes a constant angle with the leaves of the foliation, then M is part of an equidistant surface to a geodesic orthogonal to the foliation. We also consider the caustic surface associated with a family of parallel flat surfaces and prove that the caustic of such a family is also a flat surface (possibly with singularities). Finally, a rigidity result for flat surfaces with singularities and a geometrical application of Schwarz's reflection principle are shown.