Weierstrass representations for triply orthogonal and conformal Euclidean and Lorentzian systems

Starting with triply orthogonal moving frames in 3-dimensional Lie algebras, we build Weierstrass representations for triply orthogonal and conformal coordinate systems in Euclidean and Lorentian 3-space. Our constructions include spacelike and timelike Lorentzian systems. We express the first and second fundamental forms and the mean and Gaussian curvatures for the three families of surfaces in each system in terms of the Weierstrass data. Specializing to conformal mappings, we determine the Weierstrass data for all triply conformal transformations, classically known as the generalized Möbius transformations.