In many occasions throughout this work, we use an interpretation of loop group theory by Burstall−Calderbank and produce transformations of surfaces by the action of loops of flat metric connections. Such connections are gauge equivalent to the trivial flat connection, and these gauge transformations can be used to produce new surfaces, defined up to Möbius transformations. An equivalent perspective is that of change of flat metric connection, replacing the trivial one by another flat metric connection on the Lorentzian trivial bundle.