The Cartan structure equations are used to study space-like and time-like isothermic surfaces in three-dimensional Minkowski space in a unified framework. When the lines of curvature of a surface constitute an isothermal system, the surface is called isothermic. This condition serves to define a system of one-forms such that, by means of the structure equations, the Gauss-Codazzi equations for the surface are determined explicitly. A Lax pair can also be obtained from these one-forms for both cases, and, moreover, a nonhomogeneous Schrödinger equation can be associated with the set of space-like surfaces.
Bracken, Paul. "A Formulation of L-Isothermic Surfaces in Three-Dimensional Minkowski Space." International Journal of Mathematics and Mathematical Sciences 2018 (2018). https://doi.org/10.1155/2018/3713248
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International Journal of Mathematics and Mathematical Sciences