Timelike W -Surfaces in Minkowski 3-Space

Let $M$ and $M^{\ast }$ be two timelike surfaces in Minkowski 3-space ${ℝ}_1^3$ . If there exists a spacelike (timelike) Darboux line congruence between each point of $M$ and $M^{\ast }$ , then the surfaces are timelike Weingarten surfaces. It turns out their Tschebyscheff angles are solutions of the Sinh-Gordon equation, and the surfaces are related to each other by Backlund’s transformation. Finally, a method to construct new timelike Weingarten surface has been found.