On Weingarten Surfaces in Euclidean and Lorentzian 3-Space.
We study the neutral Kähler metric on the space of time-like lines in Lorentzian E3 1, which we identify with the total space of the tangent bundle to the hyperbolic plane. We find all of the infinitesimal isometries of this metric, as well as the geodesics, and interpret them in terms of the Lorentzian metric on E3 1. In addition, we give a new characterisation of Weingarten surfaces in Euclidean E3 and Lorentzian E3 1 as the vanishing of the scalar curvature of the associated normal congruence in the space of oriented lines. Finally, we relate our construction to the classical Weierstrass representation of minimal and maximal surfaces in E3 and E3 1.
Guilfoyle, Brendan & Klingenberg, Wilhelm. (2010). On Weingarten surfaces in Euclidean and Lorentzian 3-space. Differential Geometry and its Applications. 28. 454-468. 10.1016/j.difgeo.2009.12.002.