On Weingarten surfaces in Euclidean and Lorentzian 3-space

We study the neutral Kähler metric on the space of time-like lines in Lorentzian E13, 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 E13. In addition, we give a new characterisation of Weingarten surfaces in Euclidean E3E3 and Lorentzian E13 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 E3E3 and E13.