On the Equation Defining Isothermic Surfaces in Laguerre Geometry

A surface f : M 2 → E 3 oriented by a unit normal field n induces a lift F = (f, n) to the space Λ = E 3 × S 2 of contact elements of E 3 which is a Legendre immersion with respect to the canonical contact structure of Λ. Λ is a homogeneous space of the 10-dimensional group L of Laguerre contact trasformations. These are transformations on the space of oriented spheres which preserve oriented contact of spheres and take planes to planes in E 3 (cf. §2). In [13] and [14] we studied the Laguerre deformation problem for surfaces. We proved that the condition for a surface f : M → E 3, with nor umbilic nor parabolic points, being L-deformable is equivalent to the existence of local curvature line coordinates (x, y) about each point of M which are isothermic for the third fundamental form III of f. Surfaces with this property will be called L-isothermic. Examples include minimal surfaces, molding surfaces [6] and the class of Bonnet surfaces [12], that is, surfaces whose central spheres have centers in a fixed plane.