Darboux curves on surfaces I (Q520623)

In this paper a class of curves on a surface is investigated. These curves are characterized by the property that their osculating spheres are tangent to the surface. Such curves are referred to as `Darboux curves'. In this paper Darboux curves are characterized under the Lorentz group. This way, such curves can even be described as geodesics in a 3-dimensional sub-variety of some quadric. This thorough investigation of such curves is based on the space of oriented spheres which can be viewed as a hyperquadric of the 5-dimensional Lorentz space.