Darboux curves on surfaces Ⅰ

摘要

In 1872, G. Darboux defined a family of curves on surfaces of R~3 which are preserved by the action of the Moebius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric A~4 contained in the 5-dimensional Lorentz space R_1~5 naturally associated to the surface. We construct a new conformal object: the Darboux plane-field D and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that D is integrable when the surface is a special canal.