A New Projective Invariant Associated to the Special Parabolic Points of Surfaces and to Swallowtails

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we classify the godrons and present all possible local configurations of the flecnodal curve at a generic swallowtail in $R^3$. We present some global results, for instance: {\em A closed parabolic curve bounding a hyperbolic disc has a positive even number of godrons, and the flecnodal curve lying in that disc has an odd number of transverse self-intersections.}.