A Projective Invariant for Swallowtails and Godrons, and Global Theorems on the Flecnodal Curve

Abstract

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): 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 present all possible local configurations of the flecnodal curve at a generic swallowtail in R 3 . We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection).

On the characteristic curves on a smooth surface

“…By Proposition 4.1, l P , l ζ , l χ and l D are distinct unless κ 2 a 40 = 3a 31 . This non-generic case is the collapse of two cusps of Gauss of opposite index [22]. Proof.…”

“…It follows from Theorem 4.3 that the characteristic cross-ratio (ccr)-invariant is a projective invariant and related to the cr-invariant ρ defined in [22] by the linear relation ρ c = 10 − 9ρ.…”

“…In Section 3 we give precise definitions of the curves of interest via their BDEs. Sections 4 and 5 extend the analogies between the asymptotic and characteristic curves by producing results mirroring those on the asymptotic curves in [22] (and we adopt the notation of that paper). In Section 4 we consider some special curves on the surface: the parabolic set, the flecnodal curve and the characteristic inflectional curve.…”

On the affine geometry of the graph of a real polynomial

“…It is defined in [15] the concept of a simple godron which is named in the present article as a simple Gaussian cusp. This is a point such that its dual point corresponds to a swallowtail point of the dual surface, that is, to an A 3 Legendre singularity, [4].…”

“…The configuration of these sets on the surface, invariant under the action of the affine group on 3-space, is one of the basic affine differential structures of the surface. One of the goals in Differential, Affine and Projective Geometry has been the study of the parabolic curve and the basic affine structure of smooth real surfaces as well, see for instance, [2,3,5,7,[13][14][15].…”