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

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

2004-12-23

arxiv.org/abs/math/0412484v3

Mathematics

Differential Geometry

26 pages, 12 figures, the results were discovered in 2004 and reported in some seminars and conferences. Preprint of Institut

Scientific paper

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.}.

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