A New Projective Invariant Associated to the Special Parabolic Points of Surfaces and to Swallowtails
Ricardo Uribe-Vargas
备注：
26 pages, 12 figures, the results were discovered in 2004 and reported in some seminars and conferences. Preprint of Institut of Maths. of Polish Acad. of Sciences
发表信息：
A Projective Invariant for Swallowtails and Godrons, and Global Theorems on the Flecnodal Curve. Moscow Mathematical Journal, 6 :4, (2006) 731-768
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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