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Mosc. Math. J., 2006, Volume 6, Number 4, Pages 731–768 (Mi mmj267)  

This article is cited in 13 scientific papers (total in 13 papers)

A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve

R. Uribe-Vargas

Université Paris VII – Denis Diderot

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 tangent plane, the self-intersection line, the cuspidal edge and the flecnodal curve at a generic swallowtail in $\mathbb R^3$. 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).

Key words and phrases: Geometry of surfaces, tangential singularities, swallowtail, parabolic curve, flecnodal curve, cusp of Gauss, godron, wave front, Legendrian singularities.


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MSC: 14B05, 32S25, 58K35, 58K60, 53A20, 53A15, 53A05, 53D99, 70G45
Received: January 18, 2006

Citation: R. Uribe-Vargas, “A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve”, Mosc. Math. J., 6:4 (2006), 731–768

Citation in format AMSBIB
\by R.~Uribe-Vargas
\paper A~projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 4
\pages 731--768

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    1. Proc. Steklov Inst. Math., 258 (2007), 178–193  mathnet  crossref  mathscinet  zmath  elib
    2. Warder J.P., “Parallel tangency in R-3”, Mathematics of Surfaces XII, Proceedings, Lecture Notes in Computer Science, 4647, 2007, 465–479  crossref  zmath  isi
    3. Nabarro A.C., Tari F., “Families of surfaces and conjugate curve congruences”, Adv. Geom., 9:2 (2009), 279–309  crossref  mathscinet  zmath  isi  scopus
    4. Saji K., Umehara M., Yamada K., “The duality between singular points and inflection points on wave fronts”, Osaka J Math, 47:2 (2010), 591–607  mathscinet  zmath  isi
    5. Oliver J.M., “On the characteristic curves on a smooth surface”, J. London Math. Soc. (2), 83:3 (2011), 755–767  crossref  mathscinet  zmath  isi  scopus
    6. Hernandez-Martinez L.I., Ortiz-Rodriguez A., Sanchez-Bringas F., “On the Affine Geometry of the Graph of a Real Polynomial”, J. Dyn. Control Syst., 18:4 (2012), 455–465  crossref  mathscinet  zmath  isi  elib  scopus
    7. Hernandez Martinez L.I., Ortiz Rodriguez A., Sanchez-Bringas F., “On the Hessian Geometry of a Real Polynomial Hyperbolic Near Infinity”, Adv. Geom., 13:2 (2013), 277–292  crossref  mathscinet  zmath  isi  scopus
    8. Izumiya S. Fuster M. Ruas M. Tari F., “Differential Geometry From a Singularity Theory Viewpoint”, Differential Geometry From a Singularity Theory Viewpoint, World Scientific Publ Co Pte Ltd, 2016, 1–368  mathscinet  isi
    9. Sano H., Kabata Y., Silva J.L.D., Ohmoto T., “Classification of Jets of Surfaces in Projective 3-Space Via Central Projection”, Bull. Braz. Math. Soc., 48:4 (2017), 623–639  crossref  zmath  isi  scopus
    10. Angel Guadarrama-Garcia M., Ortiz-Rodriguez A., “On the Geometric Structure of Certain Real Algebraic Surfaces”, Geod. Dedic., 191:1 (2017), 153–169  crossref  zmath  isi  scopus
    11. Freitas B.R., Garcia R.A., “Inflection Points on Hyperbolic Tori of S-3”, Q. J. Math., 69:2 (2018), 709–728  crossref  mathscinet  zmath  isi  scopus
    12. Uribe-Vargas R., “On Projective Umbilics: a Geometric Invariant and An Index”, J. Singul., 17 (2018), 81–90  crossref  mathscinet  zmath  isi  scopus
    13. Sasajima T., Ohmoto T., “Thom Polynomials in a-Classification i: Counting Singular Projections of a Surface”, Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Impanga 15, Ems Series of Congress Reports, eds. Buczynski J., Michalek M., Postinghel E., European Mathematical Soc, 2018, 237–259  mathscinet  zmath  isi
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