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LEGENDRIAN GRAPHS GENERATED BY TANGENTIAL FAMILY GERMS

Published online by Cambridge University Press:  02 February 2006

Gianmarco Capitanio
Gianmarco Capitanio
Affiliation:
Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France ([email protected])
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Abstract

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We construct a Legendrian version of envelope theory. A tangential family is a one-parameter family of rays emanating tangentially from a regular plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\mathbb{R}^2$. We study the singularities of Legendrian graphs and their stability under small tangential deformations. We also find normal forms of their projections into the plane. This allows us to interpret the beak-to-beak perestroika as the apparent contour of a deformation of the double Whitney umbrella singularity $A_1^\pm$.

Keywords

Primary 14B0553C1558K25envelope theorycontact geometrytangential familiesLegendrian graphs
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 49 , Issue 1 , February 2006 , pp. 29 - 37
Copyright
Copyright © Edinburgh Mathematical Society 2006