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Evolving Surfaces and Evolving Implicit Differential Equations Via Contact Geometry and Singularities

Ricardo Uribe-Vargas

Abstract

We present the list of unavoidable local phenomena (transitions) occurring on the configuration of the parabolic and flecnodal curves of evolving smooth surfaces in R^3 (or RP^3). We also present the list of transitions occurring on the curve of inflections of the solutions of evolving implicit differential equations (IDE). Our results are based on the properties of the contours of surfaces (in a contact 3-space) for projections all whose fibres are Legendrian. Keywords: Surface, flecnodal curve, contact geometry, implicit differential equations.

Evolving Surfaces and Evolving Implicit Differential Equations Via Contact Geometry and Singularities

Abstract

We present the list of unavoidable local phenomena (transitions) occurring on the configuration of the parabolic and flecnodal curves of evolving smooth surfaces in R^3 (or RP^3). We also present the list of transitions occurring on the curve of inflections of the solutions of evolving implicit differential equations (IDE). Our results are based on the properties of the contours of surfaces (in a contact 3-space) for projections all whose fibres are Legendrian. Keywords: Surface, flecnodal curve, contact geometry, implicit differential equations.

Paper Structure

This paper contains 38 sections, 61 equations, 25 figures, 3 tables.

Table of Contents

  1. Transitions in Evolving Surfaces
  2. Flecnodal curve on propagating wave fronts
  3. Transitions in Evolving Implicit Differential Equations
  4. Equivalence of transitions and models of transitions
  5. Flecnodal Curve of Surfaces via Asymptotic IDEs
  6. Curve of Inflections and Legendrian Fibrations
  7. Some relevant references and the origin of this paper
  8. Transitions of Evolving BIDE
  9. Transitions of the curve of inflections of BIDE
  10. Transitions of IDE in a Riemannian plane
  11. IDEs as Surfaces in a Contact $3$-Space
  12. Characteristics : Geometric Solutions of IDE
  13. Duality as Geometry of Pairs of Legendrian Fibrations
  14. Inflections, Legendrian Fibrations and Contours
  15. Bi-inflections as Whitney Pleat Singularities
  16. ...and 23 more sections

Figures (25)

  • Figure 1: Transitions of the tangential singularities in generic evolving smooth surfaces. The sign $+$ or $-$ is the index (defined in p. \ref{['index-page']}) of the godrons taking part in the bifurcation.
  • Figure 2: The unavoidability of the "eight" of $F$ at the birth of a hyperbolic disc.
  • Figure 3: The four $A_3$-perestroikas of generic propagating wave fronts.
  • Figure 4: Folded umbrellas formed by the flecnodal curve of propagating wave fronts.
  • Figure 5: Transitions of the curve of inflections, discriminant and special points in generic evolving BIDE. The signs $+$ or $-$ are the indices of the folded singularities (defined in p. \ref{['indices-page']}).
  • ...and 20 more figures

Theorems & Definitions (25)

  • proof
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  • ...and 15 more