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Points and curves in the Monster tower


About this Title

Richard Montgomery, Mathematics Dept., UC Santa Cruz, Santa Cruz, CA 95064, USA and Michail Zhitomirskii, Dept. of Mathematics, Technion, 32000 Haifa, Israel

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 203, Number 956
ISBNs: 978-0-8218-4818-0 (print); 978-1-4704-0570-0 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-09-00598-5
Published electronically: August 26, 2009
MathSciNet review: 2599043
Keywords:Legendrian curve singularities, Goursat distributions, prolongation, blow-up
MSC: Primary 58A30; Secondary 53A55, 58A17, 58K50

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Table of Contents


Chapters

  • Preface
  • Chapter 1. Introduction
  • Chapter 2. Prolongations of integral curves. Regular, vertical, and critical curves and points
  • Chapter 3. RVT classes. RVT codes of plane curves. RVT and Puiseux
  • Chapter 4. Monsterization and Legendrization. Reduction theorems
  • Chapter 5. Reduction algorithm. Examples of classification results
  • Chapter 6. Determination of simple points
  • Chapter 7. Local coordinate systems on the Monster
  • Chapter 8. Prolongations and directional blow-up. Proof of Theorems A and B
  • Chapter 9. Open questions
  • Appendix A. Classification of integral Engel curves
  • Appendix B. Contact classification of Legendrian curves
  • Appendix C. Critical, singular and rigid curves

Abstract


Cartan introduced the method of prolongation which can be applied either to manifolds with distributions (Pfaffian systems) or integral curves to these distributions. Repeated application of prolongation to the plane endowed with its tangent bundle yields the Monster tower, a sequence of manifolds, each a circle bundle over the previous one, each endowed with a rank $2$ distribution. In an earlier paper (2001), we proved that the problem of classifying points in the Monster tower up to symmetry is the same as the problem of classifying Goursat distribution flags up to local diffeomorphism. The first level of the Monster tower is a three-dimensional contact manifold and its integral curves are Legendrian curves. The philosophy driving the current work is that all questions regarding the Monster tower (and hence regarding Goursat distribution germs) can be reduced to problems regarding Legendrian curve singularities. Here we establish a canonical correspondence between points of the Monster tower and finite jets of Legendrian curves. We show that each point of the Monster can be realized by evaluating the $k$-fold prolongation of an analytic Legendrian curve. Singular points arise from singular curves. The first prolongation of a point, i.e. a constant curve, is the circle fiber over that point. These curves are called vertical curves. The union of the vertical curves and their prolongations form the abnormal curves (in the sense of sub-Riemannian geometry) for the Monster distribution. Using these curves we define three types of points - regular (R), vertical (V) , and tangency (T) and from them associated singularity classes, the RVT classes.

The RVT classes corresponds to singularity classes in the space of germs of Legendrian curves. All previous classification results for Goursat flags (many obtained by long calculation) now follow from this correspondence as corollaries of well-known results in the classification of Legendrian curve germs. Using the same correspondence we go beyond known results and obtain the determination and classification of all simple points of the Monster, and hence all simple Goursat germs. Finally, as spin-off to these ideas we prove that any plane curve singularity admits a resolution via a finite number of prolongations.

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