Mobile Device Pairing

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Powered by MathJax

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 2009: Volume 203, Number 956
ISBNs: 978-0-8218-4818-0 (print); 978-1-4704-0570-0 (online)
Published electronically: August 26, 2009
MathSciNet review: 2599043
Keywords: Legendrian curve singularities, Goursat distributions, prolongation, blow-up
MSC (2000): Primary 58A30; Secondary 53A55, 58A17, 58K50

View full volume PDF

View other years and numbers:

Table of Contents


  • 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


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 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 -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.

References [Enhancements On Off] (What's this?)