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), the authors 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.

Table of Contents

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