Memoirs of the American Mathematical Society 2009; 137 pp; softcover Volume: 203 ISBN10: 0821848186 ISBN13: 9780821848180 List Price: US$72 Individual Members: US$43.20 Institutional Members: US$57.60 Order Code: MEMO/203/956
 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 threedimensional 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 blowup. 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
