# Points and Curves in the Monster Tower

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*Richard Montgomery; Michail Zhitomirskii*

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

# Table of Contents

## Points and Curves in the Monster Tower

- Preface ix10 free
- Chapter 1. Introduction 112 free
- 1.1. The Monster construction 112
- 1.2. Coordinates and the contact case 112
- 1.3. Symmetries. Equivalence of points of the Monster 213
- 1.4. Prolonging symmetries 213
- 1.5. The basic theorem 213
- 1.6. The Monster and Goursat distributions 314
- 1.7. Our approach 415
- 1.8. Proof of the basic theorem 516
- 1.9. Plan of the paper 617
- Acknowledgements 1122

- Chapter 2. Prolongations of integral curves. Regular, vertical, and critical curves and points 1324
- 2.1. From Monster curves to Legendrian curves 1324
- 2.2. Prolonging curves 1324
- 2.3. Projections and prolongations of local symmetries 1526
- 2.4. Proof of Theorem 2.2 1526
- 2.5. From curves to points 1627
- 2.6. Non-singular points 1728
- 2.7. Critical curves 1728
- 2.8. Critical and regular directions and points 2031
- 2.9. Regular integral curves 2031
- 2.10. Regularization theorem 2233
- 2.11. An equivalent definition of a non-singular point 2334
- 2.12. Vertical and tangency directions and points 2435

- Chapter 3. RVT classes. RVT codes of plane curves. RVT and Puiseux 2738
- 3.1. Definition of RVT classes 2738
- 3.2. Two more definitions of a non-singular point 2839
- 3.3. Types of RVT classes. Regular and entirely critical prolongations 2839
- 3.4. Classification problem: reduction to regular RVT classes 2940
- 3.5. RVT classes as subsets of PkR2 2940
- 3.6. Why tangency points? 3041
- 3.7. RVT code of plane curves 3142
- 3.8. RVT code and Puiseux characteristic 3344

- Chapter 4. Monsterization and Legendrization. Reduction theorems 3950
- 4.1. Definitions and basic properties 3950
- 4.2. Explicit calculation of the legendrization of RVT classes 4152
- 4.3. From points to Legendrian curves 4253
- 4.4. Simplest classification results 4354
- 4.5. On the implications and shortfalls of Theorems 4.14 and 4.15 4455
- 4.6. From points to Legendrian curve jets. The jet-identification number 4556
- 4.7. The parameterization number 4758
- 4.8. Evaluating the jet-identification number 5061
- 4.9. Proof of Proposition 4.44 5263
- 4.10. From Theorem B to Theorem 4.40 5364
- 4.11. Proof that critical points do not have a jet-identification number 5566
- 4.12. Proof of Proposition 4.26 5566
- 4.13. Conclusions. Things to come 5566

- Chapter 5. Reduction algorithm. Examples of classification results 5768
- 5.1. Algorithm for calculating the Legendrization and the parameterization number 5768
- 5.2. Reduction algorithm for the equivalence problem 5970
- 5.3. Reduction algorithm for the classification problem 6071
- 5.4. Classes of small codimension consisting of a finite number of orbits 6172
- 5.5. Classification of tower-simple points 6374
- 5.6. Classes of high codimension consisting of one or two orbits 6778
- 5.7. Further examples of classification results; Moduli 6980

- Chapter 6. Determination of simple points 7182
- Chapter 7. Local coordinate systems on the Monster 8596
- Chapter 8. Prolongations and directional blow-up. Proof of Theorems A and B 95106
- 8.1. Directional blow-up and KR coordinates 96107
- 8.2. Directional blow-up and the maps ET, EV, L 99110
- 8.3. Proof of Theorem A for Puiseux characteristics [0; 1] 100111
- 8.4. Further properties of the directional blow-up 101112
- 8.5. Proof of Theorem A for arbitrary Puiseux characteristics 104115
- 8.6. Proof of Theorem B of section 4.8 105116
- 8.7. Proof of Propositions 8.10 and 8.11 106117

- Chapter 9. Open questions 109120
- Appendix A. Classification of integral Engel curves 119130
- Appendix B. Contact classification of Legendrian curves 123134
- Appendix C. Critical, singular and rigid curves 131142
- Bibliography 135146
- Index 137148 free