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Contact Structures of Partial Differential Equations Contact Structures of Partial Differential Equations 2000 Mathematics Subject Classification: 35A30, 35L60, 35L70, 53A55 ISBN-10: 90-393-4435-3 ISBN-13: 978-90-393-4435-4 Contact Structures of Partial Differential Equations Contact Structuren voor Partiele¨ Differentiaalvergelijkingen (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof.dr. W.H. Gispen, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 10 januari 2007 des middags te 12.45 uur door Pieter Thijs Eendebak geboren op 10 september 1979 te Wageningen, Nederland Promotor: prof.dr. J.J. Duistermaat Dit proefschrift werd mede mogelijk gemaakt met financiele¨ steun van de Nederlandse Or- ganisatie voor Wetenschappelijk Onderzoek (NWO), onder het project “Contact structures of second order partial differential equations”. Contents Introduction xi 1 General theory 1 1.1 Notation . 1 1.1.1 Differential ideals . 1 1.1.2 Dual vector fields . 1 1.1.3 Vector bundles . 2 1.1.4 Jet bundles . 2 1.2 Basic geometry . 2 1.2.1 Lie groups . 2 1.2.2 Exterior differential systems . 4 1.2.3 Theory of Pfaffian systems . 6 1.2.4 Distributions . 8 1.2.5 Lie brackets modulo the subbundle . 10 1.2.6 Projections and lifting . 10 1.2.7 The frame bundle and G-structures . 11 1.2.8 The Cartan-Kahler¨ theorem . 15 1.2.9 Clean intersections . 19 1.2.10 Equivalence of coframes . 21 1.2.11 The method of equivalence . 25 1.3 Contact transformations . 29 1.3.1 Infinitesimal contact transformations . 30 1.3.2 Reeb vector fields . 31 1.3.3 Contact transformations and point transformations . 32 2 Surfaces in the Grassmannian 35 2.1 Grassmannians . 35 2.1.1 Conformal quadratic form . 36 2.1.2 Plucker¨ coordinates . 38 2.1.3 Incidence relations . 39 2.2 Hyperbolic theory . 40 2.2.1 Hyperbolic numbers . 40 vi Contents 2.2.2 Hyperbolic structures . 41 2.2.3 Almost product manifolds . 43 2.2.4 Hyperbolic groups . 44 2.3 Microlocal analysis . 46 2.3.1 Hyperbolic surfaces in the Grassmannian . 46 2.3.2 Geometrically flat surfaces . 54 2.3.3 Normal form calculations . 60 2.3.4 Moving frames . 67 3 Geometry of partial differential equations 77 3.1 Ordinary differential equations . 77 3.2 Second order scalar equations . 78 3.3 First order systems . 79 4 Contact distributions for partial differential equations 81 4.1 The contact distribution . 81 4.2 Structure on V .................................. 82 4.3 The integral elements . 87 4.4 The Nijenhuis tensor . 88 4.5 Invariant framings . 92 4.6 First order systems . 93 4.6.1 Vessiot theorem for first order systems . 95 4.6.2 The full Nijenhuis tensor . 98 4.6.3 Complexification of the tangent space . 100 4.6.4 Capˇ and Eastwood . 102 4.6.5 The flat case . 104 4.6.6 Summary . 107 4.7 Invariant framings and orders for first order systems . 108 4.7.1 Orders . 108 4.7.2 The invariant framing . 109 4.7.3 The hyperbolic case . 110 4.8 Continuous invariants . 111 4.8.1 Jet groupoids . 111 4.8.2 Invariants at order 3 . 115 5 First order systems 119 5.1 Contact transformations and symmetries . 119 5.2 Hyperbolic structure theory . 121 5.2.1 Point geometry . 121 5.2.2 Contact geometry . 130 5.2.3 Comparison with distributions . 130 5.2.4 Relations between the invariants . 131 5.2.5 Hyperbolic surfaces . 132 5.3 Local and microlocal invariants . 133 Contents vii 5.3.1 Basic idea . 134 5.3.2 Bundle map . 134 5.3.3 The local-microlocal dictionary . 135 5.3.4 Proof of the correspondence . 135 5.4 Elliptic systems . 137 5.4.1 Structure equations . 137 5.4.2 Transformations of the base foliation . 138 5.4.3 Examples . 140 5.5 Miscellaneous . 141 5.5.1 Hyperbolic exterior differential systems . 141 5.5.2 Linear hyperbolic systems . 143 6 Structure theory for second order equations 147 6.1 Contact geometry of second order equations . 147 6.1.1 Generality of solutions of a second order equation . 150 6.1.2 Monge-Ampere` invariants . 150 6.1.3 Analysis of T .............................. 151 6.2 Miscellaneous . 152 6.2.1 Jura´sˇ .................................. 152 6.2.2 Counterexample . 154 7 Systems and equations with non-generic Nijenhuis tensor 155 7.1 Pseudoholomorphic curves . 155 7.1.1 Projection to an almost complex structure . 156 7.1.2 The Darboux integrable systems . 161 7.1.3 Almost product structures . 162 7.2 Monge-Ampere` equations . 165 7.2.1 Geometry . 165 7.2.2 Equivalent definitions . 169 8 Darboux integrability 171 8.1 Some properties of Darboux integrable systems . 172 8.1.1 Base and transversal projections . 172 8.1.2 The method of Darboux . 172 8.1.3 Darboux semi-integrability . 174 8.1.4 Elliptic Darboux integrability . 175 8.1.5 Higher order Darboux integrability . 178 8.2 Hyperbolic Darboux integrability . 179 8.2.1 Relations between invariants . 179 8.2.2 Classification under contact transformations . 180 8.2.3 Point transformations . 189 8.3 Elliptic Darboux integrability . 192 8.4 Homogeneous Darboux integrable systems . 194 8.4.1 Flat case . 194 viii Contents 8.4.2 (2, 3)-Darboux integrable systems . 195 8.4.3 Affine systems . 195 8.4.4 Almost complex systems . 196 9 Pseudosymmetries 201 9.1 Pseudosymmetries of distributions . 203 9.1.1 Invariant concepts . 205 9.1.2 Formulation in differential forms . 205 9.1.3 Symmetries of pseudosymmetries . 205 9.2 Pseudosymmetries for differential equations . 206 9.2.1 Ordinary differential equations . 206 9.2.2 Second order scalar equations . 207 9.2.3 First order systems . 211 9.2.4 Monge-Ampere` equations . 213 9.3 Decomposition method . 214 9.3.1 Solution method . ..
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