A Tour of Subriemannian Geometries,Their Geodesies and Applications

Transcription

1 Mathematical Surveys and Monographs Volume 91 A Tour of Subriemannian Geometries,Their Geodesies and Applications Richard Montgomery American Mathematical Society

2 Contents Introduction Acknowledgments xi xix Part 1. Geodesies in Subriemannian Manifolds 1 Chapter 1. Dido Meets Heisenberg Dido's problem A vector potential Heisenberg geometry The definition of a subriemannian geometry Geodesic equations Chow's theorem and geodesic existence Geodesic equations on the Heisenberg group Why call it the Heisenberg group? Proof of the theorem on normal geodesies Examples Notes 21 Chapter 2. Chow's Theorem: Getting from A to B Bracket-generating distributions A heuristic proof of Chow's theorem The growth vector and canonical flag Chow and the ball-box theorem Proof of the theorem on topologies Privileged coordinates Proof of the remaining ball-box inclusion Hausdorff measure 34 Chapter 3. A Remarkable Horizontal Curve A rigid curve Martinet's genericity result The minimality theorem The minimality proof of Liu and Sussmann Failure of geodesic equations Singular curves in higher dimensions There are no H 1 -rigid curves Towards a conceptual proof? Notes 46 Chapter 4. Curvature and Nilpotentization 49

3 viii 4.1. The curvature of a distribution Dual curvature The derived ideal and the big growth vector The sheaf of Lie algebras Nilpotentization and Carnot groups Non-regular nilpotentizations Notes 54 Chapter 5. Singular Curves and Geodesies The space of horizontal paths A microlocal characterization Singularity and regularity Rank-two distributions Distributions determined by their curves Fat distributions Notes 73 Chapter 6. A Zoo of Distributions Stability and function counting The stable types Prolongation Goursat distributions Jet bundles Maximal growth and free Lie algebras Symmetries Types (3,5), (2,3,5), and rolling surfaces Type (3, 6): the frame bundle of M Type (4,7) distributions Notes 93 Chapter 7. Cartan's Approach Overview Riemannian surfaces G-structures The tautological one-form Torsion and pseudoconnections Intrinsic torsion and torsion sequence Distributions: torsion equals curvature The Riemannian case and the o(n) lemma Reduction and prolongation Subriemannian contact three-manifolds Why we need pseudo in pseudoconnection Type and growth (4, 7) 115 Chapter 8. The Tangent Cone and Carnot Groups Nilpotentization Metric tangent cones Limits of metric spaces Mitchell's theorem on the tangent cone Convergence criteria 125

4 ix 8.6. Weighted analysis Proof of Mitchell's theorem Pansu's Rademacher theorem Notes " 132 Chapter 9. Discrete Groups Tending to Carnot Geometries The growth of groups Nilpotent discrete groups A Carnot lattice Almost nilpotent Discrete converging to continuous 136 Chapter 10. Open Problems Smoothness of minimizers Sard for the endpoint map The topology of small balls Regularity of volumes Sublaplacians Popp's measure 143 Part 2. Mechanics and Geometry of Bundles 147 Chapter 11. Metrics on Bundles Ehresmann connections Metrics on principal bundles Examples 154 Chapter 12. Classical Particles in Yang-Mills Fields Nonabelian charged particles Wong's equations Circle bundles and Abelian groups Notes 171 Chapter 13. Quantum Phases The Hopf fibration in quantum mechanics The reconstruction formula A thumbnail sketch of quantum mechanics Superposition and the normal bundle Geometry of quantum mechanics Pancharatnam phase The adiabatic connection Eigenvalue degenerations and curvature Nonabelian generalizations Notes 187 Chapter 14. Falling, Swimming, and Orbiting Mechanics with symmetry The falling cat Swimming The phase in the three-body problem 200

5 x Part 3. Appendices 207 Appendix A. Geometric Mechanics 209 A.I. Natural mechanical systems 209 A.2. The TV-body problem 210 A.3. The Lagrangian side 211 A.4. The Hamiltonian side 213 A.5. Poisson bracket formalism 215 A.6. Symmetries, momentum maps, and Noether's theorem 216 A.7. Mechanics on groups 217 A.8. A mechanics dictionary 219 Appendix B. Bundles and the Hopf fibration 221 B.I. Generalities 221 B.2. Circle and line bundles 223 B.3. The Hopf fibration 223 B.4. Classification of line bundles 226 Appendix C. The Sussmann and Ambrose-Singer Theorems 229 C.I. Sussmann's theorem 229 C.2. The Ambrose-Singer theorem 231 C.3. Proof of the corollary to Ambrose-Singer 232 Appendix D. Calculus of the Endpoint Map and Existence of Geodesies 235 D.I. The chart theorem 235 D.2. Lemmas 236 D.3. Proof of the chart theorem 243 D.4. Proof of the geodesic existence theorems 244 Bibliography 247 Index 257