# A Tour of Subriemannian Geometries, Their Geodesics and Applications

Share this page
*Richard Montgomery*

Subriemannian geometries, also known as Carnot-Carathéodory
geometries, can be
viewed as limits of Riemannian geometries. They also arise in physical
phenomenon involving “geometric phases” or holonomy. Very roughly
speaking, a subriemannian geometry consists of a manifold endowed with a
distribution (meaning a \(k\)-plane field, or subbundle of the tangent
bundle), called horizontal together with an inner product on that
distribution. If \(k=n\), the dimension of the manifold, we get the
usual Riemannian geometry. Given a subriemannian geometry, we can define the
distance between two points just as in the Riemannian case, except we are only
allowed to travel along the horizontal lines between two points.

The book is devoted to the study of subriemannian geometries, their
geodesics, and their applications. It starts with the simplest
nontrivial example of a subriemannian geometry: the two-dimensional
isoperimetric problem reformulated as a problem of finding
subriemannian geodesics. Among topics discussed in other chapters of
the first part of the book the author mentions an elementary
exposition of Gromov's surprising idea to use subriemannian geometry
for proving a theorem in discrete group theory and Cartan's method of
equivalence applied to the problem of understanding invariants
(diffeomorphism types) of distributions. There is also a chapter
devoted to open problems.

The second part of the book is devoted to applications of subriemannian
geometry. In particular, the author describes in detail the following four
physical problems: Berry's phase in quantum mechanics, the problem of a falling
cat righting herself, that of a microorganism swimming, and a phase problem
arising in the \(N\)-body problem. He shows that all these problems can
be studied using the same underlying type of subriemannian geometry: that of a
principal bundle endowed with \(G\)-invariant metrics.

Reading the book requires introductory knowledge of differential geometry,
and it can serve as a good introduction to this new, exciting area of
mathematics.

#### Reviews & Endorsements

Very comprehensive and elegantly written book.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## A Tour of Subriemannian Geometries, Their Geodesics and Applications

- Contents vii8 free
- Introduction xi12 free
- Acknowledgments xix20 free
- Part 1. Geodesies in Subriemannian Manifolds 122 free
- Chapter 1. Dido Meets Heisenberg 324
- 1.1. Dido's problem 324
- 1.2. A vector potential 425
- 1.3. Heisenberg geometry 425
- 1.4. The definition of a subriemannian geometry 627
- 1.5. Geodesic equations 728
- 1.6. Chow's theorem and geodesic existence 930
- 1.7. Geodesic equations on the Heisenberg group 1031
- 1.8. Why call it the Heisenberg group? 1233
- 1.9. Proof of the theorem on normal geodesies 1334
- 1.10. Examples 1839
- 1.11. Notes 2142

- Chapter 2. Chow's Theorem: Getting from A to B 2344
- 2.1. Bracket-generating distributions 2344
- 2.2. A heuristic proof of Chow's theorem 2546
- 2.3. The growth vector and canonical flag 2647
- 2.4. Chow and the ball-box theorem 2748
- 2.5. Proof of the theorem on topologies 3152
- 2.6. Privileged coordinates 3152
- 2.7. Proof of the remaining ball-box inclusion 3455
- 2.8. Hausdorff measure 3455

- Chapter 3. A Remarkable Horizontal Curve 3960
- 3.1. A rigid curve 3960
- 3.2. Martinet's genericity result 4061
- 3.3. The minimality theorem 4061
- 3.4. The minimality proof of Liu and Sussmann 4162
- 3.5. Failure of geodesic equations 4465
- 3.6. Singular curves in higher dimensions 4465
- 3.7. There are no H[sup(1)]-rigid curves 4566
- 3.8. Towards a conceptual proof? 4566
- 3.9. Notes 4667

- Chapter 4. Curvature and Nilpotentization 4970
- Chapter 5. Singular Curves and Geodesies 5576
- Chapter 6. A Zoo of Distributions 7596
- 6.1. Stability and function counting 7697
- 6.2. The stable types 7798
- 6.3. Prolongation 7899
- 6.4. Goursat distributions 81102
- 6.5. Jet bundles 82103
- 6.6. Maximal growth and free Lie algebras 83104
- 6.7. Symmetries 85106
- 6.8. Types (3,5), (2,3,5), and rolling surfaces 86107
- 6.9. Type (3, 6): the frame bundle of M[sup(3)] 89110
- 6.10. Type (4,7) distributions 89110
- 6.11. Notes 93114

- Chapter 7. Cartan's Approach 95116
- 7.1. Overview 96117
- 7.2. Riemannian surfaces 97118
- 7.3. G-structures 99120
- 7.4. The tautological one-form 100121
- 7.5. Torsion and pseudoconnections 102123
- 7.6. Intrinsic torsion and torsion sequence 103124
- 7.7. Distributions: torsion equals curvature 104125
- 7.8. The Riemannian case and the o(n) lemma 106127
- 7.9. Reduction and prolongation 107128
- 7.10. Subriemannian contact three-manifolds 110131
- 7.11. Why we need pseudo in pseudoconnection 114135
- 7.12. Type and growth (4,7) 115136

- Chapter 8. The Tangent Cone and Carnot Groups 121142
- 8.1. Nilpotentization 121142
- 8.2. Metric tangent cones 122143
- 8.3. Limits of metric spaces 123144
- 8.4. Mitchell's theorem on the tangent cone 125146
- 8.5. Convergence criteria 125146
- 8.6. Weighted analysis 127148
- 8.7. Proof of Mitchell's theorem 131152
- 8.8. Pansu's Rademacher theorem 131152
- 8.9. Notes 132153

- Chapter 9. Discrete Groups Tending to Carnot Geometries 133154
- Chapter 10. Open Problems 139160

- Part 2. Mechanics and Geometry of Bundles 147168
- Chapter 11. Metrics on Bundles 149170
- Chapter 12. Classical Particles in Yang-Mills Fields 159180
- Chapter 13. Quantum Phases 173194
- 13.1. The Hopf fibration in quantum mechanics 174195
- 13.2. The reconstruction formula 176197
- 13.3. A thumbnail sketch of quantum mechanics 178199
- 13.4. Superposition and the normal bundle 179200
- 13.5. Geometry of quantum mechanics 181202
- 13.6. Pancharatnam phase 181202
- 13.7. The adiabatic connection 182203
- 13.8. Eigenvalue degenerations and curvature 184205
- 13.9. Nonabelian generalizations 185206
- 13.10. Notes 187208

- Chapter 14. Falling, Swimming, and Orbiting 189210

- Part 3. Appendices 207228
- Appendix A. Geometric Mechanics 209230
- A.1. Natural mechanical systems 209230
- A.2. The N-body problem 210231
- A.3. The Lagrangian side 211232
- A.4. The Hamiltonian side 213234
- A.5. Poisson bracket formalism 215236
- A.6. Symmetries, momentum maps, and Noether's theorem 216237
- A.7. Mechanics on groups 217238
- A.8. A mechanics dictionary 219240

- Appendix B. Bundles and the Hopf fibration 221242
- Appendix C. The Sussmann and Ambrose-Singer Theorems 229250
- Appendix D. Calculus of the Endpoint Map and Existence of Geodesies 235256

- Bibliography 247268
- Index 257278