Sub-Riemannian Geometry: General Theory and Examples

by Ovidiu Calin , Der-Chen Chang

• ISBN13: 9780521897303
• ISBN10: 0521897300
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-04-20
• Publisher: Cambridge University Press

Summary

Sub-Riemannian manifolds are manifolds with the Heisenberg principle built in. This comprehensive text and reference begins by introducing the theory of sub-Riemannian manifolds using a variational approach in which all properties are obtained from minimum principles, a robust method that is novel in this context. The authors then present examples and applications, showing how Heisenberg manifolds (step 2 sub-Riemannian manifolds) might in the future play a role in quantum mechanics similar to the role played by the Riemannian manifolds in classical mechanics. Sub-Riemannian Geometry: General Theory and Examples is the perfect resource for graduate students and researchers in pure and applied mathematics, theoretical physics, control theory, and thermodynamics interested in the most recent developments in sub-Riemannian geometry.

Table of Contents

Preface	p. xi
General Theory
Introductory Chapter	p. 3
Differentiable Manifolds	p. 3
Submanifolds	p. 4
Distributions	p. 4
Integral Curves of a Vector Field	p. 5
Independent One-Forms	p. 9
Distributions Defined by One-Forms	p. 11
Integrability of One-Forms	p. 13
Elliptic Functions	p. 16
Exterior Differential Systems	p. 17
Formulas Involving Lie Derivative	p. 22
Pfaff Systems	p. 24
Characteristic Vector Fields	p. 26
Lagrange-Charpit Method	p. 29
Eiconal Equation on the Euclidean Space	p. 34
Hamilton-Jacobi Equation on Rn	p. 35
Basic Properties	p. 37
Sub-Riemannian Manifolds	p. 37
The Existence of Sub-Riemannian Metrics	p. 38
Systems of Orthonormal Vector Fields at a Point	p. 39
Bracket-Generating Distributions	p. 41
Non-Bracket-Generating Distributions	p. 42
Cyclic Bracket Structures	p. 45
Strong Bracket-Generating Condition	p. 46
Nilpotent Distributions	p. 47
The Horizontal Gradient	p. 49
The Intrinsic and Extrinsic Ideals	p. 56
The Induced Connection and Curvature Forms	p. 60
The Iterated Extrinsic Ideals	p. 61
Horizontal Connectivity	p. 65
Teleman's Theorem	p. 65
Carathéodory's Theorem	p. 73
Thermodynamical Interpretation	p. 73
A Global Nonconnectivity Example	p. 75
Chow's Theorem	p. 78
The Hamilton-Jacobi Theory	p. 83
The Hamilton-Jacobi Equation	p. 83
Length-Minimizing Horizontal Curves	p. 86
An Example: The Heisenberg Distribution	p. 89
Sub-Riemannian Eiconal Equation	p. 92
Solving the Hamilton-Jacobi Equation	p. 96
The Hamiltonian Formalism	p. 98
The Hamiltonian Function	p. 98
Normal Geodesics and Their Properties	p. 102
The Nonholonomic Constraint	p. 106
The Covariant Sub-Riemannian Metric	p. 108
Covariant and Contravariant Sub-Riemannian Metrics	p. 110
The Acceleration Along a Horizontal Curve	p. 113
Horizontal and Cartesian Components	p. 113
Normal Geodesics as Length-Minimizing Curves	p. 114
Eigenvectors of the Contravariant Metric	p. 116
Poisson Formalism	p. 118
Invariants of a Distribution	p. 121
Lagrangian Formalism	p. 124
Lagrange Multipliers	p. 124
Singular Minimizers	p. 128
Regular Implies Normal	p. 130
The Euler-Lagrange Equations	p. 132
Connections on Sub-Riemannian Manifolds	p. 137
The Horizontal Connection	p. 137
The Torsion of the Horizontal Connection	p. 141
Horizontal Divergence	p. 142
Connections on Sub-Riemannian Manifolds	p. 143
Parallel Transport Along Horizontal Curves	p. 145
The Curvature of a Connection	p. 146
The Induced Curvature	p. 148
The Metrical Connection	p. 150
The Flat Connection	p. 152
Gauss' Theory of Sub-Riemannian Manifolds	p. 154
The Second Fundamental Form	p. 154
The Adapted Connection	p. 156
The Adapted Weingarten Map	p. 160
The Variational Problem	p. 163
The Case of the Sphere S3	p. 168
Examples and Applications
Heisenberg Manifolds	p. 175
The Quantum Origins of the Heisenberg Group	p. 175
Basic Definitions and Properties	p. 176
Determinants of Skew-Symmetric Matrices	p. 181
Heisenberg Manifolds as Contact Manifolds	p. 182
The Curvature Two-Form	p. 184
Volume Element on Heisenberg Manifolds	p. 189
Singular Minimizers	p. 195
The Acceleration Along a Horizontal Curve	p. 197
The Heisenberg Group	p. 199
A General Step 2 Case	p. 202
Solving the Euler-Lagrange System with ¿(x) Linear	p. 203
Periodic Solutions in the Case ¿(x) Linear	p. 209
The Lagrange Multiplier Formula	p. 211
Horizontal Diffeomorphisms	p. 213
The Darboux Theorem	p. 217
Connectivity on R2n+1	p. 218
Local and Global Connectivity	p. 224
D-Harmonic Functions	p. 226
Examples of D-Harmonic Functions	p. 229
Examples of Heisenberg Manifolds	p. 231
The Sub-Riemannian Geometry of the Sphere S3	p. 231
Connectivity on S3	p. 237
Sub-Riemannian Geodesics: A Lagrangian Approach	p. 241
Sub-Reimannian Geodesics: A Hamiltonian Approach	p. 244
The Lie Group SL(2,R)	p. 251
Liu and Sussman's Example	p. 253
Skating and Car-Like Robots as Nonholonomic Models	p. 256
An Exponential Example	p. 263
Grushin Manifolds	p. 271
Definition and Examples	p. 271
The Geometry of Grushin Operator	p. 273
Higher-Step Grushin Manifolds	p. 279
A Step 3 Grushin Manifold	p. 284
Another Grushin-Type Operator of Step 2	p. 288
Grushin Manifolds as a Limit of Riemannian Manifolds	p. 296
Hörmander Manifolds	p. 302
Definition of Hörmander Manifolds	p. 302
The Martinet Distribution	p. 303
Engel's Group and Its Lie Algebra	p. 314
The Engel Distribution	p. 316
Regular Geodesics on Engel's Group	p. 318
Singular Geodesics on Engel's Group	p. 327
Geodesic Completeness on Engel's Group	p. 329
A Step 3 Rolling Manifold: The Rolling Penny	p. 331
A Step 2(k+1) Case	p. 344
A Multiple-Step Example	p. 346
Local Nonsolvability	p. 351
Fiber Bundles	p. 354
Sub-Riemannian Fiber Bundles	p. 354
The Variational Problem	p. 358
The Hopf Fibration	p. 360
Bibliography	p. 363
Index	p. 367
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