Sub-Riemannian Geometry: General Theory and Examples (Encyclopedia of Mathematics and its Applications, Series Number 126)

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.

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.

Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; Part I General Theory; 1 Introductory Chapter; 1.1 Differentiable Manifolds; 1.2 Submanifolds; 1.3 Distributions; 1.4 Integral Curves of a Vector Field; 1.5 Independent One-Forms; 1.6 Distributions Defined by One-Forms; 1.7 Integrability of One-Forms; 1.8 Elliptic Functions; 1.9 Exterior Differential Systems; 1.10 Formulas Involving Lie Derivative; 1.11 Pfaff Systems; 1.12 Characteristic Vector Fields; 1.13 Lagrange-Charpit Method; 1.14 Eiconal Equation on the Euclidean Space; 1.15 Hamilton-Jacobi Equation on Rn 2 Basic Properties2.1 Sub-Riemannian Manifolds; 2.2 The Existence of Sub-Riemannian Metrics; 2.3 Systems of Orthonormal Vector Fields at a Point; 2.4 Bracket-Generating Distributions; 2.5 Non-Bracket-Generating Distributions; 2.6 Cyclic Bracket Structures; 2.7 Strong Bracket-Generating Condition; 2.8 Nilpotent Distributions; 2.9 The Horizontal Gradient; 2.10 The Intrinsic and Extrinsic Ideals; 2.11 The Induced Connection and Curvature Forms; 2.12 The Iterated Extrinsic Ideals; 3 Horizontal Connectivity; 3.1 Teleman's Theorem; 3.2 Carathéodory's Theorem; 3.3 Thermodynamical Interpretation 7.6 The Curvature of a Connection7.7 The Induced Curvature; 7.8 The Metrical Connection; 7.9 The Flat Connection; 8 Gauss' Theory of Sub-Riemannian Manifolds; 8.1 The Second Fundamental Form; 8.2 The Adapted Connection; 8.3 The Adapted Weingarten Map; 8.4 The Variational Problem; 8.5 The Case of the Sphere S3; Part II Examples and Applications; 9 Heisenberg Manifolds; 9.1 The Quantum Origins of the Heisenberg Group; 9.2 Basic Definitions and Properties; 9.3 Determinants of Skew-Symmetric Matrices; 9.4 Heisenberg Manifolds as Contact Manifolds; 9.5 The Curvature Two-Form 5.7 Horizontal and Cartesian Components5.8 Normal Geodesics as Length-Minimizing Curves; 5.9 Eigenvectors of the Contravariant Metric; 5.10 Poisson Formalism; 5.11 Invariants of a Distribution; 6 Lagrangian Formalism; 6.1 Lagrange Multipliers; 6.2 Singular Minimizers; 6.3 Regular Implies Normal; 6.4 The Euler-Lagrange Equations; 7 Connections on Sub-Riemannian Manifolds; 7.1 The Horizontal Connection; 7.2 The Torsion of the Horizontal Connection; 7.3 Horizontal Divergence; 7.4 Connections on Sub-Riemannian Manifolds; 7.5 Parallel Transport Along Horizontal Curves 3.4 A Global Nonconnectivity Example3.5 Chow's Theorem; 4 The Hamilton-Jacobi Theory; 4.1 The Hamilton-Jacobi Equation; 4.2 Length-Minimizing Horizontal Curves; 4.3 An Example: The Heisenberg Distribution; 4.4 Sub-Riemannian Eiconal Equation; 4.5 Solving the Hamilton-Jacobi Equation; 5 The Hamiltonian Formalism; 5.1 The Hamiltonian Function; 5.2 Normal Geodesics and Their Properties; 5.3 The Nonholonomic Constraint; 5.4 The Covariant Sub-Riemannian Metric; 5.5 Covariant and Contravariant Sub-Riemannian Metrics; 5.6 The Acceleration Along a Horizontal Curve Preface Part I General Theory 1 Introductory Chapter 2 Basic Properties 3 Horizontal Connectivity 4 The Hamilton-Jacobi Theory 5 The Hamiltonian Formalism 6 Lagrangian Formalism 7 Connections on Sub-Riemannian Manifolds 8 Gauss' Theory of Sub-Riemannian Manifolds Part II Examples and Applications 9 Heisenberg Manifolds 10 Examples of Heisenberg Manifolds 11 Grushin Manifolds 12 Hormander Manifolds A Local Nonsolvability B Fiber Bundles Bibliography Index Sub-Riemannian manifolds are manifolds with the Heisenberg principle built in. This comprehensive text and reference introduces the theory and applications of sub-Riemannian geometry for graduate students and researchers in pure and applied mathematics, theoretical physics, control theory, and thermodynamics. Potential applications include quantum mechanics and quantum field theory. A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach