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Sub-Riemannian Geometry : General Theory and Examples.
A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2009.
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| Colección: | Encyclopedia of mathematics and its applications.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 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.
- 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.
- 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.
- 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.