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A Comprehensive Introduction to Sub-Riemannian Geometry

Author: Andrei Agrachev; Davide Barilari; Ugo Boscain
Publisher: Cambridge : Cambridge University Press, 2019.
Series: Cambridge studies in advanced mathematics.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Provides a comprehensive and self-contained introduction to sub-Riemannian geometry and its applications. For graduate students and researchers.
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Agrachev, Andrei.
A Comprehensive Introduction to Sub-Riemannian Geometry.
Cambridge : Cambridge University Press, ©2019
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Andrei Agrachev; Davide Barilari; Ugo Boscain
ISBN: 9781108754514 1108754511
OCLC Number: 1126216749
Notes: 4.7.1 The Poincaré-Cartan 1-Form
Description: 1 online resource (766 pages)
Contents: Cover; Half-title; Series information; Title page; Copyright information; Dedication; Contents; Preface; Introduction; 1 Geometry of Surfaces in R[sup(3)]; 1.1 Geodesics and Optimality; 1.1.1 Existence and Minimizing Properties of Geodesics; 1.1.2 Absolutely Continuous Curves; 1.2 Parallel Transport; 1.2.1 Parallel Transport and the Levi-Civita Connection; 1.3 Gauss-Bonnet Theorems; 1.3.1 Gauss-Bonnet Theorem: Local Version; 1.3.2 Gauss-Bonnet Theorem: Global Version; 1.3.3 Consequences of the Gauss-Bonnet Theorems; 1.3.4 The Gauss Map 1.4 Surfaces in R[sup(3)] with the Minkowski Inner Product1.5 Model Spaces of Constant Curvature; 1.5.1 Zero Curvature: The Euclidean Plane; 1.5.2 Positive Curvature: The Sphere; 1.5.3 Negative Curvature: The Hyperbolic Plane; 1.6 Bibliographical Note; 2 Vector Fields; 2.1 Differential Equations on Smooth Manifolds; 2.1.1 Tangent Vectors and Vector Fields; 2.1.2 Flow of a Vector Field; 2.1.3 Vector Fields as Operators on Functions; 2.1.4 Nonautonomous Vector Fields; 2.2 Differential of a Smooth Map; 2.3 Lie Brackets; 2.4 Frobenius' Theorem; 2.4.1 An Application of Frobenius' Theorem 2.5 Cotangent Space2.6 Vector Bundles; 2.7 Submersions and Level Sets of Smooth Maps; 2.8 Bibliographical Note; 3 Sub-Riemannian Structures; 3.1 Basic Definitions; 3.1.1 The Minimal Control and the Length of an Admissible Curve; 3.1.2 Equivalence of Sub-Riemannian Structures; 3.1.3 Examples; 3.1.4 Every Sub-Riemannian Structure is Equivalent to a Free Sub-Riemannian Structure; 3.2 Sub-Riemannian Distance and Rashevskii-Chow Theorem; 3.2.1 Proof of the Rashevskii-Chow Theorem; 3.2.2 Non-Bracket-Generating Structures; 3.3 Existence of Length-Minimizers 3.3.1 On the Completeness of the Sub-Riemannian Distance3.3.2 Lipschitz Curves with respect to d vs. Admissible Curves; 3.3.3 Lipschitz Equivalence of Sub-Riemannian Distances; 3.3.4 Continuity of d with respect to the Sub-Riemannian Structure; 3.4 Pontryagin Extremals; 3.4.1 The Energy Functional; 3.4.2 Proof of Theorem 3.59; 3.5 Appendix: Measurability of the Minimal Control; 3.5.1 A Measurability Lemma; 3.5.2 Proof of Lemma 3.12; 3.6 Appendix: Lipschitz vs. Absolutely Continuous Admissible Curves; 3.7 Bibliographical Note; 4 Pontryagin Extremals: Characterization and Local Minimality 4.1 Geometric Characterization of Pontryagin Extremals4.1.1 Lifting a Vector Field from M to T*M; 4.1.2 The Poisson Bracket; 4.1.3 Hamiltonian Vector Fields; 4.2 The Symplectic Structure; 4.2.1 Symplectic Form vs. Poisson Bracket; 4.3 Characterization of Normal and Abnormal Pontryagin Extremals; 4.3.1 Normal Extremals; 4.3.2 Abnormal Extremals; 4.3.3 Codimension-1 and Contact Distributions; 4.4 Examples; 4.4.1 2D Riemannian Geometry; 4.4.2 Isoperimetric Problem; 4.4.3 Heisenberg Group; 4.5 Lie Derivative; 4.6 Symplectic Manifolds; 4.7 Local Minimality of Normal Extremal Trajectories
Series Title: Cambridge studies in advanced mathematics.

Abstract:

This comprehensive introduction to sub-Riemannian geometry proceeds from classical topics to cutting-edge theory and applications. The only prerequisites are calculus, linear algebra and differential  Read more...

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