Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 Geometry of Surfaces in R3
- 2 Vector Fields
- 3 Sub-Riemannian Structures
- 4 Pontryagin Extremals: Characterization and Local Minimality
- 5 First Integrals and Integrable Systems
- 6 Chronological Calculus
- 7 Lie Groups and Left-Invariant Sub-Riemannian Structures
- 8 Endpoint Map and Exponential Map
- 9 2D Almost-Riemannian Structures
- 10 Nonholonomic Tangent Space
- 11 Regularity of the Sub-Riemannian Distance
- 12 Abnormal Extremals and Second Variation
- 13 Some Model Spaces
- 14 Curves in the Lagrange Grassmannian
- 15 Jacobi Curves
- 16 Riemannian Curvature
- 17 Curvature in 3D Contact Sub-Riemannian Geometry
- 18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
- 19 Asymptotic Expansion of the 3D Contact Exponential Map
- 20 Volumes in Sub-Riemannian Geometry
- 21 The Sub-Riemannian Heat Equation
- Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
- References
- Index
3 - Sub-Riemannian Structures
Published online by Cambridge University Press: 28 October 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 Geometry of Surfaces in R3
- 2 Vector Fields
- 3 Sub-Riemannian Structures
- 4 Pontryagin Extremals: Characterization and Local Minimality
- 5 First Integrals and Integrable Systems
- 6 Chronological Calculus
- 7 Lie Groups and Left-Invariant Sub-Riemannian Structures
- 8 Endpoint Map and Exponential Map
- 9 2D Almost-Riemannian Structures
- 10 Nonholonomic Tangent Space
- 11 Regularity of the Sub-Riemannian Distance
- 12 Abnormal Extremals and Second Variation
- 13 Some Model Spaces
- 14 Curves in the Lagrange Grassmannian
- 15 Jacobi Curves
- 16 Riemannian Curvature
- 17 Curvature in 3D Contact Sub-Riemannian Geometry
- 18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
- 19 Asymptotic Expansion of the 3D Contact Exponential Map
- 20 Volumes in Sub-Riemannian Geometry
- 21 The Sub-Riemannian Heat Equation
- Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
- References
- Index
Summary
In this chapter we introduce the fundamental object ofthis book: the sub-Riemannian distance. We prove itsfiniteness and continuity (Rashevskii–Chow theorem)and the existence of length-minimizers. We thenintroduce Pontryagin extremals, which are curves inthe cotangent space satisfying a first-ordernecessary condition of length-minimality.
- Type
- Chapter
- Information
- A Comprehensive Introduction to Sub-Riemannian Geometry , pp. 67 - 108Publisher: Cambridge University PressPrint publication year: 2019