A Comprehensive Introduction to Sub-Riemannian Geometry

Andrei Agrachev, Davide Barilari, Ugo Boscain

• Publisher: Cambridge University Press
• Year: 2019
• Print ISBN: 9781108476355
• eBook ISBN: 9781108754514
• Topic: Mathematics
• Subtopic: Topology
• Index: Mathematics

Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory. This comprehensive introduction proceeds from classical topics to cutting-edge theory and applications, assuming only standard knowledge of calculus, linear algebra and differential equations. The book may serve as a basis for an introductory course in Riemannian geometry or an advanced course in sub-Riemannian geometry, covering elements of Hamiltonian dynamics, integrable systems and Lie theory. It will also be a valuable reference source for researchers in various disciplines.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Copyright information
• Dedication
• Contents
• Preface
• Introduction
• 1 Geometry of Surfaces in R[sup(3)]
• 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
• References
• Index