Shortest paths for sub-Riemannian metrics on rank-two distributions / Wensheng Liu, Héctor J. Sussman.

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Bibliographic Details
Main Authors:	Liu, Wensheng, 1962- (Author), Sussmann, Hector J., 1946- (Author)
Format:	eBook
Language:	English
Published:	Providence, Rhode Island, United States : American Mathematical Society, 1995.
Series:	Memoirs of the American Mathematical Society ; Volume 118, no. 564.
Subjects:	Geometry, Riemannian. Geodesics (Mathematics) Geometry, Riemannian
Online Access:	Click for online access

Table of Contents:

Table of Contents
1 Introduction
2 Three examples
2.1 Riemannian geodesies
2.2 The Heisenberg algebra case
2.3 An abnormal minimizer
3 Notational conventions and definitions
3.1 Manifolds, charts, bundles, curves and arcs
3.2 Hamiltonian functions, Hamilton vector fields, and bicharacteristics
3.3 Hamiltonian lifts and characteristics
3.4 Distributions, admissible curves, orbits
3.5 Nonholonomic distributions; regularity
4 Abnormal extremals
5 Sub-Riemannian manifolds, length minimizers and extremals
5.1 The sub-Riemannian distance, length minimization and time optimality5.2 Extremals
5.3 The relationship between minimality and extremality
6 Regular abnormal extremals for rank-two distributions
6.1 The regular abnormal foliation of a rank-two distribution
6.2 Regular abnormal biextremals of a sub-Riemannian manifold
7 Local optimality of regular abnormal extremals
7.1 The main inequality
7.2 The normal form theorem
7.3 The optimality theorem
8 Strict abnormality
9 Some special cases
9.1 2- and 3-generating distributions
9.2 3-Regular distributions9.3 The 4- and 5-dimensional cases
9.4 The three-dimensional case
9.5 A Lie group example
9.6 A nonsmooth abnormal extremal
Appendix A: The Gaveau-Brockett problem
Appendix B: Proof of Theorem 1
B.l: Control systems
B.2: The Maximum Principle
Appendix C: Local optimality of normal extremals
Appendix D: Rigid sub-Riemannian arcs and local optimality
Appendix E: A nonoptimality proof
References

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