Table of Contents
1 Introduction 1
2 Three examples 8
2.1 Riemannian geodesies 9
2.2 The Heisenberg algebra case 10
2.3 An abnormal minimizer 12
3 Notational conventions and definitions 16
3.1 Manifolds, charts, bundles, curves and arcs 16
3.2 Hamiltonian functions, Hamilton vector fields, and
bicharacteristics 17
3.3 Hamiltonian lifts and characteristics 18
3.4 Distributions, admissible curves, orbits 19
3.5 Nonholonomic distributions; regularity 20
4 Abnormal extremals 22
5 Sub-Riemannian manifolds, length minimizers and
extremals 25
5.1 The sub-Riemannian distance, length minimization and
time optimality 26
5.2 Extremals 29
5.3 The relationship between minimality and extremality 31
6 Regular abnormal extremals for rank-two distributions 33
6.1 The regular abnormal foliation of a rank-two distribution 33
6.2 Regular abnormal biextremals of a sub-Riemannian
manifold 41
7 Local optimality of regular abnormal extremals 43
7.1 The main inequality 46
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