# Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

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*Wensheng Liu; Héctor J. Sussmann*

#### Table of Contents

# Table of Contents

## Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

- Table of Contents vii8 free
- 1 Introduction 112 free
- 2 Three examples 819 free
- 3 Notational conventions and definitions 1627
- 4 Abnormal extremals 2233
- 5 Sub-Riemannian manifolds, length minimizers and extremals 2536
- 6 Regular abnormal extremals for rank-two distributions 3344
- 7 Local optimality of regular abnormal extremals 4354
- 8 Strict abnormality 6071
- 9 Some special cases 6172
- Appendix A: The Gaveau-Brockett problem 7586
- Appendix B: Proof of Theorem 1 8192
- Appendix C: Local optimality of normal extremals 89100
- Appendix D: Rigid sub-Riemannian arcs and local optimality 92103
- Appendix E: A nonoptimality proof 99110
- References 102113