**Memoirs of the American Mathematical Society**

1996;
104 pp;
Softcover

MSC: Primary 53;
Secondary 49

Print ISBN: 978-0-8218-0404-9

Product Code: MEMO/118/564

List Price: $44.00

AMS Member Price: $26.40

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-0143-6
Product Code: MEMO/118/564.E**

List Price: $44.00

AMS Member Price: $26.40

MAA Member Price: $39.60

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

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*Wensheng Liu; Héctor J. Sussmann*

This work studies length-minimizing arcs in sub-Riemannian manifolds \((M, E, G)\) where the metric \(G\) is defined on a rank-two bracket-generating distribution \(E\). The authors define a large class of abnormal extremals—the “regular” abnormal extremals—and present an analytic technique for proving their local optimality. If \(E\) satisfies a mild additional restriction-valid in particular for all regular two-dimensional distributions and for generic two-dimensional distributions—then regular abnormal extremals are “typical,” in a sense made precise in the text. So the optimality result implies that the abnormal minimizers are ubiquitous rather than exceptional.

#### Readership

Graduate students, mathematicians, physicists, engineers interested in geometry, optimal control theory, or the calculus of variations.

#### 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