## Sub-Riemannian GeometrySub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely: • control theory • classical mechanics • Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) • diffusion on manifolds • analysis of hypoelliptic operators • Cauchy-Riemann (or CR) geometry. Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub- Riemannian geometry has been recognized as a possible common framework for all these topics. This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists: • André Bellaïche: The tangent space in sub-Riemannian geometry • Mikhael Gromov: Carnot-Carathéodory spaces seen from within • Richard Montgomery: Survey of singular geodesics • Héctor J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers • Jean-Michel Coron: Stabilization of controllable systems |

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### Contents

The tangent space | 1 |

Accessibility | 10 |

Two examples | 23 |

Privileged coordinates | 30 |

The tangent nilpotent Lie algebra and the algebraic structure | 43 |

Gromovs notion of tangent space | 54 |

Why is the tangent space a group? | 73 |

MIKHAEL GROMOV | 82 |

Anisotropic connections | 302 |

Survey of singular geodesics | 325 |

The example and its properties | 331 |

Note in proof | 337 |

abnormal subRiemannian minimizers | 341 |

Abnormal extremals in dimension 4 | 351 |

An optimality lemma | 357 |

Conclusion | 363 |

Basic definitions examples and problems | 85 |

Horizontal curves and small CC balls | 112 |

Hypersurfaces in CC spaces | 152 |

CarnotCaratheodory geometry of contact manifolds | 196 |

Pfaffian geometry in the internal light | 234 |

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### Common terms and phrases

abnormal extremals approximation asymptotic asymptotically stabilized bound bundle C-C balls C-C manifolds C-C metric Carnot-Caratheodory codim cohomology commutators compact condition const contact manifolds continuous map coordinates Corollary corresponding curvature defined denote differential dimensional dimHau equations equiregular equivalent Euclidean example exists extends fact fc-dimensional fi-regular fibers fields Xi finite functions geodesic geometry given H C T(V Hausdorff dimension Hausdorff measure Heisenberg group Holder homeomorphisms homogeneous horizontal curves horizontal submanifolds immersions implies integral isoperimetric inequality left invariant lemma Lie algebra linear Lipschitz maps locally map f metric dist metric space minimizing nilpotent group nilpotent Lie group norm proof quasi-conformal rank regular restriction Riemannian manifold Riemannian metric satisfies self-similarity simply connected singular smooth map span stationary feedback law structure sub-Riemannian sub-Riemannian manifold subbundle submanifolds subset subspace tangent space topological triangulation Tv(V V C V vanishing vector fields zero