## Sub-Riemannian Geometry and Optimal TransportThe book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon. |

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### Contents

1 | |

2 SubRiemannian Geodesics | 37 |

3 Introduction to Optimal Transport | 77 |

Appendix AOrdinary Differential Equations | 120 |

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### Common terms and phrases

a.e. t e absolutely continuous absolutely continuous arc Agrachev assume BSR(x c-convex function Cauchy problem conclude easily control u e converges coordinates critical point curves defined denote differentiable dimension dsR(x End-Point mapping Example exponential map fixed following result given Hamiltonian Heisenberg group Hence infer infimum Inverse Function Theorem Kantorovitch Ker(D Ker(DoF Lebesgue measure Lie bracket linear Lipschitz in charts locally semiconcave manifold Mathematics minimizing geodesic minimizing geodesic joining Monge problem normal extremal normed vector spaces open neighborhood open set optimal transport map optimal transport plan orthonormal frame probability measures Proof of Lemma Proof Proof prove R U oo rank Remark respect Riemannian Rifford sequence shows singular horizontal paths smooth vector fields solution sub-Riemannian distance Sub-Riemannian Geometry sub-Riemannian structure sub-TWIST condition Supp(v surjective te(x Theorem totally nonholonomic distribution vector space ze(x