Sub-Riemannian Geometry and Optimal Transport

Abstract

International audienceThe 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

On the subRiemannian cut locus in a model of free two-step Carnot group

“…On sufficiently small intervals, γ is a length-minimizer (see [LS95], [Rif14] and [ABB16]). Under linear change of parameter and dilation in (2.3), γ behaves as follows…”

On the lack of semiconcavity of the subRiemannian distance in a class of Carnot groups

“…See [Mon02,Rif14,ABB16]. At the point u = u, we have P t 0 x = e tX 2 x = x 1 , x 2 + t, x 3 + tx 1 , x 4 + …”

Closing Aubry Sets I

“…For the sake of completeness, we state below the quantitative version of the inverse function theorem that we used in Appendix B. We refer the reader to [1,45] for a proof. THEOREM C.1.…”

“…Most of the results presented below cannot be found in classical references of control theory. However, we encourage the reader to have a look at the book [14] (see also the forthcoming book [45]) for more details about the material discussed in the next subsections.…”