Morse-Sard type results in sub-Riemannian geometry

Let (M, ∆, g) be a sub-Riemannian manifold and x0 ∈ M . Assuming that Chow’s condition holds and that M endowed with the subRiemannian distance is complete, we prove that there exists a dense subset N1 of M such that for every point x of N1, there is a unique minimizing path steering x0 to x, this trajectory admitting a normal extremal lift. If the distribution ∆ is everywhere of corank one, we prove the existence of a subset N2 of M of full Lebesgue measure such that for every point x of N2, there exists a minimizing path steering x0 to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x0. In particular, the image of the sub-Riemannian exponential mapping is dense in M , and in the case of corank one is of full Lebesgue measure in M .