Morse-Sard type results in sub-Riemannian geometry

Let (M,,g) be a sub-Riemannian manifold and x₀ M. Assuming that Chows condition holds and that M endowed with the sub-Riemannian 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.Mathematics Subject Classification (2000): 53C17, 49J52