Inserted: 20 sep 2019
Last Updated: 6 may 2022
This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach spaceinto a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize the abstract theory to the study of length-minimality of sub-Riemannian strictly singular curves. We conclude with the third order analysis of a specific strictly singular extremal that is not length-minimizing.