Third order open mapping theorems and applications to the end-point map

Abstract

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 space into 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.

ODE trajectories as abnormal curves in Carnot groups

“…[ABB20, Section 12.3] for the definition. Moreover the groups F can be chosen such that the abnormal lifts also satisfy even higher order abnormality conditions such as the third order condition of [BMP19].…”