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

## TL;DR

This paper advances the understanding of the end-point map in sub-Riemannian geometry by establishing third order open mapping theorems, computing higher order Taylor expansions, and analyzing length-minimality of singular curves.

## Contribution

It introduces third order open mapping results, computes third order terms in the end-point map's Taylor expansion, and applies these to study length-minimality of singular curves.

## Key findings

- Third order open mapping theorems for maps from Banach spaces to manifolds.
- Explicit computation of the third order term in the end-point map's Taylor expansion.
- Analysis showing certain singular extremals are not length-minimizing.

## 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.