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.

中文翻译：

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三阶开映射定理及其在端点映射的应用

本文致力于亚黎曼几何中端点映射的三阶研究。我们首先证明了从 Banach 空间到有限维流形的映射的三阶开放映射结果。在第二步中，我们计算端点映射的泰勒展开式中的三阶项，并将抽象理论专门用于研究亚黎曼严格奇异曲线的长度极小性。我们以非长度最小化的特定严格奇异极值的三阶分析作为结论。