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This chapter is fundamental; it is the core of this book. We consider sub-Finsler Lie groups and discuss their differential geometry and their metric geometry when equipped with Carnot-Carathéodory distances. In Sect. 7.1, we discuss the perspective of left-invariant sub-Finsler structures. We show how, in sub-Finsler geometry, quotients can be viewed as submetries, allowing for geodesic lifting. We discuss Chow’s theorem and establish a weakened version of the Ball-Box Theorem, which will be further refined in Theorem 12.5.3 . In Sect. 7.2, we discuss the endpoint map and its singular elements known as abnormal curves. In Sect. 7.3, in the framework of sub-Riemannian groups, we conduct a first-order analysis of geodesics, leading to the Pontryagin Maximum Principle. In the final Sect. 7.4, we demonstrate that geodesic homogeneous manifolds are Carnot-Carathéodory spaces.

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Sub-Finsler Lie Groups

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  • 发表日期：

    2025年

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  • Enrico Le Donne

摘要

This chapter is fundamental; it is the core of this book. We consider sub-Finsler Lie groups and discuss their differential geometry and their metric geometry when equipped with Carnot-Carathéodory distances. In Sect. 7.1, we discuss the perspective of left-invariant sub-Finsler structures. We show how, in sub-Finsler geometry, quotients can be viewed as submetries, allowing for geodesic lifting. We discuss Chow’s theorem and establish a weakened version of the Ball-Box Theorem, which will be further refined in Theorem 12.5.3 . In Sect. 7.2, we discuss the endpoint map and its singular elements known as abnormal curves. In Sect. 7.3, in the framework of sub-Riemannian groups, we conduct a first-order analysis of geodesics, leading to the Pontryagin Maximum Principle. In the final Sect. 7.4, we demonstrate that geodesic homogeneous manifolds are Carnot-Carathéodory spaces.