Why are normal sub-Riemannian extremals locally minimizing?

TL;DR

This paper presents a new geometric proof that normal extremals in sub-Riemannian geometry are locally energy-minimizing, clarifying the connection between their optimality and geometric properties.

Contribution

It offers a novel geometric proof of local optimality of normal extremals, highlighting the role of extremal regularity in sub-Riemannian geometry.

Findings

Provides a new geometric proof of local minimality

Clarifies the relation between extremal regularity and optimality

Enhances understanding of the geometric reasons for optimality

Abstract

It is well-known that normal extremals in sub-Riemannian geometry are curves which locally minimize the energy functional. Most proofs of this fact do not make, however, an explicit use of relations between local optimality and the geometry of the problem. In this paper, we provide a new proof of that classical result, which gives insight into direct geometric reasons of local optimality. Also the relation of the regularity of normal extremals with their optimality becomes apparent in our approach.