Shortest and Straightest Geodesics in Sub-Riemannian Geometry

TL;DR

This paper reviews various definitions of geodesics in sub-Riemannian geometry, explores their interrelations, and identifies classes of manifolds where shortest and straightest geodesics coincide, simplifying their construction.

Contribution

It provides a comprehensive comparison of different geodesic definitions in sub-Riemannian manifolds and identifies conditions under which shortest and straightest geodesics are equivalent.

Findings

Shortest and straightest geodesics coincide in certain principal bundle related sub-Riemannian manifolds.

In compact homogeneous sub-Riemannian manifolds, geodesic construction reduces to Riemannian geodesics.

The paper clarifies the relationship between variational and straightness-based geodesic definitions.

Abstract

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Riemannian manifolds associated with principal bundle over a Riemannian manifolds, for which shortest geodesics coincides with straightest geodesics. Using the geometry of flag manifolds, we describe some classes of compact homogeneous sub-Riemannian manifolds (including contact sub-Riemannian manifolds and symmetric sub-Riemannian manifolds) where straightest geodesics coincides with shortest geodesics. Construction of geodesics in these cases reduces to description of Riemannian geodesics of the Riemannian homogeneous manifold or left-invariant metric on a Lie group.