Riemannian and Sub-Riemannian geodesic flows

TL;DR

This paper explores the relationship between Riemannian and sub-Riemannian geodesic flows, showing their commutation under specific conditions, and characterizes sub-Riemannian geodesics via Riemannian geodesics and horizontal lifts.

Contribution

It establishes the conditions under which Riemannian and sub-Riemannian geodesic flows commute and provides a new characterization of sub-Riemannian geodesics.

Findings

Geodesic flows commute iff the extended metric is parallel.

Sub-Riemannian geodesics are horizontal lifts of Riemannian geodesics.

Characterization of sub-Riemannian geodesics on totally geodesic Riemannian submersions.

Abstract

In the present paper we show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions. As a consequence we can characterize sub-Riemannian geodesics as the horizontal lifts of projections of Riemannian geodesics.