Variations of the sub-Riemannian distance on Sasakian manifolds with applications to coupling

TL;DR

This paper investigates the behavior of certain transport maps along geodesics on Sasakian manifolds, providing bounds on sub-Riemannian distances and preliminary results on coupling sub-Riemannian Brownian motions.

Contribution

It introduces a new class of transport maps on Sasakian manifolds and uses them to derive bounds on sub-Riemannian distances and couplings.

Findings

Transport maps have well-defined limits outside the cut-locus.

Bounds on the second derivative of the sub-Riemannian distance.

Initial results on coupling sub-Riemannian Brownian motions.

Abstract

On Sasakian manifolds with their naturally occurring sub-Riemannian structure, we consider parallel and mirror maps along geodesics of a taming Riemannian metric. We show that these transport maps have well-defined limits outside the sub-Riemannian cut-locus. Such maps are not related to parallel transport with respect to any connection. We use this map to obtain bounds on the second derivative of the sub-Riemannian distance. As an application, we get some preliminary result on couplings of sub-Riemannian Brownian motions.