Sub-Riemannian structures on groups of diffeomorphisms

TL;DR

This paper develops a sub-Riemannian geometric framework on diffeomorphism groups, deriving geodesic equations, analyzing geodesics, and exploring reachability, with implications for Moser theorems in infinite-dimensional geometry.

Contribution

It introduces strong right-invariant sub-Riemannian structures on diffeomorphism groups and derives Hamiltonian geodesic equations in this infinite-dimensional setting.

Findings

Derived Hamiltonian geodesic equations for the structures

Provided examples of normal and abnormal geodesics

Established reachability properties for diffeomorphisms

Abstract

In this paper, we define and study strong right-invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold with bounded geometry. We derive the Hamiltonian geodesic equations for such structures, and we provide examples of normal and of abnormal geodesics in that infinite-dimensional context. The momentum formulation gives a sub-Riemannian version of the Euler-Arnol'd equation. Finally, we establish some approximate and exact reachability properties for diffeomorphisms, and we give some consequences for Moser theorems.