Sub-Riemannian structures on groups of 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.

References in corpus (3)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• Controllability on infinite-dimensional manifolds

Cited by in corpus (1)

• Controllability and diffeomorphism groups on manifolds with boundary