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SUB-RIEMANNIAN STRUCTURES ON GROUPS OF DIFFEOMORPHISMS

Part of: Global differential geometry Infinite-dimensional Hamiltonian systems Spaces and manifolds of mappings

Published online by Cambridge University Press:  10 August 2015

Sylvain Arguillère  and
Emmanuel Trélat
Sylvain Arguillère
Affiliation:
Johns Hopkins University, Center for Imaging Science, Baltimore, MD, USA (sarguil1@johnshopkins.edu)
Emmanuel Trélat
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, and Institut Universitaire de France, F-75005, Paris, France (emmanuel.trelat@upmc.fr)
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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.

Keywords

group of diffeomorphismssub-Riemannian geometrynormal geodesicsabnormal geodesicsreachabilityMoser theorems

MSC classification

Primary: 53C17: Sub-Riemannian geometry 58D05: Groups of diffeomorphisms and homeomorphisms as manifolds 37K65: Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 16 , Issue 4 , September 2017 , pp. 745 - 785
Copyright
© Cambridge University Press 2015 

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