Geodesic Flow on the Diffeomorphism Group of the circle

Boris Kolev (1) , Adrian Constantin (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Department of Mathematics [Lund University]

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.

Mots clés: geodesic flow; integrable systems

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Boris Kolev, Adrian Constantin. Geodesic Flow on the Diffeomorphism Group of the circle. Commentarii Mathematici Helvetici, 2003, 78, pp.787-804. ⟨10.1007/s00014-003-0785-6⟩. ⟨hal-00003261⟩