Irina Markina, University of Bergen
We consider the group of sense-preserving diffeomorphisms of the unit circle and its central extension – the Virasoro-Bott group as sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a smooth manifold M with a given sub-bundle D of the tangent bundle, and with a metric defined on the sub-bundle D. The different sub-bundles on considered groups are related to some spaces of normalized univalent functions. We present formulas for geodesics for different choices of metrics. The geodesic equations are generalizations of Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We show that any two points in these groups can be connected by a curve tangent to the chosen sub-bundle. We also discuss the similarities and peculiarities of the structure of sub-Riemannian geodesics on infinite and finite dimensional manifolds.
This is a join work with E.Grong, University of Bergen.