Irina Markina: Group of diffeomorphisms of the unit circle and sub-Riemannian geometry

Abstract:

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.