Isometries of Lie groups equipped with intrinsic distances

Enrico Le Donne, University of Jyväskylä
Fine Hall 314

We consider Lie groups equipped with distances for which every pair of points can be join with an arc with length equal to the distance of the two points. These distances are generalizations of Riemannian distances. They are completely described as subFinsler structures, by the work of Gleason, Montgomery, Zippin, and Berestowski. We are interested in studying the isometries of such metric spaces. As for the Riemannian case, we show that a (global) isometry is uniquely determined by the blown-up map at a point. The blown-up map is an isometry between the tangent metric spaces, which in this case are particular groups called Carnot groups. Generalizing a result of U.  Hamenstädt, we also show that an isometry between open sets of Carnot groups are affine maps. A key point in the argument is in showing smoothness of such isometries. The work is in collaboration with L. Capogna and A. Ottazzi.