Isometries in Lie groups

Abstract: 

Let GG be a Lie group and d a left invariant distance on GG. It is well known that if the group is RnRn and the distance Euclidean, then every isometry is affine, i.e., the composition of a translation and an (orthogonal) isomorphism. We ask whether some formulation of this fact still holds in the general setting: we provide examples of groups and distances where isometries are affine in a suitable sense and others where isometries are not necessarily affine. For most of the seminar I will focus on the example of the Heisenberg group endowed with its control distance. These results were obtained in collaboration with E. Le Donne. In the last part of the talk I shall discuss some open problems.