Isometries of Carnot Groups and Sub-Finsler Homogeneous Manifolds

We show that isometries between open sets of Carnot groups are affine. This result generalizes a result of Hamenstädt. Our proof does not rely on her proof. We show that each isometry of a sub-Riemannian manifold is determined by the horizontal differential at one point. We then extend the result to sub-Finsler homogeneous manifolds. We discuss the regularity of isometries of homogeneous manifolds equipped with homogeneous distances that induce the manifold topology.