Inserted: 26 may 2020
Last Updated: 26 may 2020
In this paper we introduce the notion of horizontally affine, $h$-affine in short, maps on step-two Carnot groups. When the group is a free step-two Carnot group, we show that such class of maps has a rich structure related to the exterior algebra of the first layer of the group. Using the known fact that arbitrary step-two Carnot groups can be written as a quotient of a free step-two Carnot group, we deduce from the free step-two case a description of $h$-affine maps in this more general setting, together with several characterizations of step-two Carnot groups where $h$-affine are affine in the usual sense, when identifying the group with a real vector space. Our interest for $h$-affine maps stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.