Intrinsic regular submanifolds in Heisenberg groups are differentiable graphs

The notion of rectifiable set is a key one in calculus of variations and in geometric measure theory. To develop a satisfactory theory of rectifiable sets inside Carnot groups has been the object of much research in the last ten years. Rectifiable sets, in Euclidean spaces, are natural generalizations of $C^1$ submanifolds and are defined, (but for a negligeable set), as a countable union of compact subsets contained in $C^1$ sub manifolds. Hence, understanding the objects that, inside Carnot groups, naturally take the role of $C^1$ submanifolds is a preliminary task in developing a satisfactory theory of rectifiable sets inside Carnot groups. In this paper we study the relations between different intrinsic notions of regular submanifolds inside Carnot groups. In particular, we characterize intrinsic regular submanifolds in Heisenberg groups as intrinsic differentiable graphs.