TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.