Stokes' theorem on positively graded groups

Abstract

This paper studies the validity of Stokes' theorem for differential subcomplexes naturally adapted to the noncommutative geometry of positively graded Lie groups, with particular emphasis on Carnot groups. We introduce geometric conditions under which Stokes-type formulae hold for the Rumin complex and for a new family of spectral complexes associated with the homogeneous weight filtration of the de Rham complex. In particular, the spectral complexes allow us to recover the validity of Stokes' theorem on locally smooth intrinsic graphs. This is achieved by showing that the corresponding Stokes' formulae are governed entirely by the degree of the underlying submanifolds. Our approach also reveals that both the Rumin complex and the spectral complexes can be interpreted directly in terms of the classical de Rham complex through the Leibniz rule and integration over suitable classes of submanifolds, namely R-manifolds and spectral manifolds, respectively. Finally, motivated by this interaction between homogeneous weights and degrees of submanifolds, we propose a notion of current naturally adapted to these subcomplexes.