Lipschitz algebras and derivations II: exterior differentiation

Abstract

Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of these cases one can define a module of measurable 1-forms and a first-order exterior derivative. We give a general construction which applies to any metric space equipped with a sigma-finite measure and produces the desired result in all of the above cases. It also applies to an important class of Dirichlet spaces, where, however, the known first-order differential calculus in general differs from ours (although the two are related).