Lipschitz Algebras and Derivations II. Exterior Differentiation

Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Wiener 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 σ-finite measure and produces the desired result in all of the above cases. It agrees with Cheegerʹs construction when the latter is defined. 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).