1998/07/17 by Nik Weaver, Weaver, Nik · 1 citation
Mathematics · #28A75 #28A80 #31C25 #46E15 #46L57 #46M20 #46M25 #53C60 #54E35 #58A10 (primary) #58A15 #58A40 #58B20 #58G32 #60J60 #60J65 (secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #math.DG #math.FA #msc:28A75 #msc:28A80 #msc:31C25 #msc:46E15 #msc:46L57 #msc:46M20 #msc:46M25 #msc:53C60 #msc:54E35 #msc:58A10 #msc:58A15 #msc:58A40 #msc:58B20 #msc:58G32 #msc:60J60 #msc:60J65
paper · pdf · doi:10.48550/arxiv.math/9807096
42 pages
arxiv created 1998/07/17 · arxiv updated 2009/11/30
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).