ABSTRACT AND CLASSICAL HODGE–DE RHAM THEORY
Nat Smale, Steve Smale · Analysis and Applications · 2012
In previous work, with Bartholdi and Schick [1], the authors developed a Hodge–de Rham theory for compact metric spaces, which defined a cohomology of the space at a scale α. Here, in the case of Riemannian manifolds at a small scale, we construct explicit chain maps between the de Rham complex of differential forms and the L2 complex at scale α, which induce isomorphisms on cohomology. We also give estimates that show that on smooth functions, the Laplacian of [1], when appropriately scaled, is a good approximation of the classical Laplacian.