Diffeological De Rham operators

Abstract

We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart Λ^1(X) of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection on the exterior algebra pseudo-bundle \bigwedge(Λ^1(X)) with the standardly defined Clifford action by Λ^1(X); the latter is therefore assumed to admit a pseudo-metric for which there exists a Levi-Civita connection. Under these assumptions, the definition is fully analogous the standard case, and our main conclusion is that this is the only way to define the De Rham operator on a diffeological space, since we show that there is not a straightforward counterpart of the definition of the De Rham operator as the sum d+d^* of the exterior differential with its adjoint. We show along the way that other connected notions do not have full counterparts, in terms of the function they are supposed to fulfill, either; this regards, for instance, volume forms, the Hodge star, and the distinction between the $k$-th exterior degree of Λ^1(X) and the pseudo-bundle of differential k-forms Λ^k(X).