A weighted de Rham operator acting on arbitrary tensor fields and their local potentials

Abstract

We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence of 2r potentials for any tensor field, where r is its form-structure number. By specialising this result to symmetric double forms, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2,3)-form potential for the Weyl tensor due to its tracelessness. This latter potential is the n-dimensional version of the double dual of the classical four dimensional (2,1)-form Lanczos potential. We also introduce a new concept of harmonic tensor fields, demonstrate that the new weighted de Rham operator has many other desirable properties and, in particular, it is the natural operator to use in the Laplace-like equation for the Riemann tensor.