A weighted de Rham operator leading to local potentials for Riemann and Weyl tensors

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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 can 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.