作用于任意张量场及其局部势的加权de Rham算子
A weighted de Rham operator acting on arbitrary tensor fields and their local potentials
摘要：
我们引入了一个加权de Rham算子，它作用于任意张量场，考虑它们的结构为r-折叠形式。因此，我们可以为所有维度的所有张量场定义相关的超势，并且从这些超势中的任何一个，我们以直接和自然的方式推导出任何张量场的2 r势的存在，其中r是它的形式结构数。通过将这个结果专门化到对称的双重形式，我们能够得到黎曼张量的一对势，以及由于其无迹性而得到外尔张量的单个（2，3）-形式势。后一种势是经典四维（2，1）-形式Lanczos势的双对偶的n维版本。我们还引入了调和张量场的一个新概念，证明了新的加权de Rham算子具有许多其他理想的性质，特别是它是用于黎曼张量的类拉普拉斯方程的自然算子.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
