Curl of Curl of Rank 2 Covariant Tensor

Question

Can anyone please tell me the expression for the curl of curl of a rank 2 covariant tensor? I've been going through a lot of books and sources and have not found an exact expression.

Answer 1

Remark

I looked for exactly the same reference after your question on curl of the metric tensor. I too have had little luck but at the moment I am going with this analogy? observe that for some $\vec{c}$:

$$ (\vec{a}\times\vec{b})\cdot\vec{c} = (\vec{a}\otimes\vec{b} - \vec{b}\otimes\vec{a})\vec{c}$$

where $\otimes$ denotes the tensor product. If we are allowed to take this analogy written in component form (Im an engineer not a mathematician so dont take me seriouly! note, $\nabla_{(.)}$ denotes covariant differentation):

$$[\nabla\times S \cdot \vec{c}] = [\nabla{S} - (\nabla{S})^T]\vec{c} $$

A little more manipulation with a similar analogy can give the form ($\epsilon$ is the levi-civita symbol):

$$ [\nabla \times S]^j_{i} = \nabla_lS_{mi}\epsilon^{lmj}$$

Going back to your previous question, this logic would mean the curl of the metric would be zero.