I recently came across the paper *Les variétés de dimension 4 à signature non nulle dont
la courbure est harmonique sont d’Einstein* by Jean Pierre Bourguignon. What he shows in §8 is that the Weitzenböck curvature operator $\mathfrak{Ric}\_\text{R}$ on $p$-forms is given by $$\mathfrak{Ric}\_\text{R}(\omega)(X\_1,\dots,X\_p) = (\hat{\omega} \circ \hat{R\_p})(X\_1,\dots,X\_p)$$
where
$$R\_p = \left(\tfrac{1}{2(p-1)}\text{Ric} \mathbin{\bigcirc\mspace{-19mu}\wedge\mspace{3mu}} \text{g} - \text{Rm}\right) \mathbin{\bigcirc\mspace{-19mu}\wedge\mspace{3mu}} \text{g}^{p-2}.$$
Here, I just can't figure out what he means by $\hat{\omega} \circ \hat{R\_p}$.


Earlier in the paper he defines (in (2.11)) for a $C \in S^2\Lambda^{2}V$, interpreted as a self-adjoint map $\Lambda^{2}V \to \Lambda^{2}V$, and a form $\eta \in \Lambda^2(V)$, that $$\hat{C}(\eta) = \sum\limits\_{i,j = 1}^n \eta(e\_i,e\_j)C(e\_{i},e\_{j})$$ for an orthonormal basis $(e\_i)\_{1 \le i \le n}$ of $V$.


I see how this can be generalized into:
For a $C \in S^2\Lambda^{p}V$, interpreted as a self-adjoint map $\Lambda^{p}V \to \Lambda^{p}V$, and a form $\eta \in \Lambda^p(V)$, define $$\hat{C}(\eta) = \sum\limits\_{i\_1, \dots, i\_p = 1}^n \eta(e\_{i\_1},\dots,e\_{i\_p})C(e\_{i\_1},\dots,e\_{i\_p})$$ for an orthonormal basis $(e\_i)\_{1 \le i \le n}$ of $V$.
And I am thinking that this is what he means here for $C = R\_k$. But this still doesn't answer what $\hat{\omega}$ would be.


I am thinking that $(\hat{\omega} \circ \hat{R\_p})$ may just mean $\hat{R\_p}(\omega)$; but why wouldn't he have written it like this then? Probably I am missing something central here..


The paper can be found here for free:
[Les variétés de dimension 4 à signature non nulle dont la courbure est harmonique sont d’Einstein by Jean Pierre Bourguignon](https://gdz.sub.uni-goettingen.de/id/PPN356556735_0063?tify=%7B%22pages%25%2022:%5B271%5D%7D)



Invariant description of the Weitzenböck curvature operator by Bourguignon
I think the idea is to think of $\hat{R}\_p$ as a mapping from $\Lambda^pM$ to itself, and $\hat{\omega}$ as a mapping from $\Lambda^pM$ to $E$ (the vector bundle in which $\omega$ takes values), and then just compose these two operators (to get something $E$ valued in the end).


In the case where $E$ is the trivial (scalar) bundle, it works out to be $\hat{R}\_p(\omega)$ since $R\_p$ is symmetric.


