Let $(M^n,g)$ be a Riemannian manifold. It is well known that the following identity holds:
$$ d(\alpha \wedge \beta) = (d\alpha) \wedge \beta + (-1)^p \alpha \wedge (d \beta) $$
where $\alpha$ is a $p$-form on $M$ and $\beta$ is a $q$-form on $M$. Is there a formula for the codifferential of a wedge product? That is, can we express $\delta(\alpha \wedge \beta)$ in terms of $\delta \alpha$ and $\delta \beta$ (and possibly $d \alpha$ and $d \beta$)? Just recalling: $\delta$ is defined as being equal to $(-1)^{n(k-1)+1}\ast d \ast$ acting on $k$-forms.