For a Riemannian manifold $(M,g)$ with exterior derivative d, the *codifferential* d$^\ast$ is defined to be the unique map for which
$$
g(\omega,d\omega') = g(d^* \omega,\omega'), ~~~ \omega,\omega' \in \Omega^{\bullet}.
$$
Now if $\ast$ is the Hodge map for $g$, then it is not too difficult to show that d$= (-1)^k\ast$ d $\ast$, when acting on $\Omega^k(M)$.

When $M$ is a complex manifold with holomorphic and anti-holomorphic partial derivatives $\partial$, and $\overline{\partial}$, we have a similarly defined $\partial^\ast$, and $\overline{\partial}^\ast$, and a similar relation between these objects and the original derivatives involving the Hodge map (well actually there's a reversal but no matter). For the Lefschetz map something similar also happens.

What I would like to know is whether the adjoint $\nabla^*$ of the Levi--Civita connection $\nabla$ has some similar re-expression? Or is this too naive?