Adjoint of the covariant derivative on a Riemannian manifold

Let $\nabla_X$ be the covariant derivative on a Riemannian manifold w.r.t. the vector field $X$. It is not clear to me what the (formal) adjoint of this operator is: I mean the operator $\tilde\nabla_X$ satisfying (for let’s say $\alpha,\beta$ 1-forms with compact support)

$\langle\nabla_X \alpha,\beta\rangle = \langle\alpha, \tilde \nabla_X \beta\rangle $

Does this operator have a special name or geometric meaning?

Many thanks for your help.

Solutions Collecting From Web of "Adjoint of the covariant derivative on a Riemannian manifold"

You can explicitly compute the adjoint by integrating by parts: the metric-compatibility of $\nabla$ gives
$$ \begin{align}
g(\nabla_X \alpha, \beta) &= X g(\alpha,\beta) – g(\alpha, \nabla_X \beta) \\
&=\text{div}(g(\alpha,\beta)X)-g(\alpha,\beta)\text{div}(X)-g(\alpha,\nabla_X \beta)
\end{align}$$

and thus integrating over a region containing the supports of $\alpha$ and $\beta$ you get

$$\langle \nabla_X \alpha, \beta \rangle = \langle\alpha,-\text{div}(X) \beta-\nabla_X\beta\rangle$$

so $\nabla_X^* = -\text{div}(X) – \nabla_X$.