# 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?

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