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.