Express covariant derivative in terms of exterior derivative I know there is an intimate relation between covariant, Lie and exterior derivative. I know that the covariant derivative requires more structure than the exterior, so it would be possible. How do I express a covariant derivative $\nabla_{X} Y$ in terms of the exterior derivative, assuming the Levi-Civita connection?

As it is well known, the Levi-Civita connection can be implicitly defined via the Koszul formula. P.Petersen in his "Riemannian Geometry" gives a nice presentation of this formula: $$ 2 g (\
abla_Y X, Z) = (L_X g) (Y, Z) + (d \theta_X) (Y, Z) $$ where $g$ is a Riemannian metric, $X, Y, Z$ are some vector fields, $\
abla$ is the corresponding Levi-Civita connection, $L$ is the Lie derivative, $d$ is the exterior derivative, $\theta_X$ is the dual covector to $X$.