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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?

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    $\begingroup$ The fact that covariant derivatives require more structure is exactly why it is not possible to express them in terms of exterior derivatives - by choosing a different metric we can get a different covariant derivative. Do I misunderstand your question? $\endgroup$ Sep 1, 2017 at 6:47
  • $\begingroup$ I think you are right and I just got confused. You pointed it you correctly $\endgroup$ Sep 1, 2017 at 6:54
  • $\begingroup$ This might be of interest: $\endgroup$ Sep 1, 2017 at 14:07
  • $\begingroup$ @AloneAndConfused thank you, very helpful. $\endgroup$ Sep 2, 2017 at 9:25

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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: $$ 2 g (\nabla_Y X, Z) = (L_X g) (Y, Z) + (d \theta_X) (Y, Z) $$ where $g$ is a Riemannian metric, $X, Y, Z$ are some vector fields, $\nabla$ is the corresponding Levi-Civita connection, $L$ is the Lie derivative, $d$ is the exterior derivative, $\theta_X$ is the dual covector to $X$.

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