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Suppose we have a manifold with a metric and a metric compatible symmetric connection.

Suppose further that we have a smooth vector field V on this manifold.

I see two ways to take the derivative of this vector field.

I can regard my vector field as a vector-valued 0-form and take the covariant exterior derivative (which, in this simple case, is simply the covariant derivative) and obtain ∇V.

Alternatively, I can use the musical isomorphisms induced by the metric to change my vector field into a one-form field and then take the regular exterior derivative, and then use the metric again to "raise the second index" to turn my 2-form into a (1,1) tensor (like ∇V above).

This operation would give me: [itex](dV^\flat)^\sharp[/itex]

Where the sharp is implied to act on the "second index".

The question is: are these two operations equivalent? It seems to me that they would not be, but it also seems to me that it's plausible that they are since both operations involved using the metric, the first in the definition of the covariant derivative, and the second explicitely in raising and lowering indices.

Are there any relation between the two? Thanks.

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# Covariant exterior derivative vs regular exterior derivative

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