How do you think that the covariant derivative in is extended over covector fields defined over a surface ?
We use the Riesz Representation’s Lemma, so if
This implies that we have:
This contrast nicely with
For a general , we use the Leibniz’s rule to get
The proof that is very fun!
You gotta remember firmly that the form the reciprocal coordinated basis, still tangent vectors but representing (à la Riesz) the coordinated covectors .