Covariant derivative, metric compatibility, Levi-Civita connections, etc.
#3
instanton
I am not sure what you are exactly meant by "metric" here.
If a vector space is equipped with "length" (I am using more intuitive notion.) it is called metric space. If you consider a set of nice functions over some domain they usually form a vector space. If this space has notion of norm, (and being complete) then it is called Banach space. With this norm you can define a derivative by usual calculus textbook way. ( In usual book way absolute value of real number plays the roll of norm). In metric space you can use this metric to define a norm.
Metric on Riemannian manifold is slightly different concept even though they are of course related. You can define a covariant derivative with any affine connections on Riemannian manifold. Fumdamental theorem of Riemannian geometry says, then, there is a unique covariant derivative that is compatible with metric. i.e. you can use metric to fix your affine connection to define covariant derivative.