Covariant Derivative: Definition & Meaning

[salparadise]

Hello everyone,

While studying properties of Riemann and tensor and Killing vectors, I found this notation/concept that I'm not sure of it's meaning.

What does it mean to have a covariant derivative, using semi-colon notation, showing in upper index position. Is it just a matter of raising the index via the metric of ordinary covariant derivative.

I tried to search in different textbooks for this, but couldn't find anything.

Thanks in advance for your help.

 

[Ben Niehoff]

$$A^{ab;c} = g^{cd} \nabla_d A^{ab} = \nabla_d (g^{cd} A^{ab})$$

Note that $\nabla_c g_{ab} = 0$.