I know that there exists a connection on a principal bundle and via parallel transport it is possible to define a a covariant derivative on the associated bundle.
However, can we also define a covariant derivative on the principal bundle. I.e. something that can differentiate a section along a vector field? Or do we need a linear structure like the one in a vector bundle to 'take derivatives'?
Best Answer
So from my point of view what a covariant derivative should be or should do I'd say that indeed you need a linear structure.
So in principle, when passing from simply vector space-valued functions $M\to W$ (or sections in the trivial bundle $M\times W$, resp.) to sections in a not-necessarily trivial bundle $E\to M$ with fiber-type $W$ you run into troubles defining a derivative of such functions. Usually this is paraphrased as that it is intrinsically "not possible to compare points in different fibers". However, it is possible to chose a covariant derivative, and thereby (at least locally) a frame which is "constant" or parallel to $M$ (cf. my comment above). Once you everywhere have distinguished such a notion of parallelity you can now indeed take derivatives and view it as "how a function changes compared to what we call constant", which is of course just the action of the covariant derivative. For this notion you need a vector space structure.
On the other hand, on a principal bundle you also have a notion of parallel in the sense of everywhere horizontal but as you don't have a vector space structure in the fibers, so "comparison with a constant section" works a bit different, particularly not by something having similar properties as a derivative in the above sense. Rather would you do something like @Deane in the comments above, but I'd say this is not a "derivative" in the sense of what a derivative should do.