The following content has been provided by the University of Erlangen-Nürnberg.

So good morning and welcome back. Today we'll be concerned with some technical aspects of

covariant derivatives and associated vector bundles and so far we discussed this topic

sketchily along the following lines. We started with some principal bundle P with some fiber G

such that there's a right action of G on there and on this principal bundle total space here we

assume there is given a connection one form omega because without a connection one form somewhere

there are no covariant derivatives either. And then we considered an associated vector bundle so that

was for some typical fiber F this total space PF in the way we defined it with some appropriate

projection map down to M and as I showed the construction of this PF and in fact the association

of such an associated vector bundle requires the information that is now in some left action

in some left action that takes a group element and acts on the fibers F from the left and in fact

you need to provide this datum of a left action in order to construct this associated bundle but

once you have provided this left action then you have here the principal bundle and here the

associated bundle. Now a connection we saw effects a parallel transport in between the fibers of the

principal bundle so if we look at some picture of this principal bundle with the base space M and

the fibers so local pictures picture the fiber G then this connection one form effected a sense

of a parallel transport along a prescribed curve in the base manifold you get a parallel transport

of this element to here and that was this parallel transport map tau. Now we saw that if we go to the

associated bundle or we argued if we go to the associated bundle which now has different fibers

so these are the fibers F so they have a different color nevertheless we can actually use the parallel

transport in the principal bundle here in order to effect again for the same curve in base space to

effect also a parallel transport between the fibers of the associated bundle and then the idea so this

was the picture so far so this is the association of the bundle by providing a left action okay on

the fiber and the idea of a covariant derivative was then to say well let F actually be a vector

space so so far no mention of that was necessary but now let's say that F is a vector space and

let's correspondingly consider only left actions that are linear so I should maybe write it like

this if the group acts on F it acts on F linearly and because we have this vector space structure

we have an addition and S multiplication we can actually compare a section if we now look at the

section Sigma that takes this point say to here and that takes this point say to there then we can

actually take this point parallel transported over here a longer curve and compare the difference

because we now have the means by this addition to take to subtract here okay and then we see

this difference and then we can construct from this difference and how far we traveled here well in

parameter space and taking derivatives you know how that works we can construct a differential

quotient and by that way we in this way we can construct a covariant derivative so that's the

idea of a covariant derivative it's very geometric it's very intuitive and it's technically a disaster

okay it's very difficult to implement technically you can and what you get is right but there is a

much neater way to do the same thing and the neater way we're now going to discuss so at least it's a

technically neater way it's the following philosophy so I again depict the elements in the game so again

we have the principal bundle and you see so this section Sigma here whose whose covariant derivative

we want to construct that's of course a section here right and the construction happens over here

in this view so far on the associated vector bundle the only thing we get from the principal

bundle is the parallel transport we push it over here okay but then the construction takes place

here so now the idea is the following so again we have the same ingredients obviously we also

need a left action we also need a left action that is linear which also requires this F that F is a

vector space so we need to need to start there and now again we wish to construct a covariant

derivative for a section Sigma over here but rather than doing this construction of the

covariant derivative on the right hand side in the associated bundle we play the following

trick we associate with this Sigma a unique function Phi Sigma that lives on this P and

is valued in the F so on P we can establish such a function and I already mentioned the

corresponding theorem and that well there is also an a G equivariance condition for this I write it