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

So good morning and welcome back. So today we are finally in the position to add some further structure to a manifold.

And we'll do this first on a principal bundle and later on we'll inherit the structure to associated bundles.

And so this comes under the section 5.4 and the topic is connections.

And more precisely this is connections on a principal bundle.

So on a principal bundle we have this right action of some Lie group G and furthermore it's a bundle and it's a principal bundle we're looking at.

And so there's some warning that usually or probably you already met the notion of a connection in say differential geometry in general relativity.

And they are very often connection the term connection is used synonymously with covariant derivative and synonymously with parallel transport.

Well you need to forget about this.

So really a connection is a structure on a principal bundle.

And then once you have a connection on a principal bundle we'll see that this implies a parallel transport also on the principal bundle.

And this also implies a parallel transport on any associated fiber bundle.

And then there is a very special case and that is if the fiber bundle is actually a vector bundle then it can be used to define a covariant derivative on a fiber bundle which is called a vector bundle.

So that is where the fiber carries a vector space structure.

So you see in decreasing order of generality we have connections, parallel transports and finally as a hyper special case covariant derivatives on a vector bundle which still is defined in terms of a connection if this vector bundle is the associated bundle of some principal bundle.

So this is important because it is often used synonymously and it could lead to confusion if one doesn't clarify it's not synonymous at all.

Now if you don't know about covariant derivatives forget what I'm going to say but for those of you who do it may be helpful.

The covariant derivative on a vector bundle if you write down in coordinates it's given in terms of some gamma symbols.

And you know that they do not transform as a tensor they transform as connection well components covariant derivative components.

Now the point is and the beauty is that a connection on a principal bundle will turn out to be something very simple namely a one form on the principal bundle.

So it's nothing special.

The point is you cannot localize you cannot give a coordinate dependent form well a local form without choosing a section and we know that on the principal bundle we can only choose a global section if the principal bundle is trivial and so on.

And this kind of thing will finally lead to some funny transformation behavior.

Okay but abstractly speaking the connection is just one form on the principal bundle.

However we're now going to start defining what a connection is and we will not immediately define it in terms of one form will define it differently and only see that this is equivalent to choosing one form with particular properties.

So but first we look at the following.

So this was just the introduction.

So now let's now look at a particular principal bundle P.

Let this be a principal G bundle.

Then this right action we have here induces a vector field on this principal bundle.

So then each A that we take from the Lie algebra of the group G induces a vector field which we may call XA.

It's a vector field X and a vector field on P on the principal bundle.

We know that a Lie algebra element induces a vector field on the group by the exponential map.

But now we want to induce a vector field on P.

This vector field on P is supposed to be called XA and we define it by defining it at each point P.

So for all P in P it's defined as follows. Well by its action on a function f on P as follows.

This is supposed to be f at the point P but the point P acted upon from the right.

We need a group element that acts from the right but we just have a Lie algebra element.

Well of course we get a group element by exponentiating the Lie algebra element.

But we want to look at the entire one parameter family of subgroup here.

And thus we get the function evaluated along a curve in P.

Because we start with a point P and this thing pushes us further along this curve.

Then we take the derivative with respect to this parameter t here.

And we evaluate at t equals to zero which means at the point P.

And this defines this vector field.

So right and it's useful to introduce or to define a map.

Let's call it I which actually effects this which takes an element in the Lie algebra.

This A and assigns to it the vector field I just defined on P.

This is a section of the tangent bundle of P.

Simply by assigning to A the entire vector field XA here defined on P.