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So good morning, happy new year and welcome back. Today we'll turn to the discussion of associated bundles.

That's section 5.3 and an associated bundle is a bundle that's associated in a certain precise sense with a principal fibre bundle.

So far we've discussed principal fibre bundles and maybe it remained a little obscure how this relates to situations or to constructions in physics in particular, as you had them before.

And in a sense one needs to introduce the associated bundles in order to start grasping the importance of principal fibre bundles.

So let's first start with the precise definition. Given a G principal bundle,

and you remember that a principal bundle is first and foremost a bundle P down, pi down to M, which as a bundle is isomorphic to P down P mod G,

and that there is a right action of a Lie group G on the total space here. So given a G principal bundle,

and so now you need something in addition and a space, well whatever it is, a smooth manifold, a topological manifold, let's say a smooth manifold,

and a smooth manifold F on which there is, on which we also have, on which we have a, now it is a left G action,

and here we use this right G action on the total space of the principal bundle, but now we consider a left G action, so I use a symbol that looks very similar,

but points to the other direction. It's a totally different map. It takes a G and an element of this new smooth manifold F,

and takes it to F, and it is a left G action with the properties that this action had. We define the associated bundle,

and so obviously whatever we're going to define will depend on the data given by the G principal bundle, by the choice of the smooth manifold F,

and by the choice of the left G action written like this. We define the associated bundle, and this associated bundle has of course,

means a bundle space, and that will be called P sub F, because it will be constructed from the F and the principal bundle.

There will be a bundle map, which we also denote by pi F, just to show it's not the projection map here, and it goes down to the same base space.

So the associated bundle will be a bundle over the same base space, okay, by, so now I need to tell you what the P F is and the pi F, by A.

Now in order to construct this pi F, we first need to introduce an equivalence relation. So let twiddle G be the relation that identifies two LH,

the relation on P Cartesian product F. So first of all we take the total space of the principal bundle and attach to each point of the total space this manifold F,

and, but that's a very big manifold then. We make it smaller, again by defining an equivalence relation, then only looking at the quotient space.

So, but that means on P cross F, we establish that any element of P cross F is supposed to be identified by this relation with an element P prime F prime,

by definition if and only if there exists a G in G. Now let's, we've got to be careful such that P prime is P acted upon from the right by this G.

Well that's all we have. We have a right action on the big bundle space and the F prime however be a left action, so that goes the other direction,

it's another map, okay, acting on F, on element in this manifold F, but now we require that it's the G inverse acting on that, okay.

Then it's quick to, you can quickly see that this twiddle G is an equivalence relation, okay,

and it being an equivalence relation we can thus consider the quotient space.

And that quotient space of course P cross F identified or modulo this equivalence relation and this guy instead of always writing this complicated guy,

we write P sub F, okay. Now very roughly speaking, you know that the, well not very roughly speaking, it's true, so the fibers of this P alone is the entire group, okay,

the fibers of the, because the P is a principal G bundle and the principal G bundle has a typical fiber, the group.

So very roughly speaking, what happens here if you then mod out the group, what is left of this first factor, well essentially only the base manifold,

very roughly speaking, and now you attach to it this fiber. So what we expect this to be is to be a fiber bundle, well we haven't written down the projection map yet,

but we expect what we constructed here to be the total space, this P F to be the total space of a bundle over M where the typical fiber is F.

But in this way here by virtue of this action, it's tied to the principal bundle. Okay, fine. So anyway, we can consider this quotient space.

And so obviously, so in other words, the elements of this P F are the equivalence classes P, F,

where P is any element in P and F is any element in F. So we will always think of this space here as these equivalence classes.

And sometimes I'll even leave out these brackets because that lightens the notation. Okay, so anyway, that's this definition.

And then B, we define the associated projection map that goes from P F down to M. Okay, we want such a guy.

Now we can practice this for the first time. So we take an element of this associated bundle, which we understand as an equivalence class of a pair of elements from the principal bundle and the fiber,

this being the identification under G, and we send this by this map pi of F. We send it to the point in M that we get by pi of P.

But now pi here, not pi down F, but pi is the projection we are handed from the principal bundle.

Now you see here, here we take an equivalence class with a P and F in there. Over here, we just take one of the representatives.

Well, not the entire representative because you need the F2, but part of it. And so we need to check well-definedness.

So this is well-defined since, okay, what do we need to check?

We need to check that if we take any other representative instead of P F and apply the map pi F to it.

So what is any other representative? Any other representative would have the form P acted upon from the right by some G

and G inverse acting from the left on F. This is clearly the same equivalence class as this guy.

So I should say this is obviously the same as this by the very definition we made here for the equivalence.

Okay, so now I apply the rule nevertheless. This is then by definition of pi F is pi of P acted upon from the right by G.

And because this is the pi, the projection of a principal bundle, we recall that, well, what does this G do?