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Good morning and welcome back. Today we'll consider parallel transport and to an extent

this is the showdown of the whole business about connections because using parallel transport,

especially on a vector bundle, we can define for instance a covariant derivative and that's

of course very useful, but as I cautioned you at the beginning, the notions of connection and

parallel transport and covariant derivative, they're all to be distinguished in general.

So what's parallel transport? Well, the basic idea is the following.

Let me illustrate it in a picture and this picture deals with a principal bundle P down

to a base manifold M and this be a G principal bundle, principal G bundle and for the purpose

of this illustration, let's say the dimension of the Lie group is just one and the dimension

of the base manifold is two because then I can draw a nice picture. Now, so assume somehow

this is our base manifold and at each point there is attached a fiber which is isomorphic

to the group, so everywhere there are these fibers and now what we're considering is a curve

in the base manifold, so this is the starting point, so say there is a point A in the base manifold

and a point B in the base manifold and we consider a curve through the base manifold,

let's call this curve gamma. So now given this curve, we would like to lift it to the principal

G bundle and there would be various ways to do that but a particularly interesting case

is the one where the principal G bundle is equipped with a connection one form. So let's

also assume that we have a connection one form omega which of course induces or defines

at each point of each fiber a little horizontal space, so if this is the point P in the fiber

on the fiber bundle then this green guy is the horizontal space at this point but of

course at other points in this fiber there are other horizontal spaces. So this is what

a connection one form induces in each fiber and so this continues, so at this point may

look like this, then at this point look like that and here may look like this and so on

and this is of course throughout the principal bundle, so this comes from this connection

one form omega and now we have a special way to lift this curve gamma from the base manifold

up to the principal bundle, namely simply by virtue of going to a particular point P

here and asking is there a tangent vector, is there a curve up here, so let me draw it,

it may look like this, such that I use the bundle projection pi here to map the curve

down, so pi projects down along this fiber that the projection of the blue curve is the

curve we started with, that's the first condition, second condition that this blue curve goes

through a specific point here because I could also look at the point that goes through here

and has this property and further that the tangent vector to this lifted curve has zero

vertical component and that it lies entirely within this horizontal subspace in such a

way that the push forward of this tangent vector here in this horizontal subspace under

the pi map is the tangent vector down here, so these are four conditions that must be

met and then we have a unique lift of this curve down here to the blue curve up there

and in that case and I will write this down formally but that's the idea, in that case

this blue curve will be called the horizontal lift of gamma and we denote this by a little

arrow here. The idea will be that of course I can do this now for every point in a fiber

and this way given a curve I get, we'll see, I'll get an isomorphism between all the fibers

where the curve along whose base points the curve passes, okay, but this isomorphism,

so how this point is being mapped to this point is being mapped to this point crucially

depends on how the curve connects the two fibers. If the curve connects the same two

fibers in another fashion there may be a different map of this point to say this point, okay,

so this mapping between the fibers will depend on what path in the base manifold I choose

and that this is dependent on the path in the base manifold will lead to a notion of

holonomy which we're going to talk about, okay, so that's the basic idea and once we

have that we have such an isomorphism between various fibers along a certain curve. If the

fiber, we can also push this, this is a principle bundle picture, a very special principle bundle