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Good morning and welcome back. Today we'll come to the last section of the present chapter,

and actually that will be the end of the last purely mathematical chapter,

and then from next Tuesday onwards we'll look at physics applications in various fields.

And so today we consider curvature and torsion. So that's 5.7.

So in differential geometry if you look at covariant derivatives on the tension bundle and the frame bundle,

curvature and torsion are usually mentioned together as properties of a covariant derivative.

And so one might get the impression that the curvature, which we defined quite generally on the principal bundle,

that there will also be a torsion, which is quite generally defined on the principal bundle,

if there is a connection one form defined, but that is absolutely not the case.

So torsion requires additional structure beyond the structure you have on a principal bundle that's equipped with a connection.

Now in the context where curvature and torsion are usually mentioned together,

so that's the covariant derivative of an affine covariant derivative, there the frame bundle,

which is the natural principal bundle in that setting, carries a canonical structure,

which allows you to define torsion. So sometimes it seems like there is no extra structure required,

but in more general cases it is. So if I say curvature and torsion here,

I mean on a principal bundle and torsion with an extra structure.

So the key technical definition we have here is that of a covariant exterior derivative.

The definition. If we start with a principal G bundle,

P over base base M with a group G acting from the right, be a principal bundle,

and we consider also a K form, let phi be a K form, well an arbitrarily valued,

so a flower valued K form.

And so this has to do with the remark I made last time that if we talk about a Lie algebra valued K form,

or a vector space valued K form, or a whatever valued K form,

we should make sure that the thing still deserves the name K form for all the operations we perform on it.

And so I make this explicit here, so let phi be whatever valued K form,

then this object capital D acting on phi is again, no, is a K plus 1 form,

so it actually takes K vectors here on the principal bundle,

and it maps to, well whatever it is, it maps to the flower space,

and it's defined particularly simply as D phi as a K plus 1 form, it eats K vectors,

so X1, vector fields in this case, up to XK, and this is simply defined as the ordinary exterior derivative,

whose definition doesn't depend at all on what the space flower is,

and the connection, ah, I forgot the connection, a principal bundle with a connection 1 form, that's important,

with a connection, well I just write connection, but I mean connection 1 form omega that lives on this principal bundle,

then I plug in all the X's here, but only the horizontal parts,

so this is the horizontal part of X1 up to the horizontal part of XK plus 1,

you must say K plus 1, K plus 1, and that's it, and of course you remember that in the definition of this horizontal projection,

the 1 form or the connection make their appearance, alright,

so that is the covariant bit of the covariant derivative,

so then this guy, then this phi, this map is called the covariant exterior derivative,

so the covariant being the qualifier, covariant exterior derivative, well of the K form phi,

okay, so this is an incredibly useful definition, and of course for actual calculations,

we'll figure out how to represent this horizontal projection by some more explicit expression.

Now, the key notion of curvature is defined as follows,

so again we have a principal bundle on which there sits a connection 1 form,

then the curvature of the connection, so that's the property of the connection,

and if I say connection, technically of course this is equivalent to having a connection 1 form,

so the curvature of the connection 1 form is the Lie algebra valued,

the Lie algebra valued 2 form that goes by the name capital omega,

well it eats 2 vectors on the principal bundle, so 2 form on P,