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So good morning and welcome back. Today we will further develop some technology, but along the way we'll also collect a number of interesting results.

The two most important ones are first the Whitney embedding theorems, which clarify whether there is really a difference between the intrinsic definition of a manifold and especially its tangent spaces and so on,

and the extrinsic definition, which we didn't follow, but which I hinted at, that you think of the manifold as being embedded in a higher dimensional Rn and then that you construct the tangent spaces as planes in this higher embedding space.

And I argued for the intrinsic point of view because simply for physics later on, especially if the manifold is supposed to represent the universe, it'd be rather, well, funny to think of an even bigger space into which the universe is embedded.

So we followed the intrinsic approach, but the question remains maybe something could be gained by taking the embedding approach, and the Whitney embedding theorems tell you no, they're equivalent.

Okay, so that's good news, so we'll collect that result along the way, but then for further, so we will remain to hold the intrinsic point of view, and then we will go on to develop one key point, and probably only next time we'll then look at the full implications, namely that we will define vector fields,

but we will find that the space of all vector fields is not a vector space. The space of all vectors was a vector space, but the space of all vector fields is merely a module.

So that's a vector space over a commutative ring, and in order to fully explore the implications of that, in the next lecture then, I will talk briefly about modules and what you can expect of them and what not.

All right, but first we have a very brief section 4.4, cotangent spaces to a manifold, cotangent spaces, and an important notion, the so-called gradient of a function.

Now having constructed last time the tangent spaces, it's actually quite simple to come up with the cotangent space, so let M be a smooth manifold, suitably equipped with a topology and a smooth atlas, then the cotangent space,

and normally it's given the name Tp star M, so you put the star there, that's the cotangent space, and well, this is more terminology, this is simply the vector space dual of the tangent space.

The tangent space is a vector space, so this star makes sense to use the vector space dual with the thus inherited addition and s multiplication, and this is simply notation that you let the star jump here, it means just this.

Okay, so then the cotangent space is this vector space dual, and of course the cotangent space as the vector space dual is again a vector space, there is no secret here, and since we have a finite dimensional manifold, so remark,

if the manifold is finite dimensional, and the way we define manifolds, they are always finite dimensional, because you always map into some Rd, so we didn't include in our definition the possibility to have an infinite dimensional manifold,

that can be seen as a defect, and in fact if you want to do this, you have to go through some more trouble, not much more trouble, but some more trouble, so I decided to only look at finite dimensional manifolds, and if or since the dimension of the manifold is less than infinity,

it follows that the dimension of the vector space, tangent vector space, okay dimension is less than infinity, since they're the same, these two dimensions, we showed that last time, and hence it follows that the tangent space and the cotangent space as vector spaces, they are isomorphic as vector spaces.

There is a word in order here, they are isomorphic, and isomorphic being isomorphic as a vector space means there exists a bijective linear map between them, there exists a vector space isomorphism between them, now you may ask which one, my answer is many,

but none of them can be constructed without extra structure, so I can provide some extra structure, and then I can construct this isomorphism, and that then presents a proper isomorphism, so it's true there exists isomorphisms,

but we call them non-canonical, because from the pure vector space or covector space structure, it cannot be constructed alone, it seems to be a very subtle point, because you can say, well usually if you provide a map, you always have to provide say the entries in the matrix that defines the linear map,

I mean it seems kind of natural, but the question is can you provide this map without picking arbitrarily such a matrix, the answer is here no you can't, but nevertheless it's isomorphic.

It was quite a different thing with the vector space and the dual of the dual, and that's on the problem sheet you're currently working on I guess, so the vector space isomorphic to the dual of the dual,

and for that there is a canonical isomorphism, you don't need to provide extra structure in order to construct such an isomorphism.

Anyway, so because this is often discussed canonical and non-canonical, canonical means you don't need to provide extra structure beyond the level at which you're discussing this, and we're discussing this at the level of bare vector spaces.

Alright, fine, so you have this, so you see we had the nuts and bolts approach if you want to construct the tangent space at a point with the curves, and then the directional derivative operator to the curve, we collected this,

you see we constructed everything really step by step, but for the cotangent space it's simply not necessary because you just take the vector space dual.

So what we did at this point is we, so far we look at our manifold M, we got smooth manifold M or differentiable at least, we go to a point P, and at this point P, really only at the point P,

we constructed the tangent space and hence quite trivially the cotangent space, and obviously I could now also construct the entire tensor space here, right,

so yeah, similarly one can construct a space that we may call TPM,

aha, how did you call this before, we called this TPQV, right, so to stay in this notation we should call this TRS of TPM, so TPM is just the underlying vector space, you see,

and we could construct this as the set of all maps, okay let's call them small t, too many capital T's around, small t that goes from R copies of TP star M, R copies taken together with the Cartesian product,

and then you take of the tangent space without the star, so there's a star, you take S copies and you look at the multilinear maps from here to the real numbers,

and you can also require that they're smooth and so on, and not smooth, I'm sorry, no, no, it's pure vector space thing, sorry, it's a multilinear map from this collection of vector spaces at the same point,

here the cotangent space, and right, and this would be the tensor, so this T is the tangent T, and this is the tensor T, they're different T's, alright,

okay, according to our previous notation that we take TRS of some given vector space, and if we take the tangent space as a foundation, and here we have the cotangent space,

then we can construct all the tensors at this point, all the RS tensors at this point, we can construct this comma, the set, and we know even vector space,

because it can be equipped pointwise with addition and S multiplication, even vector space of all RS tensors at P,

of course, if you have vector space, you have its dual, you can construct every tensor, so really the only time we have to really construct this from scratch,

this kind of object, is if you construct the tangent space with the curves and the tangents to the curve, and everything else by near linear algebra,

so you refer to the, I've referred to the previous, second previous subsection, by pure linear algebra, everything else is generated,

so tensors have nothing to do with differential geometry, tensors have all to do with linear algebra and these multilinear maps,

the only thing we have to construct was an underlying vector space over which then the tensor space is built, is constructed,

but we're always at this point, at this one point P.

Alright, so now that we're speaking of the tangent space here, we have another important definition, definition.

Let f be a smooth function on our smooth manifold, then at every point P in M, we have a linear map,

we have a linear map and this linear map has the name D. Now if you want to emphasize that we're looking at this D only at one point,

and we need to emphasize this at the moment, I write D sub P, so for every point you have a different map.

Okay, that's one thing. What does the map do? Well it acts on the, I'm sorry, it acts on the smooth functions

and it takes you to the cotangent space at that point. That is why it depends on the point, okay.

Take a smooth map which lives on the entire manifold and you map it there. So how is this guy defined?