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

So good morning and welcome back. So far we considered the tangent spaces to a manifold

but only at one point and only last time we considered the tangent bundle which is the

disjoint union of all the tangent spaces to all the points. Now it is the tangent bundle

and the fact that we saw that it is a smooth bundle of smooth manifolds that now allows

us to define tensor fields. And the moment we define tensor fields there is little surprise

because as long as we only considered the tangent space to one point that tangent space

was a vector space. Now we are going to consider not only elements of such a vector space but

in fact sections of the whole tangent bundle which intuitively can be understood as vector

fields and we will call them vector fields. And it will turn out that we can add vector

fields and we can also scale vector fields. But other than the vectors in one tangent

space the collection of vector fields does not constitute a vector space in itself. The

collection of all vector fields will actually only be a module. And so today we will study

this connection so that is 4.7 tensor fields and modules. So unlike mathematicians most

people who apply mathematics consider sometimes rings and modules a somewhat esoteric subject

and today we will see that it is not esoteric at all and that actually the funny properties

of modules that come up at least if you compare them to vector spaces are of direct geometric

relevance and make us understand the subject better. So that is the reason why we emphasize

this. So last time we defined the tangent bundle and that prepares us for the definition

of what a vector field is. Definition. So let M be a smooth manifold and Tm its tangent

bundle which we saw as a smooth bundle so Tm down to M with the bundle projection pi,

pi is smooth. That is what we mean by the tangent bundle being a smooth bundle. A vector

field as opposed to only being a vector, a vector field is a smooth section, a smooth

section of the tangent bundle and I remind you of what a smooth section was or what a

section was in the first place. If you have a base space M and you have the total space

M there is the bundle map pi that sends you down. That is the bundle and if the bundle

map pi is smooth and M is a smooth manifold I showed you last time how to construct a

smooth atlas and there is also a problem about that on the problem sheet. But now what if

you go in the opposite direction and you call this a map sigma, if the map sigma is such

that if you first apply sigma but then you go back by pi, so you consider pi after sigma,

if this brings you back to from where you started, so if this is the identity map on

M, then the sigma is called a section of that bundle, in this case the tangent bundle and

the smooth section is simply the restriction that this map is smooth. Now we already saw

before when we talked about sections that if I think locally a section is just can be

thought of as to every point you assign a vector, an element in the fiber over that

point, so if you have a manifold M and you go to this point P, then this sigma because

of this property pi after sigma is in M, this map assigns to this point not a vector in

another tangent space at point Q but it assigns to this P a vector, so sigma of P is an element

of TPM, it assigns a vector in that tangent space at the point where you are. That's of

course the idea of a vector field that you provide a vector at that point, at another

point you might provide another vector but it's not that at this point you provide a

vector that lies in another vector space. So this is this section property. Now you can

say couldn't we have had that simpler, couldn't we have written down a map from M to the tension

space at that point. Well yeah but you don't know at which point you are if you start in

all of M and then you have the other problem you need to say what does it mean that a vector

here, if I have a whole vector field, so that's the intuition of this section, have a whole

vector field, what does it mean that this is a smooth vector field so that you can differentiate

this stuff. Well it's very difficult to say if you think about this vector at this point

then another vector at the other point you would have to introduce differentiation of

vector fields in a particular new manner but you don't need to because if this is a smooth