"

 Pullback of Sections vs. Pullback of Bundles 

Actually, there is a subtle point that this article gets wrong, and in fact many books get wrong (or fail to emphasize, although I believe it's because everyone learns it this way). Yes, I know that if it is wrong, I could just update it myself; but it looks like this will require a bit of rearrangement which I'm not quite in the mood to sort out now, and so it is best for me to give a brief statement of the issue here. It's also way past my bedtime, so, if anyone wants to help out while I'm sleeping, be my guest.

The problem is a terminological confusion which arises from using the term pullback for different (but related) things. For f  M → N a smooth map, there is usually not a smooth bundle map from T*N to T*M (unless f is a diffeomorphism). There is a pointwise map of individual cotangent spaces f*p  Tf(p) N → Tp M for every p, defined by operation with the pushforward of a vector in the corresponding tangent space (it is the transpose of the pushfoward on individual tangent spaces). But for the general point q in N we can't pull back to M unless we have a way of choosing a point in f-1(q) (which is especially problematic if f is not onto...). However, we can pull back sections of the cotangent bundle (i.e. covector field, and this applies in general to covariant tensor fields). The ""pullback"" everyone talks about is a map from covector fields on N to covector fields on M, that is, given ω: N → T*N a section, we define f*ω  M → T*M  by

where f*p is just the pointwise pullback defined before. This is always well-defined. Note that this is just the opposite of the problem with vectors and their fields (and in general, contravariant tensor fields): in that case, there is a pushforward which is a smooth bundle map of the tangent bundles, but in general vector field does not push forward unless the map is a diffeomorphism. It has been remarked in several books that the co-objects seem to have this wonderful extra structure, of being able to pull back sections. What really happened is that it gave up the capability to induce a smooth bundle map of the bundles in the process, thus restoring one aspect of ""duality"" that seems to have been broken when dealing with this subject."