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 its 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 Im 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 Its also way past my bedtime so if anyone wants to help out while Im 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 TN to TM unless f is a diffeomorphism There is a pointwise map of individual cotangent spaces fp  Tfp 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 cant pull back to M unless we have a way of choosing a point in fq which is especially problematic if f is not onto However we can pull back sections of the cotangent bundle ie 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  N  TN a section we define f  M  TM  by    where fp is just the pointwise pullback defined before This is always welldefined 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 coobjects 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