Talk:Pullback (differential geometry)

message: 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
:(f^*\omega)(p) = f^{*p}(\omega(f(p)))
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.
71.136.49.68 12:32, 11 August 2006 (UTC)
thread_title:  Pullback of Sections vs. Pullback of Bundles 

message: The notion of a pullback in mathematics is really fundamental and general. In my view, it separates into two distinct (but related) meanings: the notion of a categorical pullback including as a special case, the notion of a pullback bundle, and (second) the idea of precomposition with a map. Almost all notions of pullback are combinations of these ideas (and the dual notion of pushforward). I would like to propose that this article be renamed Pullback (differential geometry) or something similar, and that a short article should be written introducing the general idea of pullback and linking to the other articles on the subject. I would be happy to make a first draft of such an article.
In the long term, we might considering merging this article with the pushforward article. However, if we ackowledge that the present article needs a substantial clean-up/re-write, as several people have remarked on this page, then this is even more true for the pushforward article! Geometry guy 22:31, 9 February 2007 (UTC)
replies: text: Moving this material to Pullback (differential geometry) is a good idea. Pullback can then be, at the least, a disambiguation page, or better an overview of the concept and how the various cases tie together. I would argue against merging this with pushforward though. There is plenty of material for two separate articles. Some overlap is fine. -- Fropuff 23:46, 9 February 2007 (UTC)

text: I've gone ahead and moved it. For now I'll set up pullback as a disambig page. To be expanded later if you see fit. -- Fropuff 23:53, 9 February 2007 (UTC)
thread_title:  Pullbacks in general, and pushforwards 

message: There is no need to restrict this to covariant tensors - it works with arbitrary tensors as well and actually should be defined on arbitrary tensors. See Abraham, Marsden, Ratiu, in case someone is interested in details.  — Preceding unsigned comment added by 87.163.203.178 (talk) 18:23, 9 March 2022 (UTC)
thread_title:  Covariant Restriction