# Talk:Pullback (differential geometry)

## Categorical Pullback

The pullback of a vector bundle is a categorical pullback. It is the pullback of the diagram

 f      p


M--->N<----VN

Where VN is the vector bundle over N. So if we replace VN by T*N we see that a pullback of a differential form is a general pullback. So they are more than loosely related.

I disagree. The arrow in the diagram for the pullback of differential forms goes the wrong way for it to be a categorical pullback. Furthermore, the pullback map for differential forms can have a kernel, but the categorical pullback is an isomorphism of the fibres.151.204.6.171

## Diffeomorphism

Linas, regarding the statement:

If the map from M to N is bijective, then the pushforward can be defined as ${\displaystyle f_{*}=\left(f^{-1}\right)^{*}}$.

I should have stated that I removed it for more reasons then bijectivity. The pushforward is defined on tangent vectors and contravariant tensors while the pullback is defined on cotangent vectors and covariant tensors. So the statement doesn't really make any sense. These two thing can't be equal if they operate on different domains. -- Fropuff 04:52, 2005 Apr 10 (UTC)

OK, yes, sorry, you're right; I was being sloppy and careless here, used to having a metric around, was just going for the quick edit. Went awry. For this to work, M and N would also need to have metrics; I guess this is sufficient, not sure. It has been a *very long time* since I've worked with diff geom in general, and so I'm dredging from memory in a rather incorrect fashion, never mind I was an indifferent student to begin with. I guess saying f has to be a diffeomorphism rather than saying 'bijective' would also be safer. linas 00:58, 12 Apr 2005 (UTC)

The statement may make sense if M and N have metrics, but is it true? It certainly isn't true if the metrics are arbitrary and f is an arbitrary diffeomorphism. It may work if f is an isometry, although I am not sure. Can you check your sources? -- Fropuff 16:59, 2005 Apr 12 (UTC)

Ugh, OK, I'll see if I can do that; yes, its possible that f may need to be an isometry. I'll have to put my thinking cap on, this may take a little while; feel free to then delete this formula until that time. I guess I should review the material, I wasn't planning to this week. My apologies, again. linas 15:12, 13 Apr 2005 (UTC)
OK, well, a beer has fortified me. Lets see if I can do this now. 1) Having a metric has nothing to do with anything; forget it ever got mentioned. 2) f being a diffeomorphism is key. Since f is a diffeomorphism, the tangent spaces at a given point are isomorphic. That is, ${\displaystyle T_{p}M}$ for a point p in M is isomorphic to a ${\displaystyle T_{f(p)}N}$. In particular, ${\displaystyle f_{*}(p)}$ is that isomorphism; in particular, its linear, its invertible. If M is n-dimensional, then we can visualize ${\displaystyle f_{*}(p)}$ as an nxn matrix. i.e. ${\displaystyle f_{*}(p)\in GL(T_{p}M,T_{f(p)}N)}$ with GL(V,W) being the group of linear isomorphisms from n-dimensional vector space ${\displaystyle V=T_{p}M}$ to the n-dimensional vector space ${\displaystyle W=T_{f(p)}N}$.
Consider now the cotangent spaces. We can define these just fine, without any appeal to a metric (which was my earlier mind-bending mistake). At ${\displaystyle p\in M}$ define the cotangent space ${\displaystyle T_{p}^{*}M}$ of one-forms ω by defining them as dual to the basis vectors e of ${\displaystyle T_{p}M}$. Think plain-old vector dot-product. Now, from plain-old vector algebra, if v,w are plain-old vectors, and A is a matrix, then ${\displaystyle w\cdot (Av)=(A^{T}w)\cdot v}$ where ${\displaystyle A^{T}}$ is the transpose of the matrix. See where I'm going with this? So if we take ${\displaystyle A=f_{*}(p)\in GL(T_{p}M,T_{f(p)}N)}$ then the transpose is going to act on the 1-forms: ${\displaystyle A^{T}=[f_{*}(p)]^{T}\in GL(T_{f(p)}^{*}N,T_{p}^{*}M)}$. We use the transpose to map the cotangent spaces. In essence, we can say that the pullback is just the plain-old matrix transpose of the push-forward.
OK, now consider the following questions: 1) can we define a meaningful pullback of a vector? 2) can we define a meaningful pushforward of a covector? The answer to both is yes if f is a diffeomorphism. This makes f invertible, so that given a ${\displaystyle q=f(p)\in N}$ we can meaingfully produce a ${\displaystyle p=f^{-1}(q)\in M}$. It also means we can invert the matrix. OK, the beer is now taking effect; I'm not sure how much more I can write.
I think I need to find some more elegant way of explaining the above and adding it to some article somewhere. The upshot is that we define the pullback of a vector as the pushforward of the inverse of f. Similarly, we define the pushforward of a covector as the pullback of the inverse of f. This now allows us to define the pullbacks and pushforwards of arbitrary mixed tensors.
I think there are additional requirements that all of this has to be resrticted to Banach spaces or probably Sobolev spaces for the functions and Hausdorf spaces for the manifolds (I guess this is implied by the word diffeo-). But I'm out of my depth; I have a vague idea of what a Sobolev space is, but I'd have to work very hard to tell how that affects the definition of vector fields on manifolds. I've never seen counter-examples for diff geom definitions. linas 06:24, 20 Apr 2005 (UTC)

Yes, okay, I see your point now. What should be said is that the pullback of contravariant tensors is undefined for an arbitrary f, however if f is a diffeomorphism then the pullback (of contravariant tensors) can be defined as the pushforward of f−1. And conversely for pushforwards of covariant tensors. This makes perfect sense. In general, if f is a diffeomorphism then one can either pushforward or pullback tensors of mixed rank.

Your first point is actually unrelated to your second but is also true. For an arbitrary f, the pullback by f is the transpose of the pushforward by f. This should also be mentioned.

Regarding your last comment: this article currently deals only with finite-dimensional manifolds which are Hausdorff by definition. -- Fropuff 14:40, 2005 Apr 20 (UTC)

OK, I psyched myself up to expanding this article sometime soon. Plan: define pullback on vector spaces; define pullback on manifolds in general; point out properties for diffeomorphisms; point out properties for symplectic maps. BTW, I am planning on working some of the symplectic topics real soon now; I notice that this is where your working. I've got this painfully simple fractal Hamiltonian that is eluding my ability to grasp it, and so I need to step through symplectic topics one at a time. Question: have you heard of anything like this: if one knows (some of) the geodesics of a system, is there a way to go back and find the corresponding hamiltonian and/or the metric? FWIW, its a 2D phase space and a 1D hamiltonian with a probably insane geometry of some kind; I can't put my finger on it though.

## Pullback of vector bundles

I added a section on the pullback of vector bundles (which is the categorical pullback in the category of vector bundles). This is necessary to clarify the article on Chern classes in which this type of pullback is used.

Also, I think it is misleading to indicate that pullbacks of differential forms are detailed in the pullback_(category theory) article. The two notions of pullback really are a bit different. 151.204.6.171

Okay, there needs to be some cleanup here. We already have an article on the pullback of bundles at pullback bundle (which is a special case of the categorical pullback). This article should just be on pullback of differential forms which (as far as I can tell) has absolutely nothing to do with categorical pullbacks or pullbacks of bundles.
I've marked the article for cleanup until I (or someone else) can sort out this material. -- Fropuff 15:23, 3 November 2005 (UTC)
Thanks. You are quite right, so I just updated the link on the referring page Chern class and removed my edit. Probably the only thing that needs to be cleaned up is the Pullbacks in category theory section, since it is quite misleading. A few more words in the intro may help those of us who were misdirected here.151.204.6.171

## 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 : MN 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) NTp 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: NT*N a section, we define f*ω : MT*M by

${\displaystyle (f^{*}\omega )(p)=f^{*p}(\omega (f(p)))}$

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)

I've tried to clear up this point for pullbacks of cotangent vectors and 1-forms Geometry guy 22:04, 9 February 2007 (UTC)

## Pullbacks in general, and pushforwards

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)

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)
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)

Great!! Thanks for the support! I agree that it may not be appropriate to merge pullback and pushforward: I just wanted to flag the possibility - I think it is too early to call it yet. I'm looking forward to see what happens here... Geometry guy 00:02, 10 February 2007 (UTC)

I've now done the work I suggested, and will remove the clean-up tag: probably a more specific tag should be put in place somewhere. There is still more to be done: in particular the representation theoretic pullback by diffeomorphisms, and the relation to Lie derivative (for one parameter groups of diffeomorphism). Geometry guy 22:52, 11 February 2007 (UTC)

## Edit request: is this a typo?

I am so far out of my depth here but the following seems to me is wrong: "This linear map is known as the pullback (by φ), and is frequently denoted by φ*. More generally, any covariant tensor field – in particular any differential form – on N may be pulled back to M using φ." Short version: shouldn't the φ at the end of the quote be " φ* "  ??? Longer version: if φ:M→N then I don't see any way to get from N to M using it...this seems to totally violate the meaning of the mapping symbol " → " in a way we really do NOT want to countenance...doesn't it?173.189.73.72 (talk) 00:02, 26 January 2014 (UTC)