# Talk:Pushforward (differential)

## Pushforward verses pushout

I wish I remembered more, but isn't this also a term in category theory, where stuff is being pushed along functions? As I said, I really wish I remembered more of those days. --SPUI (talk) 18:39, 2 Mar 2005 (UTC)

It could be. My background is mostly differential geometry, and I never met a categorial push forward at graduate school. I wonder what the requester intended. I suspect it probably wasn't what this article covers. mat_x 22:06, 2 Mar 2005 (UTC)
Ah, yes, there is pushout. Maybe that was intended. Never heard it called pushforward though. mat_x 22:10, 2 Mar 2005 (UTC)
Yes, that was it. --SPUI (talk) 03:36, 3 Mar 2005 (UTC)

There are things like f* in sheaf theory that are probably pronouced as pushforward or direct image. Charles Matthews 10:38, 3 Mar 2005 (UTC)

I'm not sure if the 'introductory' additions by the preceding user are that helpful. Couldn't they be buried a bit deeper in the article, as a reward for interested readers who get that far? mat_x 19:05, 4 Mar 2005 (UTC)

I've taught this stuff that way, and at least with pictures it's an OK thing to do. Charles Matthews 19:10, 4 Mar 2005 (UTC)
I'm sorry if you misunderstood me, there was certainly no questioning your experience. I was just wondering (and not in a passive-aggressive sense) whether an encyclopaedia article should begin with a jargon-blast. mat_x 21:31, 4 Mar 2005 (UTC)

I didn't take it that way. A jet, intuitively, is a kind of ghost of a curve drawn on a manifold at a point P. That is, we think of a tangent vector as a stubby kind of curve c, mapping a short interval in R to M with 0 → P. The value of that is to make the variance clear: if we have a smooth mapping f from M to N, the composite foc is going to be a curve at f(P). All this does for you is to give a picture why tangent vectors 'push forward'. Nothing gets differentiated, of course - that's hidden away in various equivalence relations.

Some day the jet article will be more sorted out, and this will be a better way to consolidate the area, with cotangent vectors defined as jets of functions from M to R, also. For some reason the calculus-on-manifolds stuff isn't yet all written properly. Charles Matthews 21:43, 4 Mar 2005 (UTC)

The push forward is a categorical notion. It has nothing more to do with differential geometry then it does with anything other subject in mathematics.

Categorical pushforwards are usually called pushouts. I believe that this page should be devoted to pushforward of vector fields, the term pushforward should be mentioned briefly on the pushout page, there should be a separate page on pushforward measures, and a disambiguation page for pushforward. Do others agree? Geometry guy 16:44, 8 February 2007 (UTC)

## Relation to other types of derivatives, etc

Hi, User:Fropuff, your recent edits removed a bunch of text I'd recently added, text that was trying to explain in simpler, more accessible terms what this was all about. Why? If you think some of it was incorrect, then please fix it, rather than chopping it out ... Here's one of the parts that disappeared:

Here, ${\displaystyle X(f)={\mathcal {L}}_{X}f}$ is the Lie derivative (or, more simply, the directional derivative) of the function f along the direction X.
Equivalently, given a vector field X on M, the pushforward defines a vector field Y on N, given by ${\displaystyle Y=F_{*}X}$ with
${\displaystyle Y(p)=dF(X(F^{-1}(p)))}$
Here, ${\displaystyle F^{-1}(p)}$ maps the point p back from the manifold N to the manifold M. Then ${\displaystyle X(F^{-1}(p))}$ is the vector field at the point ${\displaystyle F^{-1}(p)}$ on M. Finally, dF is the differential of F.
Thus, the pushfoward defines a mapping of vector fields. The analogous mapping for differential forms is the pullback; indeed, one can define the pullback as ${\displaystyle F^{*}=\left(F^{-1}\right)_{*}}$. When N=M is the same manifold, then the pullback and pushforward define the transformation properties of the covariant and contravariant indices of a tensor defined on the manifold M.

Instead, you just left the definition in terms of curves, which I think is bogus, and then said "variety of notations" which is even more bogus, since these other notations have distinct meanings and definitions which can be distinctly related to the definitions in this article. i.e. they are not just 'different notations' , they represent different approaches to the topic (of which the definition in terms of curves is just one). linas 19:48, 9 Apr 2005 (UTC)

Problems. Vector fields don't push forward, for one thing. The pushforward of a vector field only exists under some conditions on F, which are expressed generally in terms of DF. So that's all a bit circular. This can be mentioned, but not as a definition: tangent vectors do push forward individually, and the curve approach to seeing that is fine. Charles Matthews 20:05, 9 Apr 2005 (UTC)
Hmm, well, can we amend the article to elaborate this point? If you've got a Hausdorff topology on the thing, with an open cover and some coordinate charts, then as far as I know, the curve definition and the vector field definition are one and the same. There might be subtle aspects for non-hausdorff spaces, but I don't know what these are (and these are never covered in introductory texts). linas 20:19, 9 Apr 2005 (UTC)

I'm also considerably un-nerved by the addition of the following:

Although one can always push forward tangent vectors, the push forward of a vector field does not always make sense. For example, if the map F is not surjective how should one define the vector outside the range of F?

Well, there is an easy and simple answer to this, its called a coordinate chart, see atlas (topology). Pullbacks and pushforwards are always defined in terms of coordiante charts on top of open coverings, as are vector fields; this has nothing to do with surjectivity. So this whole section is in fact incorrect.

When one defines vector fields in terms of coordinate charts, then the definition in terms of vector fields and curves are completely identical, that's the whole point of a Lie derivative, after all. Topoliogy is same as geometry for smooth manifolds. Now, there are some subtle points here ... weird shit happens for non-standard topologies, but the current edits are not conducive to exploring/explaining the "weird shit". linas 20:19, 9 Apr 2005 (UTC)

Sorry, you really are missing some basic points, and it's not anything tricky about topology. You can't push forward a vector field by a many-to-one mapping because if two points x and x' both map to y, you can't know at y whether the pushed-forward vector field is the arrow from x or the arrow from x' - which need not have any relation. So this is a picture you seem to be missing. I agree that this could usefully go into the article, but the other matters you are bringing up aren't very relevant. Charles Matthews 21:02, 9 Apr 2005 (UTC)

Well, yes, of course ... but, how, exactly, is it that the "equivalence class of curves" definition avoids this problem? Given that vector fields on manifolds are usually defined in terms of the equivalence classes of curves, I just don't see the practical difference between these two definitions.

Maybe the problem is with the definition of what a "vector field" is? I'm used to thinking of a vector field as being that thing which is restricted to an open subset. Then, on the open subset, the vector field is defined to be the set of equivalence classes of curves going through the points of the open set. At least, that is one possible definition of what a vector field is. With this definition, the equivelance class of curves in one space induces an equivalence class in the other.

If the map is not surjective, e.g. if the other space is of higher dimension, one then has the codomain is vertical part of a fibre bundle, and the horizontal part is .. ill defined (it would be the kernel of the projection going in the other direction). By talking about curves instead of vector fields, one gets lead to the same conclusions, right?

If the map is not injective ... there are two ways in which this may be the case. One case is handled via universal cover. I don't quite see how the curve definition case rescues one there. The other case is where the target manifold is of lower dimension ... in which case one has potential consistency problems with the 'equivalence class of curves' defintion as well. Yes?

There may well be some circularity in these definitions; however, the problem of circularity should be attacked in a distinct article. I'm just thinking that in practical terms, for the common cases of smooth, differentialble manifolds, all the different definitions are more or less equivalent, and the article should first focus on the similarities before facing the differences.

Either that, or I'm just plain confused; its been a few decades since I really thought even this much about this topic; I may have slipped some gears here. I'll think a bit harder on this, maybe I'm missing something simple. linas 21:25, 9 Apr 2005 (UTC)

Your language isn't clear enough enough to pick this up. You say

vector fields on manifolds are usually defined in terms of the equivalence classes of curves,

and that is incorrect. Tangent vectors are defined that way. Vector fields are not. The rest of the jargon isn't helping you. Just think 'one arrow = ghost of one curve going through a fixed point P'. This all works fine for one arrow at a point, but fails when you try to do an arrow varying over a small neighbourhood. Unless the mapping does no 'folding' at all. You can push forward via a local diffeomorphism, no problem. Charles Matthews 21:31, 9 Apr 2005 (UTC)

Hmm, sorry, I'm not trying to jargonize, I was trying to be reasonably precise as best I could. At this point, I don't understand what difference you are trying to make. How is a vector field defined, then? I thought a vector field is just a bunch of arrows, one arrow each for each point p, for a bunch of points p (that comprise an open subset of M). A pushforward takes a vector at point p and maps it somewhere, on this we agree. How does this differ from saying that a pushforward maps a "bunch of vectors" all at once? To me, mapping a bunch of arrows at a bunch of points is a synonym for saying that one is mapping a vector field.
Put it a different way: the article states: "The push forward of F induces in an obvious manner a vector bundle morphism from the tangent bundle of M to the tangent bundle of N". Is this statement correct? I think it is. Then, as far as I know, a vector field is a (not necessarily smooth) section of a tangent bundle, right? (That's what the article on vector field says). So if I've got a morphism of vector bundles, don't I also get a morphism of bundle sections ('for free')? linas 22:30, 9 Apr 2005 (UTC)
OK, never mind, I guess I concede the point. I see what you're driving at. I was just envisioning a multi-valued vector field on the target manifold, and I now realize that is a mangling of the definition. The multi-valuedness is there, its just that poking a vector field into the midst of it is a bad idea without further defining the expected outcome. Feeling slightly chastened; thanks for the patience. linas 04:24, 10 Apr 2005 (UTC)

Right, I think you have most of this now. The vector bundle morphism formulation is fine. The fact that such a morphism does not mean that a section can be 'mapped forward' is something quite general, not to do with the details of this particular case. Charles Matthews 09:14, 10 Apr 2005 (UTC)

I think what linas is referring to as a vector field is usually called a local vector field. Just as non-trivial principal bundles don't have sections but do have local sections. Linas is talking about local sections of the tangent bundle. Sections are just special local sections and we might avoid some problems. --MarSch 10:01, 22 Jun 2005 (UTC)
No Charles was absolutely right; I was having a brain-fart of some kind. I think I did amended either this article or pull-back later on to make this clear. linas 00:53, 23 Jun 2005 (UTC)
  Isn't there some ambiguity, when using the curve definition of tangent vector, in saying that:


" ...In other words, the pushforward of the tangent vector to the curve γ at 0 is just the tangent vector to the curve φ∘γ at 0."? by referring to "the" tangent vector? I don't mean mathematical ambiguity; I mean I think it is not likely clear to the

 lay reader, who may not know which tangent vector is _the_ tangent vector to the curve, since in higher dimensions there are infinitely-many possible vectors/directions.  — Preceding unsigned comment added by 146.96.33.197 (talk) 02:20, 16 May 2013 (UTC)


## Pushforwards and pullbacks

One of the reasons the notion of a pushforward of a vector field is confusing, is that it is actually a section of a pullback bundle. If V->N is a vector bundle over N and F:M->N is a smooth map, then the pullback bundle F*V -> M is the bundle whose fibre at x in M is VF(x). (This is an example of a categorical pullback.) The differential of F, dF (or DF if you prefer), is then a bundle homomorphism from TM to F*TN, or equivalently a section of the bundle Hom(TM,F*TN): these are bundles over M, not N, so I think the notation F* for the differential can be misleading (as is the term pushforward). If X is a vector field on M, then dF(X) (aka F*X) is a section of F*TN. This leads to the notion of a vector field along a map: a section of F*TN is called a vector field along F.

The cleanest way to understand the relation between vector fields on M and N is via the notion of F-related vector fields. If s is a section of a vector bundle V->N, then F*s, the composite of s with F, is a section of F*V over M: thus (F*s)(x)=s(F(x)) in (F*V)x = VF(x). In particular, if Y is a vector field on N, then F*Y is a vector field along F. Vector fields X on TM and Y on TN are said to be F-related if dF(X) = F*Y (both of which are vector fields along F).

Under certain conditions - for example if F is a diffeomorphism - there is a unique Y which is F-related to X, and this is the pushforward "which does not always exist" in the above talk. I think this article could be cleaned up by clarifying all this. Maybe I will find time to do it, if others agree. Geometry guy 17:15, 8 February 2007 (UTC)

hello. those seem to be reasonable comments and incorporating them into the article would be a good idea. Mct mht 00:46, 10 February 2007 (UTC)

Thanks for your support. I've tagged this page, and will try to edit it when I have the time and energy. If anyone else wants to have a shot at incorporating these ideas, then please go for it. Geometry guy 01:28, 10 February 2007 (UTC)

Okay, I've now done a first draft of these edits. Thanks Mct mht, I noticed from the history how much you improved an earlier version. This made my job a lot easier! I hope the article is now close to something which is at least mathematically solid, and we can soon remove the clean-up tag. I made quite a few independent changes, so if there are some you don't like please discuss and/or change them individually. Geometry guy 01:37, 11 February 2007 (UTC)

## Title and clean-up

Fropuff: thanks for moving this page (I've only just got move rights), but how about we move it to Pushforward (differential), i.e., pushforward by the differential of a smooth map, and replace this by a disambiguation page which can link to pushforward measure and direct image sheaf as well? Geometry guy 17:49, 11 February 2007 (UTC)

Okay, I've made some further edits to this page, so it compares well with pullback (differential geometry). I think the mathematics is okay now, although there is plenty of scope for improving the exposition and adding examples, references and citations. So, I'm going to be bold and remove the clean-up tag: I wasn't able to find a more specific one to put in its place. I'll also carry out the above suggested move. Anyone who thinks this needs some sort of tag still, please add it. Geometry guy 20:48, 11 February 2007 (UTC)

This has now been moved, and I've updated a lot of links to this page. I hope this works for everyone!! Geometry guy 23:39, 11 February 2007 (UTC)

## Image too subtle?

I feel it's too easy to chalk the squishing and change of angle in the elipsoid point's tangent space up to roughness or perspective. Should the image be redone so that it's more dramatic, with a lower and nearer-facing point? ᛭ LokiClock (talk) 22:36, 20 July 2011 (UTC)

## Nice Image

The image is really cool. Do you have the source code for that? — Preceding unsigned comment added by 68.147.37.61 (talk) 10:09, 29 October 2012 (UTC)

The image is an svg, so the source can be obtained like you would for a normal HTML page. ᛭ LokiClock (talk) 23:04, 25 November 2013 (UTC)

## dφ(x) arrow in the image

The red and yellow vectors form a basis for the tangent space in the image, and dφ(x) is supposed to be the best linear transform between the tangent spaces. Shouldn't that map red coordinate a in one tangent space to the red coordinate a in the another, and same for yellow coordinates? Yet, the yellow coordinate for the example point being transformed by dφ(x) is a large positive number in the left image and a large negative number in the right image. How does that work? -- 212.149.196.26 (talk) 23:23, 23 November 2013 (UTC)

The black arrow is not transforming a specific point, but showing the spaces the function is going from and to. The yellow arrows and red arrows are the only specific vectors whose images are depicted. ᛭ LokiClock (talk) 17:16, 24 November 2013 (UTC)