Talk:Almost complex manifold

Incorrect statement

"An ordinary almost complex structure is a choice of a half-dimensional subspace of each fiber of the complexified tangent bundle TM." -- This is not correct.

i somewhat dislike this sentence: "equipped with a structure that defines a multiplication by i on each tangent space". the almost complex structure is an object defined on a real manifold, whose tangent spaces are real vector spaces. what meaning does multiplication by i have on such a space? perhaps it is a good description to give an intuitive understanding, perhaps it just needs a qualifier?Lethe

also, i am starting to think that it may be better if almost complex structure and almost complex manifold were separate articles. what do you think? (you=whoever is reading this)Lethe

I was just trying to give a informal introductory sentence or two on what an almost complex structure is before defining it. Technically, the introduction of an almost complex structure, does allow for multiplication by i on T_pM via

i v = J_pv

This turns T_p from a 2n dimensional real vector space into a n dimensional complex vector space.

At this point, I don't see a reason to separate the manifold page from the structure that defines it. It seems the two go hand in hand. Perhaps if the article gets overly large. -- Fropuff 20:32, 2004 Mar 12 (UTC)

I believe that the discussion of almost complex structure should be replaced with a link to the "almost complex" page. That page is well-developed and the discussion here is now superfluous. Lost-n-translation (talk) 23:26, 1 March 2010 (UTC)

Complex/Almost complex

This part seems to be missleading, instead of talking about ... for complex structure on an almost complex mnfld you did just test for almost complex structure to be non complex

I'm not a specialist, so can not fix it correctly but maybe you can, or at least clearify it.

existence of an almost complex structure on a real manifold is a necessary, but not sufficient, condition for the existence of a complex structure. The obstruction to the existence of a complex structure can be codified in a rank (1,2) tensor, called the Nijenhuis tensor.

again it is not an obstruction, one can take a complex structure and deform it locally and get an almost complex and it should be noted that not any almcomplex mnfld are of that type

Also it showld be a couple of words on simplectic-Riemannian connection with use of almost complex structure...

Tosha 01:30, 13 Mar 2004 (UTC)

In the formal definition section, the following appeared:

dω=0 if and only if ∇J=0

I'm not sure what the gradient of an almost-complex structure is, so I'm moving it here (after writing the requested subsuming section on compatible triples) hoping someone would see this and explain. Orthografer 03:14, 11 October 2006 (UTC)

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Tosha - your text makes it seem like vanishing Nijenhuis tensor is the definition of an integrable almost complex structure. i think a more appropriate definition is closure of the Lie bracket on the space of holomorphic vector fields. this is what the word integrable means. then the vanishing of the Nijenhuis tensor is a useful theorem about integrable almost complex structures, rather than a definition. in your text, you seem to want to give two different definitions of integrable.

i think the most clear way to convey the situation is to state these equivalent conditions: 1. an almost complex manifold admits a complex structure (which is then uniquely determined by the almost complex structure), 2. the Nijenhuis tensor vanishes, 3. the almost complex structure is integrable (the Lie bracket closes on holomorphic vectors) 4. the almost complex structure is covariantly constant

(i think sometimes 4 is taken as the definition of integrable instead of 3)

also, you deleted my comment about how the existence of a J tensor implies that the tangent space (and therefore also the manifold) must be even dimensional. - Lethe

I partly agree, change it if you want, it is not exacly my text, I just compactified an older one. For me integrable means that it defines complex structure, I think it should be the very base definition and it stated in the first par. of the subsection

Tosha 15:20, 13 May 2004 (UTC)

I tend to agree Lethe. We should be sure of what the word integrable actually means in this context. Although, I think point (4) in your list should be excluded. It applies only after the manifold is equipped with a metric. -- Fropuff 16:31, 2004 May 13 (UTC)

I think integrable used to mean the integrability condition - but a long time ago (eg Weil's book on Kahler manifolds, from the 1950s). So perhaps now things have moved on. Charles Matthews 16:54, 13 May 2004 (UTC)

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I am a little uncomfortable with the phrase "linear map J: TM→TM". a tangent bundle is not a vector space. it shouldn't have linear maps. maybe it should read "bundle morphism" instead, or else we should just say "restricts to linear map J_p: T_pM→T_pM at each point p in M"?Lethe

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OK, i have added some stuff:

• complex structure on a vector space. i was thinking about having this for a while, this is why i thought it might be a good idea to separate complex structures from complex manifolds, in a discussion with Fropuff at the top of this page. because you can have a complex structure on other spaces than manifolds. but there isn't really enough about complex structures on vector spaces alone to warrant its own article, plus that has limited (none?) applicability outside of complex manifolds.

• i changed the stuff about J being a linear map to it being a bundle morphism, as per my above comment.

• i tried to make clear in precisely what sense J can be thought of as "multiplication by i". i.e. only on the holomorphic sector of the complexified space, not on the antiholomorphic sector, nor on the underlying real space.

• i added a section called "differential topology". right now this section is kinda rough, but i actually have to get some work done today, so i will have to come back later. i don't feel like my explanation of $\Omega ^{m,n}(M)$ is very lucid. nor my explanation of the Doubeault operators.

• i added another equivalent condition to the list of conditions which determine when an almost complex structure is induced by a complex structure. namely, $d=\partial +{\overline {\partial }}$. this implies that $\partial ^{2}=0$ and so the almost complex structure is integrable.

• the problem with my explanation of the Doubeault operators is that it assumes an integrable almost complex structure (so that $d=\partial +{\overline {\partial }}$), whereas the Doubeault operators can be defined for a nonintegrable almost complex structure. my definition, in addition to being unclear, assumes (without stating) integrability. this should be fixed.

Holomorphic?

The section on differential topology defines holomorphic and antiholomorphic vector fields. Holomorphic vector bundles are usually defined only over complex manifolds. It seems wrong to me to call TM⁺ a holomorphic vector bundle when M is only an almost complex manifold. In what sense in the projection map TM⁺ → M holomorphic? Is this naming standard? -- Fropuff 07:05, 2005 Feb 27 (UTC)

An almost differential structure is all that is required to define a locally holomorphic vector field. It's built in to the definition. Lethe | Talk 08:58, Mar 1, 2005 (UTC)

I know these vector fields exist (although I'm not sure what you mean by a locally holomorphic vector field). My concern is whether or not they should be called holomorphic/antiholomorphic unless J is integrable. Better to call them vector fields of type (1,0) and type (0,1) or something similiar. I have found one author [1] who calls them J-holomorphic vector fields. -- Fropuff Fropuff 02:38, 2005 Mar 3 (UTC)

1. I guess what I meant by "locally holomorphic vector field" was redundant. I just meant an eigenvector of J.
2. My previous reply was kinda drunken. Isn't there a rule about using wikipedia while drunk?
3. I'm not averse to a different choice of definition of holomorphicity. But I can provide references which use the convention in this article, I believe. Let's take a survey of sources.
4. I like that you moved the linear complex structure to a different article. That's actually what I had in mind a year ago, about splitting the article, which you advised against, I think because I didn't state very clearly that that's what I had in mind.
5. So, for a year now, I've left this article in a half finished state, including knowingly leaving false information! Shame on me! I should get back to work on my wikipedia tasks, I've become lazy about those. I think I had more momentum to write wikipedia articles when I was new. But that's a pretty lousy excuse.
6. I think I'm going to have trouble tracking down one of the books that I was using as a reference when I wrote some of this stuff. There are plenty of other books though. I guess I'll be OK.

-Lethe | Talk 02:51, Mar 3, 2005 (UTC)

[I must confess to drinking and editing myself — funny how it all seemed so clear at the time.] Of the sources I've checked (mostly symplectic topology books) I've only seen Nakahara call these fields holomorphic, and I've never really trusted that book when it comes to rigor. I've been thinking about almost complex structures lately so I may do some more work on this article. I'll try to fix the problems you mentioned above (assuming you don't get around to it first). -- Fropuff 03:36, 2005 Mar 3 (UTC)

what does integrable mean?

I'm currently fixing this article (about time). But I'm having a problem. I think we've discussed this in the past, but we have to decide what "integrable" actually means. Right now, the article states that an almost complex structure is integrable if it arises from a bona fide complex structure, a view that Tosha was advocating a long time ago. But with that convention, the Newlander-Nirenberg theorem becomes tautologous sounding (an almost complex structure is integrable iff it is integrable). I've seen different books use different definitions of integrable: Nijenhuis vanishes, Lie bracket closes, exterior derivative decomposes. I propose that the first of those be taken as the def. -Lethe | Talk 20:41, July 14, 2005 (UTC)

I strongly advocate returning to the definition of integrable that I had before. That is, an almost complex structure is integrable iff one can find local holomorphic coordinates around every point. The almost complex structure then arises from an honest complex structure. It is a simple and intuitive definition which captures the essence of what it means to be integrable: an almost complex structure is integrable iff it comes from an honest complex structure. One doesn't need any extra constructions or definitions.

The constructions like the Nijenhuis tensor are superfluous as far as the basic idea is concerned. Trying to define integrability in terms of this tensor and then claiming (without proof) that the vanishing of this tensor guarentees the existence of an honest complex structure seems very circuitous and unnecessarily complicated. The Newlander-Nirenberg theorem states that J is integrable (i.e. local holomorphic coordinates exist) if and only if the Nijenhuis tensor vanishes. This to me is very non-obvious statement and not the least bit tautological.

Take an analogy with symplectic manifolds. Every vector space with a nondegenerate 2-form is a symplectic vector space. However, a manifold with a global nondegenerate 2-form ω is not always a symplectic manifold, but rather an almost symplectic manifold. Every almost symplectic manifold looks like a symplectic vector space at a point (i.e. the tangent space is symplectic). But in order to be an honest symplectic manifold ω must be closed. One can't determine if ω is closed merely by looking at it at a point, one must look at ω in a neighborhood of that point. This is the essence of integrability: if one can move from the infinitesimal description to the local description. An almost complex manifold looks like a complex vector space at a point. It is a complex manifold iff it looks like a complex vector space in every neighborhood of that point. This is integrability.

-- Fropuff 22:57, 14 July 2005 (UTC)

Fropuff, I agree that the weird looking constructions like the Nijenhuis tensor hide what it means to be integrable. The reason for my change is that I'm having a hard time stating the N-N theorem without having some local differential definition of integrability. That's what makes N-N a deep theorem, right? The local differential equation yields global equivalence to a complex structure? -Lethe

I'm not entirely sure what you mean. The construction of introducing complex analytic coordinates is a purely local one. The N-N theorem should by stated as

An almost complex structure is integrable if and only if the Nijenhuis tensor vanishes.

I should point out that Newlander & Nirenberg defined integrability in their original article just as I have: in terms of the existence of local holomorphic coordinates (see the first paragraph of that article). See also McDuff's book Introduction to Symplectic Topology (searchable on Amazon). Again both integrability and the statement of N-N are as I propose. -- Fropuff 04:30, 15 July 2005 (UTC)

OK, I could get on board. -Lethe | Talk 17:34, July 15, 2005 (UTC)

This doesn't seem right

This statement:

"Any even dimensional vector space always admits a complex structure. Therefore an even dimensional manifold always locally admits a (1,1) rank tensor (which is just a linear transformation on each tangent space) such that Jp2 = −1 at each point p. Only when this local tensor can be patched together to be defined globally does the almost complex structure yield a complex structure, which is then uniquely determined."

doesn't seem right to me. For example, any almost complex manifold has a globally defined J tensor, while it may not be complex. See current revision for a better attempt. It's quite possible that I wrote that wrong statement. -Lethe | Talk 20:48, July 14, 2005 (UTC)

That statement is clearly not true. I'm can't figure out why I wrote it. I must have been typing something different than I was actually thinking. Thanks. -- Fropuff 23:04, 14 July 2005 (UTC)

I was surprised to see you taking the blame for a mistake which I was sure must be mine, but after careful inspection, I see that it really was in fact me, as expected, who made the egregious statement. So stop taking credit for my blunders! -Lethe | Talk 03:31, July 15, 2005 (UTC)

Oh good, I thought I was losing my mind. I guess I checked the history too quickly. Anyway, it doesn't really matter so long as it's fixed. However, the uniqueness part of the statement still bothers me. Even on an vector space the complex structure is not unique (not even up to homotopy). Do you mean something I'm not grasping? -- Fropuff 04:35, 15 July 2005 (UTC)

rewrite

OK, I've rewritten a bunch of stuff, and I believe I have removed all the stuff that was known to be incorrect. -Lethe | Talk 20:49, July 14, 2005 (UTC)

Remaining errors

The page is still not correct. It gives two different definitions of integrable (as the existence of local holomorphic coordinates, and then as the vanishing of the Nijenhuis tensor). I believe that simply removing the sentence "An almost complex structure is called integrable if this tensor vanishes for all smooth vector fields X and Y on M (here [·, ·] denotes the Lie bracket of vector fields)." will solve the problem. Then there is one definition of integrable, followed by the statement of the NN theorem.

As a matter of opinion, I strongly think that integrable should not be defined (as here) as the existence of local holomorphic coordinates. Rather, integrable means that the holomorphic tangent space is closed under the Lie bracket (as Charles Matthews says above). Then the NN theorem states that a manifold with an integrable almost complex structure admits local holomorphic coordinates. -Chris 3 April 2006

deletion of nonexample

prior to my edit, the page claimed that surfaces of general type failing to meet certain restrictions on their chern numbers furnish examples of almost complex manifolds with non-integrable complex structures on their tangent bundles. while it is probably true that one can deform the complex structure on the tangent bundle of such a surface given to you by the definition of complex surfaces to give a nonintegrable structure, that's not something that's going to be detected by the chern classes. bottom line is, the sentence is either flat out nonsense or the editor who wrote it made some sort of error in committing it to writing. since it's unclear what was meant and no one seems to know, seems like it ought to be removed. it would be nice to have some examples of the sort of thing the sentence tried to address, but i'm going to have to leave that to someone else. 140.254.93.115 (talk) 16:33, 15 October 2008 (UTC)

Changes

I made these changes: Given any linear map A on each tangent space of M; i.e., A is a tensor field of rank (1,1), then the Nijenhuis tensor is a tensor field of rank (1,2) given by

$N_{A}(X,Y)=-A^{2}[X,Y]+A([AX,Y]+[X,AY])-[AX,AY].$

The individual expressions on the right depend on the choice of the smooth vector fields X and Y, but the left side actually depends only on the pointwise values of X and Y, which is why N_A is a tensor. This is also clear from the component formula

$(N_{A})_{ij}^{k}=A_{i}^{m}\partial _{m}A_{j}^{k}-A_{j}^{m}\partial _{m}A_{i}^{k}-A_{m}^{k}(\partial _{i}A_{j}^{m}-\partial _{j}A_{i}^{m}).$

In terms of the Frolicher-Nijenhuis bracket, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor N_A is just one-half of [A,A].

First of all, the statement about smooth X and Y is not correct. The local values suffice: that is essential for the concept of a tensor. (Too bad, the page on tensors does not deal with this matter.) Second. the text I propose places the N tensor in the context of the F-N bracket. The page on the F-N bracket only casually mentions the relationship. MY10SOR (talk) 18:58, 13 September 2010 (UTC) MY10SOR (I am new here, please forgive the intrusion.)

The Lie bracket of vector fields is defined in terms of the Lie derivative, so the vector fields have to at least be differentiable. The point the article was making was that it's not immediately obvious that the RHS is tensorial. Its not even obvious that it transforms as a tensor under a change of basis. 67.198.37.16 (talk) 03:09, 8 May 2019 (UTC)

6-sphere

I started a discussion here that relates to this article as well... Franp9am (talk) 20:34, 23 December 2015 (UTC)

Higgs and spontaneous symmetry breaking

The above-mentioned link in the 6-sphere has this (to me) astounding paragraph:

A complex structure on a complex manifold can always be re-interpreted as a spontaneously broken classical vacuum solution in a non-linear version of a Yang-Mills-Higgs theory formulated on the underlying manifold. This is because an almost complex tensor field J is mathematically the same as a Higgs field Φ. Although the terminology comes from theoretical physics, the corresponding mathematical structures are well-defined.

I find this to be quite remarkable, and worth a detailed exposition here, or somewhere. I have a good-enough imagination that I can imagine how this might be true (plus I know what Higgs fields are, I'm unclear why J is the Higgs field, instead of being just the classical solution to one, or some patch near the minimum...) But I find that imagination is unsuitable, when something more explicit can be stated. — Preceding unsigned comment added by 67.198.37.16 (talk) 21:44, 7 May 2019 (UTC)

Worked examples needed.

This article could be improved by adding sections on the following:

• Examples of almost-complex structures that are not complex. The only one given is S^6 and that one is clearly too complicated, and too ambiguous.
• A worked example of integrability, and an example where integrability fails.
• A more precise definition of integrability. I imagine it means "smooth integral curves on a neighborhood" which works only if J is no-where degenerate in the neighborhood and satisfies a few other properties.... But this is just my imagination...
• Articulation of the compatible triple in these example settings.

67.198.37.16 (talk) 02:55, 8 May 2019 (UTC)

Link clarity

In the Jean’s complex manifold solution the reader was sent to explanations about manifolds and complex numbers at first, without much information how does it correspond to real manifolds. In my solution the reader is immediately directed to the word “integrable”. Incnis Mrsi (talk) 11:57, 11 September 2019 (UTC)