Talk:Almost complex manifold

message: 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 TpM via
: i v = Jpv
This turns Tp 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)
thread_title:  

message: 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)
thread_title:  deletion of nonexample 

message: 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^k_j
-A_j^m\partial_mA^k_i-A^k_m(\partial_iA^m_j-\partial_jA^m_i).
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)
replies: text: 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)
thread_title: Changes

message: I started a discussion here that relates to this article as well... Franp9am (talk) 20:34, 23 December 2015 (UTC)
thread_title:  6-sphere 

message: 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)
thread_title:  Higgs and spontaneous symmetry breaking 

message: 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)
thread_title:  Worked examples needed. 

message: 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  the reader is immediately directed to the word “integrable”. Incnis Mrsi (talk) 11:57, 11 September 2019 (UTC)
thread_title:  Link clarity