Talk:Hermitian manifold

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"E bar" Undefined?Edit

In the "Formal definition" section, the notation   isn't explicitly defined or referenced anywhere. Presumably, this denotes the conjugate bundle? Some clarification here would help, or if someone can confirm that this was the intended meaning, I can try to edit it in. Dzackgarza (talk) 02:16, 29 July 2020 (UTC)

In the introduction, shouldn't the almost complex structure preserve the metric, rather than the other way around? —Preceding unsigned comment added by 130.102.0.171 (talk) 01:11, 16 January 2009 (UTC)

Distinction between Hermitian Metric and Hermitian StructureEdit

I think this article is a bit confusing at the moment. Should we not make a distinction between a Hermitian metric (given by the formula in the text) and a Hermitian structure (that is a smoothly varying choice of Hermitian form)? We could then show how each was related to the other. It's a simple enough point, but potentially confusing for newcomers.78.151.55.190 (talk) 00:02, 7 November 2013 (UTC)

Topological manifold?Edit

Shouldn't the intro say smooth manifold? How exactly does one define an almost complex structure on a topological manifold? -- Fropuff 03:40, 10 October 2006 (UTC)

You're right and I changed it. VectorPosse 06:49, 10 October 2006 (UTC)

Either confusing notation or completely wrong.Edit

Either the notation in section 2 is very misleading, or many of the statements are completely wrong. A Hermitian  -by-  matrix   can indeed be decomposed into two parts   where   is a real, symetric  -by-  matrix and   is a real skew-symmetric  -by-  matrix. The article imples that the metric   is to be identified with  , and   with  . It then states that one can be obtained from the other by means of the complex structure  . This cannot be true. Hermiticity requires no relation between   and  .

What is actually true is that there is a Riemann metric   on the underlying  -dimensional real smooth manifold, where

 

is a real symmetric  -by-  matrix, while

 

is a skew-symmetric  -by-  matrix. Here

 

is the  -by-  matrix representing the complex structure in the underlying real vector space.   commutes with   because it is simply multiplication by   in the original complex basis. Mike Stone (talk) 19:01, 9 December 2016 (UTC)

The in the "Formal definition" section is not definedEdit

I think defining it explicitly would ease the understanding of the article. Luca (talk) 14:32, 11 February 2021 (UTC)