Talk:Hermitian manifold

"E bar" Undefined?

In the "Formal definition" section, the notation ${\overline {E}}$ 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 Structure

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?

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.

Either the notation in section 2 is very misleading, or many of the statements are completely wrong. A Hermitian $n$-by-$n$ matrix $h$ can indeed be decomposed into two parts $h=a+ib$ where $a=(h+{\bar {h}})/2$ is a real, symetric $n$-by-$n$ matrix and $b=(h-{\bar {h}})/2i$ is a real skew-symmetric $n$-by-$n$ matrix. The article imples that the metric $g$ is to be identified with $a$, and $\omega$ with $b$. It then states that one can be obtained from the other by means of the complex structure $J$. This cannot be true. Hermiticity requires no relation between $a$ and $b$.

What is actually true is that there is a Riemann metric $g$ on the underlying $2n$-dimensional real smooth manifold, where

$g=\left({\begin{matrix}a&b\\-b&a\end{matrix}}\right)$

is a real symmetric $2n$-by-$2n$ matrix, while

$\omega =\left({\begin{matrix}b&-a\\a&b\end{matrix}}\right)=\left({\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix}}\right)\left({\begin{matrix}a&b\\-b&a\end{matrix}}\right)=\left({\begin{matrix}a&b\\-b&a\end{matrix}}\right)\left({\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix}}\right)$

is a skew-symmetric $2n$-by-$2n$ matrix. Here

$J=\left({\begin{matrix}0&-I_{n}\\I_{n}&0\end{matrix}}\right)$

is the $2n$-by-$2n$ matrix representing the complex structure in the underlying real vector space. $J$ commutes with $g$ because it is simply multiplication by ${\sqrt {-1}}$ in the original complex basis. Mike Stone (talk) 19:01, 9 December 2016 (UTC)