# Talk:Hermitian manifold

Active discussions
WikiProject Mathematics (Rated Start-class, Low-priority)

## "E bar" Undefined?

In the "Formal definition" section, the notation ${\displaystyle {\overline {E}}}$  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 ${\displaystyle n}$ -by-${\displaystyle n}$  matrix ${\displaystyle h}$  can indeed be decomposed into two parts ${\displaystyle h=a+ib}$  where ${\displaystyle a=(h+{\bar {h}})/2}$  is a real, symetric ${\displaystyle n}$ -by-${\displaystyle n}$  matrix and ${\displaystyle b=(h-{\bar {h}})/2i}$  is a real skew-symmetric ${\displaystyle n}$ -by-${\displaystyle n}$  matrix. The article imples that the metric ${\displaystyle g}$  is to be identified with ${\displaystyle a}$ , and ${\displaystyle \omega }$  with ${\displaystyle b}$ . It then states that one can be obtained from the other by means of the complex structure ${\displaystyle J}$ . This cannot be true. Hermiticity requires no relation between ${\displaystyle a}$  and ${\displaystyle b}$ .

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

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

is a real symmetric ${\displaystyle 2n}$ -by-${\displaystyle 2n}$  matrix, while

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

is a skew-symmetric ${\displaystyle 2n}$ -by-${\displaystyle 2n}$  matrix. Here

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

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

## The ${\displaystyle \Gamma }$ in the "Formal definition" section is not defined

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