View source for Hermitian connection

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An [[Affine connection|affine connection]] on a Hermitian manifold  $  M $
relative to which the tensor  $  \phi $
defined by the complex structure and the fundamental  $  2 $-
form  $  \Omega = ( 1 / 2 ) g _ {\alpha \beta }  \omega  ^  \beta  \wedge \overline \omega \; {}  ^  \alpha  $
are parallel, implying the same property for the Hermitian form  $  ds  ^ {2} = g _ {\alpha \beta }  \overline \omega \; {}  ^  \alpha  \omega  ^  \beta  $. 
If the affine connection on  $  M $
is given by local connection forms  $  \omega _  \beta   ^  \alpha  = \Gamma _ {\beta \gamma }   ^  \alpha  \omega  ^  \gamma  + \Gamma _ {\beta {\overline \gamma \; }  }  ^  \alpha  \overline \omega \; {}  ^  \gamma  $, 
then these conditions can be expressed as

$$ 
\omega _ {down - 3 {\overline \beta \; }  }  ^  \alpha   =  \omega _ {down - 3 {\overline \beta \; }  } ^ {\overline \alpha \; }  =  0,\ \ 
\omega _ {down - 3 {\overline \beta \; }  } ^ {\overline \alpha \; }  = \ 
\overline \omega \; {} _  \beta   ^  \alpha  ,
$$

$$ 
d g _ {\alpha \beta }   =  \overline \omega \; {} _  \alpha   ^  \gamma  g _ {\gamma \beta }  + g _ {\alpha \gamma }  \omega _  \beta   ^  \gamma  .
$$

On a given Hermitian manifold  $  M $
there is one and only one Hermitian connection for which  $  \Gamma _ {\beta \overline \gamma \; }   ^  \alpha  = 0 $.

A generalization is an almost-Hermitian connection, which is defined by similar conditions on the tensors  $  \phi _ {j}  ^ {i} $
and  $  g _ {ij} $
with  $  g _ {kl} \phi _ {i}  ^ {k} \phi _ {l}  ^ {l} = g _ {ij} $
on an almost-Hermitian manifold  $  \widetilde{M}  $. 
An almost-Hermitian connection on a given  $  \widetilde{M}  $
exists. It is uniquely defined by its torsion tensor: If the torsion tensors of two almost-Hermitian connections are the same, then so are the connections. For example, there is one and only one almost-Hermitian connection for which the torsion forms are sums of  "pure"  forms (that is, forms of type  $  ( 2 , 0 ) $
and  $  ( 0 , 2 ) $), 
namely, the second canonical Lichnerowicz connection.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Lichnerowicz,   "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  K. Yano,   "Differential geometry on complex and almost complex spaces" , Pergamon  (1965) {{ZBL|0127.12405}}</TD></TR>
</table>

====Comments====
The first and second canonical connections on an almost-Hermitian manifold are described in [[#References|[1]]], p. 192 and pp. 194-195, respectively.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi, K. Nomizu,   "Foundations of differential geometry" , '''2''' , Interscience  (1969) {{ZBL|0175.48504}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.O. Wells jr., "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR>
</table>