View source for Hermitian connection

An [[Affine connection|affine connection]] on a Hermitian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470401.png" /> relative to which the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470402.png" /> defined by the complex structure and the fundamental <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470403.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470404.png" /> are parallel, implying the same property for the Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470405.png" />. If the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470406.png" /> is given by local connection forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470407.png" />, then these conditions can be expressed as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470408.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h0470409.png" /></td> </tr></table>

On a given Hermitian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704010.png" /> there is one and only one Hermitian connection for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704011.png" />.

A generalization is an almost-Hermitian connection, which is defined by similar conditions on the tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704014.png" /> on an almost-Hermitian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704015.png" />. An almost-Hermitian connection on a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047040/h04704018.png" />), 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)</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)</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>