View source for Hermitian structure

''on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471001.png" />''

A pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471002.png" /> consisting of a [[Complex structure|complex structure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471003.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471004.png" /> and a [[Hermitian metric|Hermitian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471005.png" /> in the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471006.png" />, that is, a Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471007.png" /> that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471008.png" />:

<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/h047100/h0471009.png" /></td> </tr></table>

for any vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710012.png" />. A Hermitian structure specifies in any tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710013.png" /> the structure of a Hermitian vector space (see [[Hermitian metric|Hermitian metric]]). A manifold with a Hermitian structure is called a Hermitian manifold. A Hermitian structure defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710014.png" /> a differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710015.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710016.png" />, which is called the canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710018.png" />-form of the Hermitian manifold. Any structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710020.png" /> can be completed by some Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710021.png" /> to a Hermitian structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710022.png" />: for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710023.png" /> one can take the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710025.png" /> is an arbitrary metric. The canonical Hermitian connection of a Hermitian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710026.png" /> can be regarded as an affine connection with torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710028.png" /> relative to which the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710030.png" /> are parallel. Among all affine connections satisfying these conditions it is uniquely characterized by the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710031.png" />, which is valid for its [[Torsion tensor|torsion tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710032.png" /> and any vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710034.png" />. The [[Curvature tensor|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710035.png" /> of the canonical connection satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710036.png" />. A Hermitian manifold is a [[Kähler manifold|Kähler manifold]] if and only if the canonical Hermitian connection has no torsion and hence is the same as the [[Levi-Civita connection|Levi-Civita connection]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710037.png" />.

A natural generalization of the concept of a Hermitian structure is that of an almost-Hermitian structure, which is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710038.png" /> consisting of an [[Almost-complex structure|almost-complex structure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710040.png" /> and a [[Riemannian metric|Riemannian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710041.png" /> that is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710042.png" />. If the fundamental <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710043.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710044.png" /> is closed, then an almost-Hermitian structure is called almost Kählerian. The specification of an almost-Hermitian structure is equivalent to a reduction of the structure group of the tangent bundle to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710046.png" />. Any non-degenerate differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710047.png" />-form on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710048.png" /> is the fundamental <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h04710049.png" />-form of some almost-Hermitian structure.

For references see [[Hermitian metric|Hermitian metric]].