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| − | ''on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047100/h0471001.png" />'' | + | {{TEX|done}} |
| | + | ''on a manifold $M$'' |
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| − | 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" />: | + | A pair $(J,g)$ consisting of a [[Complex structure|complex structure]] $J$ on $M$ and a [[Hermitian metric|Hermitian metric]] $g$ in the tangent space $TM$, that is, a Riemannian metric $g$ that is invariant under $J$: |
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| − | <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>
| + | $$g(JX,JY)=g(X,Y)$$ |
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| − | 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" />. | + | for any vector fields $X$ and $Y$ on $M$. A Hermitian structure specifies in any tangent space $T_pM$ 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 $M$ a differential $2$-form $\omega(X,Y)=g(X,JY)$, which is called the canonical $2$-form of the Hermitian manifold. Any structure $J$ on $M$ can be completed by some Riemannian metric $g$ to a Hermitian structure $(J,g)$: for $g$ one can take the metric $g(X,Y)=g_0(X,Y)+g_0(JX,JY)$, where $g_0$ is an arbitrary metric. The canonical Hermitian connection of a Hermitian metric $g$ can be regarded as an affine connection with torsion $T$ on $M$ relative to which the fields $J$ and $g$ are parallel. Among all affine connections satisfying these conditions it is uniquely characterized by the identity $T(JX,Y)=T(X,JY)$, which is valid for its [[Torsion tensor|torsion tensor]] $T$ and any vector fields $X$ and $Y$. The [[Curvature tensor|curvature tensor]] $R$ of the canonical connection satisfies the condition $R(JX,JY)=R(X,Y)$. 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 $g$. |
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| − | 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. | + | A natural generalization of the concept of a Hermitian structure is that of an almost-Hermitian structure, which is a pair $(J,g)$ consisting of an [[Almost-complex structure|almost-complex structure]] $J$ on $M$ and a [[Riemannian metric|Riemannian metric]] $g$ that is invariant under $J$. If the fundamental $2$-form $\omega(X,Y)=g(X,JY)$ 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 $U(n)$, where $n=\dim M/2$. Any non-degenerate differential $2$-form on a manifold $M$ is the fundamental $2$-form of some almost-Hermitian structure. |
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| | For references see [[Hermitian metric|Hermitian metric]]. | | For references see [[Hermitian metric|Hermitian metric]]. |
Latest revision as of 07:16, 3 October 2014
on a manifold $M$
A pair $(J,g)$ consisting of a complex structure $J$ on $M$ and a Hermitian metric $g$ in the tangent space $TM$, that is, a Riemannian metric $g$ that is invariant under $J$:
$$g(JX,JY)=g(X,Y)$$
for any vector fields $X$ and $Y$ on $M$. A Hermitian structure specifies in any tangent space $T_pM$ the structure of a Hermitian vector space (see Hermitian metric). A manifold with a Hermitian structure is called a Hermitian manifold. A Hermitian structure defines on $M$ a differential $2$-form $\omega(X,Y)=g(X,JY)$, which is called the canonical $2$-form of the Hermitian manifold. Any structure $J$ on $M$ can be completed by some Riemannian metric $g$ to a Hermitian structure $(J,g)$: for $g$ one can take the metric $g(X,Y)=g_0(X,Y)+g_0(JX,JY)$, where $g_0$ is an arbitrary metric. The canonical Hermitian connection of a Hermitian metric $g$ can be regarded as an affine connection with torsion $T$ on $M$ relative to which the fields $J$ and $g$ are parallel. Among all affine connections satisfying these conditions it is uniquely characterized by the identity $T(JX,Y)=T(X,JY)$, which is valid for its torsion tensor $T$ and any vector fields $X$ and $Y$. The curvature tensor $R$ of the canonical connection satisfies the condition $R(JX,JY)=R(X,Y)$. A Hermitian manifold is a Kähler manifold if and only if the canonical Hermitian connection has no torsion and hence is the same as the Levi-Civita connection of $g$.
A natural generalization of the concept of a Hermitian structure is that of an almost-Hermitian structure, which is a pair $(J,g)$ consisting of an almost-complex structure $J$ on $M$ and a Riemannian metric $g$ that is invariant under $J$. If the fundamental $2$-form $\omega(X,Y)=g(X,JY)$ 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 $U(n)$, where $n=\dim M/2$. Any non-degenerate differential $2$-form on a manifold $M$ is the fundamental $2$-form of some almost-Hermitian structure.
For references see Hermitian metric.
How to Cite This Entry:
Hermitian structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_structure&oldid=12107
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article