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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
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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/h047040/h0470408.png" /></td> </tr></table>
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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/h047040/h0470409.png" /></td> </tr></table>
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An [[Affine connection|affine connection]] on a Hermitian manifold  $  M $
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relative to which the tensor  $  \phi $
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defined by the complex structure and the fundamental  $  2 $-
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form  $  \Omega = ( 1 / 2 ) g _ {\alpha \beta }  \omega  ^  \beta  \wedge \overline \omega \; {}  ^  \alpha  $
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are parallel, implying the same property for the Hermitian form  $  ds  ^ {2} = g _ {\alpha \beta }  \overline \omega \; {}  ^  \alpha  \omega  ^  \beta  $.
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If the affine connection on  $  M $
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is given by local connection forms  $  \omega _  \beta  ^  \alpha  = \Gamma _ {\beta \gamma }  ^  \alpha  \omega  ^  \gamma  + \Gamma _ {\beta {\overline \gamma \; }  }  ^  \alpha  \overline \omega \; {}  ^  \gamma  $,
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then these conditions can be expressed as
  
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" />.
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$$
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\omega _ {down - 3 {\overline \beta \; }  }  ^  \alpha  = \omega _ {down - 3 {\overline \beta \; }  } ^ {\overline \alpha \; }  = 0,\ \
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\omega _ {down - 3 {\overline \beta \; }  } ^ {\overline \alpha \; }  = \
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\overline \omega \; {} _  \beta  ^  \alpha  ,
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$$
  
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.
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$$
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d g _ {\alpha \beta }  \overline \omega \; {} _  \alpha  ^  \gamma  g _ {\gamma \beta }  + g _ {\alpha \gamma }  \omega _  \beta  ^  \gamma .
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$$
  
====References====
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On a given Hermitian manifold  $  M $
<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>
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there is one and only one Hermitian connection for which  $ \Gamma _ {\beta \overline \gamma \; }   ^ \alpha = 0 $.
  
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A generalization is an almost-Hermitian connection, which is defined by similar conditions on the tensors  $  \phi _ {j}  ^ {i} $
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and  $  g _ {ij} $
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with  $  g _ {kl} \phi _ {i}  ^ {k} \phi _ {l}  ^ {l} = g _ {ij} $
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on an almost-Hermitian manifold  $  \widetilde{M}  $.
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An almost-Hermitian connection on a given  $  \widetilde{M}  $
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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 ) $
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and  $  ( 0 , 2 ) $),
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namely, the second canonical Lichnerowicz connection.
  
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====References====
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<table>
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<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>
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<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>
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</table>
  
 
====Comments====
 
====Comments====
Line 18: Line 52:
  
 
====References====
 
====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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.O. Wells jr., "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR>
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</table>

Latest revision as of 13:42, 17 March 2023


An 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

[1] A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[2] K. Yano, "Differential geometry on complex and almost complex spaces" , Pergamon (1965) Zbl 0127.12405

Comments

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

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969) Zbl 0175.48504
[a2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
How to Cite This Entry:
Hermitian connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_connection&oldid=13208
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article