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Linear connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108002.png" /> such that for the corresponding operators of [[Covariant differentiation|covariant differentiation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108004.png" /> there holds
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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/a/a010/a010800/a0108005.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108007.png" /> are arbitrary vector fields, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108008.png" /> is a quadratic form (i.e. a symmetric bilinear form), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a0108009.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080010.png" />-form (or covector field). One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080012.png" /> are adjoint with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080013.png" />. In coordinate form (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080017.png" />),
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Linear connections  $  \Gamma $
 +
and $  \widetilde \Gamma  $
 +
such that for the corresponding operators of [[Covariant differentiation|covariant differentiation]]  $  \nabla $
 +
and $  \widetilde \nabla  $
 +
there holds
  
<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/a/a010/a010800/a01080018.png" /></td> </tr></table>
+
$$
 +
Z B ( X , Y )  = B
 +
( \nabla _ {Z} X , Y ) + B
 +
( X , {\widetilde \nabla  } _ {Z} Y ) +
 +
2 \omega (Z) B ( X , Y ) ,
 +
$$
  
For the curvature operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080020.png" /> and torsion operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080022.png" /> of the connections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080024.png" />, respectively, the following relations hold:
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where  $  X , Y $
 +
and $  Z $
 +
are arbitrary vector fields,  $  B ( \cdot , \cdot ) $
 +
is a quadratic form (i.e. a symmetric bilinear form), and $  \omega ( \cdot ) $
 +
is a $  1 $-
 +
form (or covector field). One also says that  $  \nabla $
 +
and $  \widetilde \nabla  $
 +
are adjoint with respect to  $  B $.  
 +
In coordinate form (where  $  X , Y , Z \Rightarrow \partial  _ {i} $,
 +
$  B \Rightarrow b _ {ij} $,
 +
$  \omega \Rightarrow \omega _ {i} $,  
 +
$  \nabla \Rightarrow \Gamma _ {ij}  ^ {k} $),
  
<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/a/a010/a010800/a01080025.png" /></td> </tr></table>
+
$$
 +
\partial  _ {k} b _ {ij} -
 +
\Gamma _ {ki}  ^ {s} b _ {sj} -
 +
{\widetilde \Gamma  } _ {kj}  ^ {s} b _ {is}  = \
 +
2 \omega _ {k} b _ {ij} .
 +
$$
  
<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/a/a010/a010800/a01080026.png" /></td> </tr></table>
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For the curvature operators  $  R $
 +
and  $  \widetilde{R}  $
 +
and torsion operators  $  T $
 +
and  $  \widetilde{T}  $
 +
of the connections  $  \nabla $
 +
and  $  \widetilde \nabla  $,
 +
respectively, the following relations hold:
  
<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/a/a010/a010800/a01080027.png" /></td> </tr></table>
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$$
 +
B ( R ( U , Z ) X , Y ) +
 +
B ( X , \widetilde{R}  ( U , Z ) Y ) =
 +
$$
  
<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/a/a010/a010800/a01080028.png" /></td> </tr></table>
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$$
 +
= \
 +
2 \{ \omega ( [ U , Z ] ) - U \omega
 +
(Z) + Z \omega (U) \} B ( X , Y ) ,
 +
$$
 +
 
 +
$$
 +
B ( Z , \Delta T ( X , Y ) ) - B ( \Delta T ( Z , Y ) X ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
B ( \Delta T ( Z , X ) , Y ) ,\  \Delta T  = \widetilde{T}  - T .
 +
$$
  
 
In coordinate form,
 
In coordinate form,
  
<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/a/a010/a010800/a01080029.png" /></td> </tr></table>
+
$$
 +
R _ {rsj}  ^ {m} b _ {im} +
 +
{\widetilde{R}  } _ {rsi}  ^ {m} b _ {jm}  = \
 +
- 2 ( \partial  _ {r} \omega _ {s} -
 +
\partial  _ {s} \omega _ {r} ) b _ {ij} ,
 +
$$
  
<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/a/a010/a010800/a01080030.png" /></td> </tr></table>
+
$$
 +
\Delta T _ {ij}  ^ {s} b _ {sk} - \Delta T _ {kj}  ^ {s}
 +
b _ {si} - \Delta T _ {ki}  ^ {s} b _ {sj}  = 0 .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Norden,  "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Norden,  "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Instead of the name adjoint connections one also encounters conjugate connections.
 
Instead of the name adjoint connections one also encounters conjugate connections.
  
Sometimes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080031.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010800/a01080032.png" /> is not mentioned in the notion of adjoint connections. Strictly speaking this notion of an  "adjoint connection"  should be called  "adjoint with respect to B and w" .
+
Sometimes the $  1 $-
 +
form $  \omega $
 +
is not mentioned in the notion of adjoint connections. Strictly speaking this notion of an  "adjoint connection"  should be called  "adjoint with respect to B and w" .

Latest revision as of 16:09, 1 April 2020


Linear connections $ \Gamma $ and $ \widetilde \Gamma $ such that for the corresponding operators of covariant differentiation $ \nabla $ and $ \widetilde \nabla $ there holds

$$ Z B ( X , Y ) = B ( \nabla _ {Z} X , Y ) + B ( X , {\widetilde \nabla } _ {Z} Y ) + 2 \omega (Z) B ( X , Y ) , $$

where $ X , Y $ and $ Z $ are arbitrary vector fields, $ B ( \cdot , \cdot ) $ is a quadratic form (i.e. a symmetric bilinear form), and $ \omega ( \cdot ) $ is a $ 1 $- form (or covector field). One also says that $ \nabla $ and $ \widetilde \nabla $ are adjoint with respect to $ B $. In coordinate form (where $ X , Y , Z \Rightarrow \partial _ {i} $, $ B \Rightarrow b _ {ij} $, $ \omega \Rightarrow \omega _ {i} $, $ \nabla \Rightarrow \Gamma _ {ij} ^ {k} $),

$$ \partial _ {k} b _ {ij} - \Gamma _ {ki} ^ {s} b _ {sj} - {\widetilde \Gamma } _ {kj} ^ {s} b _ {is} = \ 2 \omega _ {k} b _ {ij} . $$

For the curvature operators $ R $ and $ \widetilde{R} $ and torsion operators $ T $ and $ \widetilde{T} $ of the connections $ \nabla $ and $ \widetilde \nabla $, respectively, the following relations hold:

$$ B ( R ( U , Z ) X , Y ) + B ( X , \widetilde{R} ( U , Z ) Y ) = $$

$$ = \ 2 \{ \omega ( [ U , Z ] ) - U \omega (Z) + Z \omega (U) \} B ( X , Y ) , $$

$$ B ( Z , \Delta T ( X , Y ) ) - B ( \Delta T ( Z , Y ) X ) = $$

$$ = \ B ( \Delta T ( Z , X ) , Y ) ,\ \Delta T = \widetilde{T} - T . $$

In coordinate form,

$$ R _ {rsj} ^ {m} b _ {im} + {\widetilde{R} } _ {rsi} ^ {m} b _ {jm} = \ - 2 ( \partial _ {r} \omega _ {s} - \partial _ {s} \omega _ {r} ) b _ {ij} , $$

$$ \Delta T _ {ij} ^ {s} b _ {sk} - \Delta T _ {kj} ^ {s} b _ {si} - \Delta T _ {ki} ^ {s} b _ {sj} = 0 . $$

References

[1] A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)

Comments

Instead of the name adjoint connections one also encounters conjugate connections.

Sometimes the $ 1 $- form $ \omega $ is not mentioned in the notion of adjoint connections. Strictly speaking this notion of an "adjoint connection" should be called "adjoint with respect to B and w" .

How to Cite This Entry:
Adjoint connections. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_connections&oldid=45035
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article