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A torsion-free [[Affine connection|affine connection]] on a [[Riemannian space|Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w0976901.png" /> which is a generalization of the [[Levi-Civita connection|Levi-Civita connection]] in the sense that the corresponding [[Covariant differential|covariant differential]] of the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w0976902.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w0976903.png" /> is not necessarily equal to zero, but is proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w0976904.png" />. If the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w0976905.png" /> is given by the matrix of local connection forms
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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/w/w097/w097690/w0976906.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w0976907.png" />, it will be a Weyl connection if and only if
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A torsion-free [[Affine connection|affine connection]] on a [[Riemannian space|Riemannian space]]  $  M $
 +
which is a generalization of the [[Levi-Civita connection|Levi-Civita connection]] in the sense that the corresponding [[Covariant differential|covariant differential]] of the metric tensor  $  g _ {ij} $
 +
of  $  M $
 +
is not necessarily equal to zero, but is proportional to  $  g _ {ij} $.  
 +
If the affine connection on  $  M $
 +
is given by the matrix of local connection forms
  
<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/w/w097/w097690/w0976908.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
\left .
 +
 
 +
and  $  ds  ^ {2} = g _ {ij} \omega  ^ {i} \omega  ^ {j} $,
 +
it will be a Weyl connection if and only if
 +
 
 +
$$ \tag{2 }
 +
dg _ {ij}  = g _ {kj} \omega _ {i}  ^ {k} +
 +
g _ {ik} \omega _ {j}  ^ {k} + \theta g _ {ij} .
 +
$$
  
 
Another, equivalent, form of this condition is:
 
Another, equivalent, form of this condition is:
  
<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/w/w097/w097690/w0976909.png" /></td> </tr></table>
+
$$
 +
Z \langle  X, Y > = < \nabla _ {Z} X, Y \rangle + \langle  X, \nabla _ {Z} Y \rangle +
 +
\theta ( Z) \langle  X, Y\rangle ,
 +
$$
 +
 
 +
where  $  \nabla _ {Z} X $,
 +
the [[Covariant derivative|covariant derivative]] of  $  X $
 +
with respect to  $  Z $,
 +
is defined by the formula
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769010.png" />, the [[Covariant derivative|covariant derivative]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769011.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769012.png" />, is defined by the formula
+
$$
 +
\omega  ^ {i} ( \nabla _ {Z} X)  = \
 +
Z \omega  ^ {i} ( X) + \omega _ {k}  ^ {i} ( Z) \omega  ^ {k} ( X).
 +
$$
  
<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/w/w097/w097690/w09769013.png" /></td> </tr></table>
+
With respect to a local field of orthonormal coordinates, where  $  g _ {ij} = \delta _ {ij} $,
 +
the following equation is valid:
  
With respect to a local field of orthonormal coordinates, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769014.png" />, the following equation is valid:
+
$$
 +
\omega _ {i}  ^ {j} + \omega _ {j}  ^ {i} + \delta _ {j}  ^ {i} \theta  = 0,
 +
$$
  
<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/w/w097/w097690/w09769015.png" /></td> </tr></table>
+
i.e. any torsion-free affine connection whose [[Holonomy group|holonomy group]] is the group of similitudes or one of its subgroups is a Weyl connection for some Riemannian metric on  $  M $.
  
i.e. any torsion-free affine connection whose [[Holonomy group|holonomy group]] is the group of similitudes or one of its subgroups is a Weyl connection for some Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769016.png" />.
+
If in (1)  $  \omega  ^ {i} = dx  ^ {i} $,
 +
then for a Weyl connection
  
If in (1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769017.png" />, then for a Weyl connection
+
$$
 +
\Gamma _ {jk}  ^ {i} =
 +
\frac{1}{2}
 +
g  ^ {il}
 +
\left (  
 +
\frac{\partial  g _ {lj} }{\partial  x  ^ {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/w/w097/w097690/w09769018.png" /></td> </tr></table>
+
+
 +
\frac{\partial  g _ {lk} }{\partial  x  ^ {j} }
 +
-
  
<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/w/w097/w097690/w09769019.png" /></td> </tr></table>
+
\frac{\partial  g _ {jk} }{\partial  x  ^ {l} }
 +
\right ) -  
 +
\frac{1}{2}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097690/w09769020.png" />. Since
+
g  ^ {il} g _ {jk} \theta _ {l} +
 +
$$
  
<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/w/w097/w097690/w09769021.png" /></td> </tr></table>
+
$$
 +
+
 +
 
 +
\frac{1}{2}
 +
( \delta _ {j}  ^ {i} \phi _ {k} + \delta _ {k}  ^ {i} \phi _ {j} ) ,
 +
$$
 +
 
 +
where  $  \theta = \theta _ {k}  dx  ^ {k} $.
 +
Since
 +
 
 +
$$
 +
g _ {kj} \Omega _ {i}  ^ {k} +
 +
g _ {ik} \Omega _ {j}  ^ {k} +
 +
g _ {ij}  d \theta  = 0,
 +
$$
  
 
the tensor
 
the tensor
  
<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/w/w097/w097690/w09769022.png" /></td> </tr></table>
+
$$
 +
F _ {ij,kl}  = \
 +
g _ {im} R _ {jkl}  ^ {m} +
 +
\frac{1}{2}
 +
 
 +
g _ {ij} ( \nabla _ {k} \theta _ {l} - \nabla _ {l} \theta _ {k} ) ,
 +
$$
  
 
called the directional curvature tensor by H. Weyl, is anti-symmetric with respect to both pairs of indices:
 
called the directional curvature tensor by H. Weyl, is anti-symmetric with respect to both pairs of indices:
  
<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/w/w097/w097690/w09769023.png" /></td> </tr></table>
+
$$
 +
F _ {ij,kl} + F _ {ji,kl}  = 0 .
 +
$$
  
 
Weyl connections were introduced by Weyl [[#References|[1]]].
 
Weyl connections were introduced by Weyl [[#References|[1]]].

Revision as of 08:29, 6 June 2020


A torsion-free affine connection on a Riemannian space $ M $ which is a generalization of the Levi-Civita connection in the sense that the corresponding covariant differential of the metric tensor $ g _ {ij} $ of $ M $ is not necessarily equal to zero, but is proportional to $ g _ {ij} $. If the affine connection on $ M $ is given by the matrix of local connection forms

$$ \tag{1 } \left . and $ ds ^ {2} = g _ {ij} \omega ^ {i} \omega ^ {j} $, it will be a Weyl connection if and only if $$ \tag{2 } dg _ {ij} = g _ {kj} \omega _ {i} ^ {k} + g _ {ik} \omega _ {j} ^ {k} + \theta g _ {ij} . $$ Another, equivalent, form of this condition is: $$ Z \langle X, Y > = < \nabla _ {Z} X, Y \rangle + \langle X, \nabla _ {Z} Y \rangle + \theta ( Z) \langle X, Y\rangle , $$ where $ \nabla _ {Z} X $, the [[Covariant derivative|covariant derivative]] of $ X $ with respect to $ Z $, is defined by the formula $$ \omega ^ {i} ( \nabla _ {Z} X) = \ Z \omega ^ {i} ( X) + \omega _ {k} ^ {i} ( Z) \omega ^ {k} ( X). $$ With respect to a local field of orthonormal coordinates, where $ g _ {ij} = \delta _ {ij} $, the following equation is valid: $$ \omega _ {i} ^ {j} + \omega _ {j} ^ {i} + \delta _ {j} ^ {i} \theta = 0, $$ i.e. any torsion-free affine connection whose [[Holonomy group|holonomy group]] is the group of similitudes or one of its subgroups is a Weyl connection for some Riemannian metric on $ M $. If in (1) $ \omega ^ {i} = dx ^ {i} $, then for a Weyl connection $$ \Gamma _ {jk} ^ {i} = \frac{1}{2}

g  ^ {il}

\left ( \frac{\partial g _ {lj} }{\partial x ^ {k} }

+ \frac{\partial g _ {lk} }{\partial x ^ {j} }

-

\frac{\partial g _ {jk} }{\partial x ^ {l} }

\right ) - 

\frac{1}{2}

g ^ {il} g _ {jk} \theta _ {l} + $$ $$ +

\frac{1}{2}

( \delta _ {j}  ^ {i} \phi _ {k} + \delta _ {k}  ^ {i} \phi _ {j} ) ,

$$ where $ \theta = \theta _ {k} dx ^ {k} $. Since $$ g _ {kj} \Omega _ {i} ^ {k} + g _ {ik} \Omega _ {j} ^ {k} + g _ {ij} d \theta = 0, $$ the tensor $$ F _ {ij,kl} = \ g _ {im} R _ {jkl} ^ {m} + \frac{1}{2}

g _ {ij} ( \nabla _ {k} \theta _ {l} - \nabla _ {l} \theta _ {k} ) , $$ called the directional curvature tensor by H. Weyl, is anti-symmetric with respect to both pairs of indices: $$ F _ {ij,kl} + F _ {ji,kl} = 0 . $$

Weyl connections were introduced by Weyl [1].

References

[1] H. Weyl, "Reine Infinitesimalgeometrie" Math. Z. , 2 (1918) pp. 384–411
[2] A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[3] G.B. Folland, "Weyl manifolds" J. Differential Geom. , 4 (1970) pp. 145–153
How to Cite This Entry:
Weyl connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_connection&oldid=15696
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article