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An [[Affine connection|affine connection]] on a [[Riemannian space|Riemannian space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821101.png" /> with respect to which the [[Metric tensor|metric tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821102.png" /> of the space is covariantly constant. If the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821103.png" /> is given by a 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/r/r082/r082110/r0821104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
and the metric form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821105.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821106.png" />, then the latter condition is expressed as
+
An [[Affine connection|affine connection]] on a [[Riemannian space|Riemannian space]]  $  M $
 +
with respect to which the [[Metric tensor|metric tensor]]  $  g _ {ij} $
 +
of the space is covariantly constant. If the affine connection on  $  M $
 +
is given by a 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/r/r082/r082110/r0821107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
\left .
  
It can be also expressed as follows: Under [[Parallel displacement(2)|parallel displacement]] along any curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821108.png" />, the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821109.png" /> of two arbitrary vectors preserves its value, i.e. for vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211011.png" /> the following equality holds:
+
\begin{array}{c}
 +
\omega  ^ {i}  = \Gamma _ {k}  ^ {i}  dx  ^ {k} ,\  \mathop{\rm det}  | \Gamma _ {k}  ^ {i} |
 +
\neq  0, \\
 +
\omega _ {j}  ^ {i= \Gamma _ {jk}  ^ {i}  dx  ^ {k}  \\
 +
\end{array}
  
<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/r/r082/r082110/r08211012.png" /></td> </tr></table>
+
\right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211013.png" /> is the vector field, called the [[Covariant derivative|covariant derivative]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211014.png" /> relative to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211015.png" />, defined by the formula
+
and the metric form on  $  M $
 +
is $  ds  ^ {2} = g _ {ij} \omega  ^ {i} \omega  ^ {j} $,  
 +
then the latter condition is expressed as
  
<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/r/r082/r082110/r08211016.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
dg _ {ij}  = g _ {kj} \omega _ {i}  ^ {k} + g _ {ik} \omega _ {j}  ^ {k} .
 +
$$
  
If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211017.png" /> one goes over to a local field of orthonormal frames, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211018.png" /> (if one restricts to the case of a positive-definite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211019.png" />) and condition (2) takes the form
+
It can be also expressed as follows: Under [[Parallel displacement(2)|parallel displacement]] along any curve in $  M $,
 +
the scalar product  $  \langle  X, Y\rangle = g _ {ij} \omega  ^ {i} ( X) \omega  ^ {j} ( Y) $
 +
of two arbitrary vectors preserves its value, i.e. for vector fields  $  X, Y, Z $
 +
on  $  M $
 +
the following equality 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/r/r082/r082110/r08211020.png" /></td> </tr></table>
+
$$
 +
Z\langle  X, Y \rangle  = \langle \nabla _ {Z} X, Y\rangle + \langle  X, \nabla _ {Z} Y\rangle,
 +
$$
  
i.e. the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211021.png" /> of forms (1) takes values in the Lie algebra of the group of motions of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211022.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211023.png" />. Thus, a Riemannian connection can be interpreted as a connection in the fibre space of orthonormal frames in the Euclidean spaces tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211024.png" />. The [[Holonomy group|holonomy group]] of a Riemannian connection is a subgroup of the group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211025.png" />; a Riemannian connection for some Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211026.png" /> is any affine connection whose holonomy group is the group of motions or some subgroup of it.
+
where  $  \nabla _ {Z} X $
 +
is the vector field, called the [[Covariant derivative|covariant derivative]] of the field  $  X $
 +
relative to the field  $  Z $,
 +
defined by the formula
  
If in (1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211027.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211028.png" /> is considered with respect to the field of natural frames of a local coordinate system), then
+
$$
 +
\omega  ^ {i} ( \nabla _ {Z} X) = Z \omega  ^ {i} ( X) + \omega _ {k}  ^ {i} ( Z)
 +
\omega  ^ {k} ( X).
 +
$$
  
<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/r/r082/r082110/r08211029.png" /></td> </tr></table>
+
If in  $  M $
 +
one goes over to a local field of orthonormal frames, then  $  g _ {ij} = \delta _ {ij} $(
 +
if one restricts to the case of a positive-definite  $  ds  ^ {2} $)
 +
and condition (2) takes the form
 +
 
 +
$$
 +
\omega _ {i}  ^ {j} + \omega _ {j}  ^ {i}  = 0,
 +
$$
 +
 
 +
i.e. the matrix  $  \omega $
 +
of forms (1) takes values in the Lie algebra of the group of motions of the Euclidean space  $  E  ^ {n} $
 +
of dimension  $  n = \mathop{\rm dim}  M $.  
 +
Thus, a Riemannian connection can be interpreted as a connection in the fibre space of orthonormal frames in the Euclidean spaces tangent to  $  M $.  
 +
The [[Holonomy group|holonomy group]] of a Riemannian connection is a subgroup of the group of motions of  $  E  ^ {n} $;
 +
a Riemannian connection for some Riemannian metric on  $  M $
 +
is any affine connection whose holonomy group is the group of motions or some subgroup of it.
 +
 
 +
If in (1)  $  \omega  ^ {i} = dx  ^ {i} $(
 +
i.e.  $  M $
 +
is considered with respect to the field of natural frames of a local coordinate system), then
 +
 
 +
$$
 +
 
 +
\frac{\partial  g _ {ij} }{\partial  x  ^ {l} }
 +
  =  g _ {kj} \Gamma _ {il}  ^ {k} +
 +
g _ {ik} \Gamma _ {jl}  ^ {k} ,
 +
$$
  
 
and
 
and
  
<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/r/r082/r082110/r08211030.png" /></td> </tr></table>
+
$$
 +
\Gamma _ {ij}  ^ {k}  = \left \{ \begin{array}{c}
 +
k \\
 +
ij
 +
\end{array}
 +
\right \} -
 +
\frac{1}{2}
 +
S _ {ij}  ^ {k} - g  ^ {kl} g _ {m(} i S _ {j)} l  ^ {m} ,
 +
$$
  
 
where
 
where
  
<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/r/r082/r082110/r08211031.png" /></td> </tr></table>
+
$$
 +
\left \{ \begin{array}{c}
 +
k \\
 +
ij
 +
\end{array}
 +
\right \}  = \
  
is the so-called [[Christoffel symbol|Christoffel symbol]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211032.png" /> is the [[Torsion tensor|torsion tensor]] of the Riemannian connection. There exists one and only one Riemannian connection without torsion (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211033.png" />); it is determined by the forms
+
\frac{1}{2}
 +
g  ^ {kl} \left (  
 +
\frac{\partial  g _ {li} }{\partial  x  ^ {j} }
 +
+
 +
\frac{\partial
 +
g _ {lj} }{\partial  x  ^ {i} }
 +
-
 +
\frac{\partial  g _ {ij} }{\partial  x  ^ {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/r/r082/r082110/r08211034.png" /></td> </tr></table>
+
\right )
 +
$$
 +
 
 +
is the so-called [[Christoffel symbol|Christoffel symbol]] and  $  S _ {ij}  ^ {k} = \Gamma _ {ij}  ^ {k} - \Gamma _ {jk}  ^ {k} $
 +
is the [[Torsion tensor|torsion tensor]] of the Riemannian connection. There exists one and only one Riemannian connection without torsion (i.e. such that  $  S _ {ij}  ^ {k} = 0 $);
 +
it is determined by the forms
 +
 
 +
$$
 +
\omega _ {j}  ^ {i} = \left \{ \begin{array}{c}
 +
i \\
 +
jk
 +
\end{array}
 +
\right \}  dx  ^ {k} ,
 +
$$
  
 
and it is called the [[Levi-Civita connection|Levi-Civita connection]].
 
and it is called the [[Levi-Civita connection|Levi-Civita connection]].
Line 41: Line 133:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Lichnerowicz,  "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Lichnerowicz,  "Global theory of connections and holonomy groups" , Noordhoff  (1976)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:21, 12 January 2021


An affine connection on a Riemannian space $ M $ with respect to which the metric tensor $ g _ {ij} $ of the space is covariantly constant. If the affine connection on $ M $ is given by a matrix of local connection forms

$$ \tag{1 } \left . \begin{array}{c} \omega ^ {i} = \Gamma _ {k} ^ {i} dx ^ {k} ,\ \mathop{\rm det} | \Gamma _ {k} ^ {i} | \neq 0, \\ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} dx ^ {k} \\ \end{array} \right \} $$

and the metric form on $ M $ is $ ds ^ {2} = g _ {ij} \omega ^ {i} \omega ^ {j} $, then the latter condition is expressed as

$$ \tag{2 } dg _ {ij} = g _ {kj} \omega _ {i} ^ {k} + g _ {ik} \omega _ {j} ^ {k} . $$

It can be also expressed as follows: Under parallel displacement along any curve in $ M $, the scalar product $ \langle X, Y\rangle = g _ {ij} \omega ^ {i} ( X) \omega ^ {j} ( Y) $ of two arbitrary vectors preserves its value, i.e. for vector fields $ X, Y, Z $ on $ M $ the following equality holds:

$$ Z\langle X, Y \rangle = \langle \nabla _ {Z} X, Y\rangle + \langle X, \nabla _ {Z} Y\rangle, $$

where $ \nabla _ {Z} X $ is the vector field, called the covariant derivative of the field $ X $ relative to the field $ Z $, defined by the formula

$$ \omega ^ {i} ( \nabla _ {Z} X) = Z \omega ^ {i} ( X) + \omega _ {k} ^ {i} ( Z) \omega ^ {k} ( X). $$

If in $ M $ one goes over to a local field of orthonormal frames, then $ g _ {ij} = \delta _ {ij} $( if one restricts to the case of a positive-definite $ ds ^ {2} $) and condition (2) takes the form

$$ \omega _ {i} ^ {j} + \omega _ {j} ^ {i} = 0, $$

i.e. the matrix $ \omega $ of forms (1) takes values in the Lie algebra of the group of motions of the Euclidean space $ E ^ {n} $ of dimension $ n = \mathop{\rm dim} M $. Thus, a Riemannian connection can be interpreted as a connection in the fibre space of orthonormal frames in the Euclidean spaces tangent to $ M $. The holonomy group of a Riemannian connection is a subgroup of the group of motions of $ E ^ {n} $; a Riemannian connection for some Riemannian metric on $ M $ is any affine connection whose holonomy group is the group of motions or some subgroup of it.

If in (1) $ \omega ^ {i} = dx ^ {i} $( i.e. $ M $ is considered with respect to the field of natural frames of a local coordinate system), then

$$ \frac{\partial g _ {ij} }{\partial x ^ {l} } = g _ {kj} \Gamma _ {il} ^ {k} + g _ {ik} \Gamma _ {jl} ^ {k} , $$

and

$$ \Gamma _ {ij} ^ {k} = \left \{ \begin{array}{c} k \\ ij \end{array} \right \} - \frac{1}{2} S _ {ij} ^ {k} - g ^ {kl} g _ {m(} i S _ {j)} l ^ {m} , $$

where

$$ \left \{ \begin{array}{c} k \\ ij \end{array} \right \} = \ \frac{1}{2} g ^ {kl} \left ( \frac{\partial g _ {li} }{\partial x ^ {j} } + \frac{\partial g _ {lj} }{\partial x ^ {i} } - \frac{\partial g _ {ij} }{\partial x ^ {l} } \right ) $$

is the so-called Christoffel symbol and $ S _ {ij} ^ {k} = \Gamma _ {ij} ^ {k} - \Gamma _ {jk} ^ {k} $ is the torsion tensor of the Riemannian connection. There exists one and only one Riemannian connection without torsion (i.e. such that $ S _ {ij} ^ {k} = 0 $); it is determined by the forms

$$ \omega _ {j} ^ {i} = \left \{ \begin{array}{c} i \\ jk \end{array} \right \} dx ^ {k} , $$

and it is called the Levi-Civita connection.

References

[1] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2] A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)

Comments

Instead of "Riemannian connection" one also uses metric connection.

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Riemannian connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_connection&oldid=17394
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article