Difference between revisions of "Riemannian connection"

Revision as of 14:53, 7 June 2020

An affine connection on a Riemannian space with respect to which the metric tensor of the space is covariantly constant. If the affine connection on is given by a matrix of local connection forms

(1)

and the metric form on is , then the latter condition is expressed as

(2)

It can be also expressed as follows: Under parallel displacement along any curve in , the scalar product of two arbitrary vectors preserves its value, i.e. for vector fields on the following equality holds:

where is the vector field, called the covariant derivative of the field relative to the field , defined by the formula

If in one goes over to a local field of orthonormal frames, then (if one restricts to the case of a positive-definite ) and condition (2) takes the form

i.e. the matrix of forms (1) takes values in the Lie algebra of the group of motions of the Euclidean space of dimension . Thus, a Riemannian connection can be interpreted as a connection in the fibre space of orthonormal frames in the Euclidean spaces tangent to . The holonomy group of a Riemannian connection is a subgroup of the group of motions of ; a Riemannian connection for some Riemannian metric on is any affine connection whose holonomy group is the group of motions or some subgroup of it.

If in (1) (i.e. is considered with respect to the field of natural frames of a local coordinate system), then

and

where

is the so-called Christoffel symbol and is the torsion tensor of the Riemannian connection. There exists one and only one Riemannian connection without torsion (i.e. such that ); it is determined by the forms

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=49405

This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article

@@ Line 1: / Line 1: @@
-<!--
+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
-r0821101.png
-$#A+1 = 34 n = 0
-$#C+1 = 34 : ~/encyclopedia/old_files/data/R082/R.0802110 Riemannian connection
-Automatically converted into TeX, above some diagnostics.
-Please remove this comment and the {{TEX|auto}} line below,
-if TeX found to be correct.
--->
-{{TEX|auto}}
+<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>
-{{TEX|done}}
-An [[Affine connection|affine connection]] on a [[Riemannian space|Riemannian space]]  $  M $
+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
-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
-$$ \tag{1 }
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821107.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
-\left .
-and the metric form on  $  M $
+It can be also expressed as follows: Under [[Parallel displacement(2)|parallel displacement]] along any curve in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821108.png" />, the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r0821109.png" /> of two arbitrary vectors preserves its value, i.e. for vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211011.png" /> the following equality holds:
-is  $  ds  ^ {2} = g _ {ij} \omega  ^ {i} \omega  ^ {j} $,
-then the latter condition is expressed as
-$$ \tag{2 }
+<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>
-dg _ {ij}  =  g _ {kj} \omega _ {i}  ^ {k} + g _ {ik} \omega _ {j}  ^ {k} .
-$$
-It can be also expressed as follows: Under [[Parallel displacement(2)|parallel displacement]] along any curve in  $  M $,
+where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211013.png" /> is the vector field, called the [[Covariant derivative|covariant derivative]] of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211014.png" /> relative to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211015.png" />, defined by the formula
-the scalar product  $  \langle  X, Y\rangle = g _ {ij} \omega  ^ {i} ( X) \omega  ^ {j} ( Y) $
-of two arbitrary vectors preserves its value, i.e. for vector fields  $  X, Y, Z $
-on  $  M $
-the following equality holds:
-$$
+<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/r08211016.png" /></td> </tr></table>
-Z\langle  X, Y>  =  < \nabla _ {Z} X, Y\rangle + \langle  X, \nabla _ {Z} Y\rangle,
-$$
-where  $  \nabla _ {Z} X $
+If in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211017.png" /> one goes over to a local field of orthonormal frames, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211018.png" /> (if one restricts to the case of a positive-definite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211019.png" />) and condition (2) takes the form
-is the vector field, called the [[Covariant derivative|covariant derivative]] of the field  $  X $
-relative to the field  $  Z $,
-defined by the formula
-$$
+<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>
-\omega  ^ {i} ( \nabla _ {Z} X)  =  Z \omega  ^ {i} ( X) + \omega _ {k}  ^ {i} ( Z)
-\omega  ^ {k} ( X).
-$$
-If in  $  M $
+i.e. the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211021.png" /> of forms (1) takes values in the Lie algebra of the group of motions of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211022.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211023.png" />. Thus, a Riemannian connection can be interpreted as a connection in the fibre space of orthonormal frames in the Euclidean spaces tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211024.png" />. The [[Holonomy group|holonomy group]] of a Riemannian connection is a subgroup of the group of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211025.png" />; a Riemannian connection for some Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211026.png" /> is any affine connection whose holonomy group is the group of motions or some subgroup of it.
-one goes over to a local field of orthonormal frames, then  $  g _ {ij} = \delta _ {ij} $(
-if one restricts to the case of a positive-definite  $  ds  ^ {2} $)
-and condition (2) takes the form
-$$
+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}  ^ {j} + \omega _ {j}  ^ {i}  =  0,
-$$
-i.e. the matrix  $  \omega $
+<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082110/r08211029.png" /></td> </tr></table>
-of forms (1) takes values in the Lie algebra of the group of motions of the Euclidean space  $  E  ^ {n} $
-of dimension  $  n =  \mathop{\rm dim}  M $.
-Thus, a Riemannian connection can be interpreted as a connection in the fibre space of orthonormal frames in the Euclidean spaces tangent to  $  M $.
-The [[Holonomy group|holonomy group]] of a Riemannian connection is a subgroup of the group of motions of  $  E  ^ {n} $;
-a Riemannian connection for some Riemannian metric on  $  M $
-is any affine connection whose holonomy group is the group of motions or some subgroup of it.
-If in (1)  $  \omega  ^ {i} = dx  ^ {i} $(
-i.e.  $  M $
-is considered with respect to the field of natural frames of a local coordinate system), then
-$$
-\frac{\partial  g _ {ij} }{\partial  x  ^ {l} }
-  =  g _ {kj} \Gamma _ {il}  ^ {k} +
-g _ {ik} \Gamma _ {jl}  ^ {k} ,
-$$
 and
-$$
+<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} ,
-$$
 where
-$$
+<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 \}  = \
-\frac{1}{2}
+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
- 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 )
+<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>
-$$
-is the so-called [[Christoffel symbol|Christoffel symbol]] and  $  S _ {ij}  ^ {k} = \Gamma _ {ij}  ^ {k} - \Gamma _ {jk}  ^ {k} $
-is the [[Torsion tensor|torsion tensor]] of the Riemannian connection. There exists one and only one Riemannian connection without torsion (i.e. such that  $  S _ {ij}  ^ {k} = 0 $);
-it is determined by the forms
-$$
-\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]].
@@ Line 124: / Line 41: @@
 ====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>
 ====Comments====

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Difference between revisions of "Riemannian connection"

Revision as of 14:53, 7 June 2020

References

Comments

References