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An [[Affine connection|affine connection]] on a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582301.png" /> that is a [[Riemannian connection|Riemannian connection]] (that is, a connection with respect to which the [[Metric tensor|metric tensor]] is covariantly constant) and has zero [[Torsion|torsion]]. An affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582302.png" /> is determined uniquely by these conditions, hence every Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582303.png" /> has a unique Levi-Civita connection. This concept first arose in 1917 with T. Levi-Civita [[#References|[1]]] as the concept of [[Parallel displacement(2)|parallel displacement]] of a vector in Riemannian geometry. The idea itself goes back to F. Minding, who in 1837 introduced the concept of the involute of a curve on a surface.
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Automatically converted into TeX, above some diagnostics.
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With respect to a local coordinate system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582305.png" />, the Levi-Civita connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582306.png" /> is defined by the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582307.png" />, where
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{{TEX|auto}}
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{{TEX|done}}
  
<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/l/l058/l058230/l0582308.png" /></td> </tr></table>
+
An [[Affine connection|affine connection]] on a Riemannian space  $  M $
 +
that is a [[Riemannian connection|Riemannian connection]] (that is, a connection with respect to which the [[Metric tensor|metric tensor]] is covariantly constant) and has zero [[Torsion|torsion]]. An affine connection on  $  M $
 +
is determined uniquely by these conditions, hence every Riemannian space  $  M $
 +
has a unique Levi-Civita connection. This concept first arose in 1917 with T. Levi-Civita [[#References|[1]]] as the concept of [[Parallel displacement(2)|parallel displacement]] of a vector in Riemannian geometry. The idea itself goes back to F. Minding, who in 1837 introduced the concept of the involute of a curve on a surface.
  
its [[Curvature tensor|curvature tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l0582309.png" /> is defined by the formula
+
With respect to a local coordinate system in  $  M $,
 +
where  $  d s  ^ {2} = g _ {ij}  d x  ^ {i}  d x  ^ {j} $,
 +
the Levi-Civita connection on  $  M $
 +
is defined by the forms  $  \omega _ {j}  ^ {i} = \{ _ {jk} ^ { i } \}  d x  ^ {k} $,
 +
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/l/l058/l058230/l05823010.png" /></td> </tr></table>
+
$$
 +
\left \{ \begin{array}{c}
 +
i \\
 +
jk
 +
\end{array}
 +
\right \}
 +
=
 +
\frac{1}{2}
 +
g  ^ {il}
 +
\left (
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823011.png" />; then
+
\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/l/l058/l058230/l05823012.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/l/l058/l058230/l05823013.png" /></td> </tr></table>
+
\frac{\partial  g _ {jk} }{\partial  x  ^ {l} }
 +
 
 +
\right ) ;
 +
$$
 +
 
 +
its [[Curvature tensor|curvature tensor]]  $  R _ {jkl}  ^ {i} $
 +
is defined by the formula
 +
 
 +
$$
 +
d \omega _ {j}  ^ {i} + \omega _ {k}  ^ {i} \wedge
 +
\omega _ {j}  ^ {k}  =
 +
\frac{1}{2}
 +
 
 +
R _ {jkl}  ^ {i}  d x  ^ {k} \wedge d x  ^ {l} .
 +
$$
 +
 
 +
Let  $  R _ {ij,kl} = g _ {im} R _ {jkl}  ^ {m} $;  
 +
then
 +
 
 +
$$
 +
R _ {ij,kl}  =
 +
\frac{1}{2}
 +
 
 +
\left \{
 +
 
 +
\frac{\partial  ^ {2} g _ {jk} }{\partial  x  ^ {i} \partial  x  ^ {l} }
 +
-
 +
 
 +
\frac{\partial  ^ {2} g _ {jl} }{\partial  x  ^ {i} \partial  x  ^ {k} }
 +
-
 +
 
 +
\frac{\partial  ^ {2} g _ {ik} }{\partial  x  ^ {j} \partial  x  ^ {l} }
 +
+
 +
 
 +
\frac{\partial  ^ {2} g _ {il} }{\partial  x  ^ {j} \partial  x  ^ {k} }
 +
 
 +
\right \} +
 +
$$
 +
 
 +
$$
 +
+
 +
g _ {pq} \left ( \left \{ \begin{array}{c}
 +
p \\
 +
il
 +
\end{array}
 +
 
 +
\right \} \left \{ \begin{array}{c}
 +
q \\
 +
jk
 +
\end{array}
 +
\right \}
 +
- \left \{ \begin{array}{c}
 +
p \\
 +
ik
 +
\end{array}
 +
\right \}
 +
\left \{ \begin{array}{c}
 +
q \\
 +
jl
 +
\end{array}
 +
\right \} \right ) ;
 +
$$
  
 
thus:
 
thus:
  
<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/l/l058/l058230/l05823014.png" /></td> </tr></table>
+
$$
 +
R _ {ij,kl}  = - R _ {ij,lk} ,\ \
 +
R _ {ij,kl}  = R _ {kl,ij} ,
 +
$$
  
<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/l/l058/l058230/l05823015.png" /></td> </tr></table>
+
$$
 +
R _ {ij,kl} + R _ {ik,lj} + R _ {il,jk}  = 0 .
 +
$$
  
The curvature tensor of the Levi-Civita connection has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823016.png" /> essential components, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823017.png" />. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823018.png" /> there is only one essential component: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823020.png" /> is the [[Gaussian curvature|Gaussian curvature]].
+
The curvature tensor of the Levi-Civita connection has $  n  ^ {2} ( n  ^ {2} - 1 ) / 12 $
 +
essential components, where $  n = \mathop{\rm dim}  M $.  
 +
For example, for $  n = 2 $
 +
there is only one essential component: $  R _ {12,12} = K  \mathop{\rm det}  | g _ {ij} | $,  
 +
where $  K $
 +
is the [[Gaussian curvature|Gaussian curvature]].
  
If a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823021.png" /> is isometrically immersed in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823022.png" />, then its Levi-Civita connection is characterized as follows: For two arbitrary vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823025.png" /> the [[Covariant derivative|covariant derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823026.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823027.png" /> is the orthogonal projection on the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823028.png" /> of the ordinary differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823029.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823031.png" /> with respect to the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058230/l05823032.png" />. In other words, the mapping of a neighbouring infinitely close tangent plane onto the original tangent plane is accomplished by orthogonal projection.
+
If a Riemannian space $  M $
 +
is isometrically immersed in a Euclidean space $  E  ^ {N} $,  
 +
then its Levi-Civita connection is characterized as follows: For two arbitrary vector fields $  X $,  
 +
$  Y $
 +
on $  M \subset  E  ^ {N} $
 +
the [[Covariant derivative|covariant derivative]] $  ( \nabla _ {Y} X ) _ {x} $
 +
at a point $  x \in M $
 +
is the orthogonal projection on the tangent plane $  T _ {x} ( M) \subset  E  ^ {N} $
 +
of the ordinary differential $  ( d _ {Y} X ) _ {x} $
 +
of the field $  X $
 +
in $  E  ^ {N} $
 +
with respect to the vector $  Y _ {x} \in T _ {x} ( M) $.  
 +
In other words, the mapping of a neighbouring infinitely close tangent plane onto the original tangent plane is accomplished by orthogonal projection.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Levi-Civita,  "Nozione di parallelismo in una varietá qualunque e consequente specificazione geometrica della curvatura riemanniana"  ''Rend. Circ. Math. Palermo'' , '''42'''  (1917)  pp. 173–205</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T. Levi-Civita,  "Nozione di parallelismo in una varietá qualunque e consequente specificazione geometrica della curvatura riemanniana"  ''Rend. Circ. Math. Palermo'' , '''42'''  (1917)  pp. 173–205</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Gromoll,  W. Klingenberg,  W. Meyer,  "Riemannsche Geometrie im Grossen" , Springer  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,  "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>

Latest revision as of 22:16, 5 June 2020


An affine connection on a Riemannian space $ M $ that is a Riemannian connection (that is, a connection with respect to which the metric tensor is covariantly constant) and has zero torsion. An affine connection on $ M $ is determined uniquely by these conditions, hence every Riemannian space $ M $ has a unique Levi-Civita connection. This concept first arose in 1917 with T. Levi-Civita [1] as the concept of parallel displacement of a vector in Riemannian geometry. The idea itself goes back to F. Minding, who in 1837 introduced the concept of the involute of a curve on a surface.

With respect to a local coordinate system in $ M $, where $ d s ^ {2} = g _ {ij} d x ^ {i} d x ^ {j} $, the Levi-Civita connection on $ M $ is defined by the forms $ \omega _ {j} ^ {i} = \{ _ {jk} ^ { i } \} d x ^ {k} $, where

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

its curvature tensor $ R _ {jkl} ^ {i} $ is defined by the formula

$$ d \omega _ {j} ^ {i} + \omega _ {k} ^ {i} \wedge \omega _ {j} ^ {k} = \frac{1}{2} R _ {jkl} ^ {i} d x ^ {k} \wedge d x ^ {l} . $$

Let $ R _ {ij,kl} = g _ {im} R _ {jkl} ^ {m} $; then

$$ R _ {ij,kl} = \frac{1}{2} \left \{ \frac{\partial ^ {2} g _ {jk} }{\partial x ^ {i} \partial x ^ {l} } - \frac{\partial ^ {2} g _ {jl} }{\partial x ^ {i} \partial x ^ {k} } - \frac{\partial ^ {2} g _ {ik} }{\partial x ^ {j} \partial x ^ {l} } + \frac{\partial ^ {2} g _ {il} }{\partial x ^ {j} \partial x ^ {k} } \right \} + $$

$$ + g _ {pq} \left ( \left \{ \begin{array}{c} p \\ il \end{array} \right \} \left \{ \begin{array}{c} q \\ jk \end{array} \right \} - \left \{ \begin{array}{c} p \\ ik \end{array} \right \} \left \{ \begin{array}{c} q \\ jl \end{array} \right \} \right ) ; $$

thus:

$$ R _ {ij,kl} = - R _ {ij,lk} ,\ \ R _ {ij,kl} = R _ {kl,ij} , $$

$$ R _ {ij,kl} + R _ {ik,lj} + R _ {il,jk} = 0 . $$

The curvature tensor of the Levi-Civita connection has $ n ^ {2} ( n ^ {2} - 1 ) / 12 $ essential components, where $ n = \mathop{\rm dim} M $. For example, for $ n = 2 $ there is only one essential component: $ R _ {12,12} = K \mathop{\rm det} | g _ {ij} | $, where $ K $ is the Gaussian curvature.

If a Riemannian space $ M $ is isometrically immersed in a Euclidean space $ E ^ {N} $, then its Levi-Civita connection is characterized as follows: For two arbitrary vector fields $ X $, $ Y $ on $ M \subset E ^ {N} $ the covariant derivative $ ( \nabla _ {Y} X ) _ {x} $ at a point $ x \in M $ is the orthogonal projection on the tangent plane $ T _ {x} ( M) \subset E ^ {N} $ of the ordinary differential $ ( d _ {Y} X ) _ {x} $ of the field $ X $ in $ E ^ {N} $ with respect to the vector $ Y _ {x} \in T _ {x} ( M) $. In other words, the mapping of a neighbouring infinitely close tangent plane onto the original tangent plane is accomplished by orthogonal projection.

References

[1] T. Levi-Civita, "Nozione di parallelismo in una varietá qualunque e consequente specificazione geometrica della curvatura riemanniana" Rend. Circ. Math. Palermo , 42 (1917) pp. 173–205
[2] D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[3] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)

Comments

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
How to Cite This Entry:
Levi-Civita connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Levi-Civita_connection&oldid=14415
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article