View source for Levi-Civita connection

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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.

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|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|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|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====
<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====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR></table>