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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

$$ \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(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:

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

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

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

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|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|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]].

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

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