Riemannian connection

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

$$ \tag{1 } \left . and the metric form on $ M $ is $ ds ^ {2} = g _ {ij} \omega ^ {i} \omega ^ {j} $, then the latter condition is expressed as $$ \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.

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)