# 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)