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

$$ \tag{1 } \left . \begin{array}{c} \omega ^ {i} = \Gamma _ {k} ^ {i} dx ^ {k} ,\ \mathop{\rm det} | \Gamma _ {k} ^ {i} | \neq 0, \\ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} dx ^ {k} \\ \end{array} \right \} $$

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 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:

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

where $ \nabla _ {Z} X $ is the vector field, called the 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 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 and $ S _ {ij} ^ {k} = \Gamma _ {ij} ^ {k} - \Gamma _ {jk} ^ {k} $ is the 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)