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)