# Weyl connection

A torsion-free affine connection on a Riemannian space $M$ which is a generalization of the Levi-Civita connection in the sense that the corresponding covariant differential of the metric tensor $g _ {ij}$ of $M$ is not necessarily equal to zero, but is proportional to $g _ {ij}$. If the affine connection on $M$ is given by the matrix of local connection forms

$$\tag{1 } \left . \begin{array}{rcl} \omega^i &=& \Gamma^i_k (x) dx^k\,,\ \ \det|\Gamma^i_k| \ne 0 \\ \omega^i_j &=& \Gamma^i_{jk} \omega^k \end{array} \right\rbrace$$

and $ds ^ {2} = g _ {ij} \omega ^ {i} \omega ^ {j}$, it will be a Weyl connection if and only if

$$\tag{2 } dg _ {ij} = g _ {kj} \omega _ {i} ^ {k} + g _ {ik} \omega _ {j} ^ {k} + \theta g _ {ij} .$$

Another, equivalent, form of this condition is:

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

where $\nabla _ {Z} X$, the covariant derivative of $X$ with respect to $Z$, is defined by the formula

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

With respect to a local field of orthonormal coordinates, where $g _ {ij} = \delta _ {ij}$, the following equation is valid:

$$\omega _ {i} ^ {j} + \omega _ {j} ^ {i} + \delta _ {j} ^ {i} \theta = 0,$$

i.e. any torsion-free affine connection whose holonomy group is the group of similitudes or one of its subgroups is a Weyl connection for some Riemannian metric on $M$.

If in (1) $\omega ^ {i} = dx ^ {i}$, then for a Weyl connection

$$\Gamma _ {jk} ^ {i} = \frac{1}{2} g ^ {il} \left ( \frac{\partial g _ {lj} }{\partial x ^ {k} } + \frac{\partial g _ {lk} }{\partial x ^ {j} } - \frac{\partial g _ {jk} }{\partial x ^ {l} } \right ) - \frac{1}{2} g ^ {il} g _ {jk} \theta _ {l} +$$

$$+ \frac{1}{2} ( \delta _ {j} ^ {i} \phi _ {k} + \delta _ {k} ^ {i} \phi _ {j} ) ,$$

where $\theta = \theta _ {k} dx ^ {k}$. Since

$$g _ {kj} \Omega _ {i} ^ {k} + g _ {ik} \Omega _ {j} ^ {k} + g _ {ij} d \theta = 0,$$

the tensor

$$F _ {ij,kl} = \ g _ {im} R _ {jkl} ^ {m} + \frac{1}{2} g _ {ij} ( \nabla _ {k} \theta _ {l} - \nabla _ {l} \theta _ {k} ) ,$$

called the directional curvature tensor by H. Weyl, is anti-symmetric with respect to both pairs of indices:

$$F _ {ij,kl} + F _ {ji,kl} = 0 .$$

Weyl connections were introduced by Weyl .

How to Cite This Entry:
Weyl connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_connection&oldid=51349
This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article