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

$$ \tag{1 } \left . and $ ds ^ {2} = g _ {ij} \omega ^ {i} \omega ^ {j} $, it will be a Weyl connection if and only if $$ \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 > = < \nabla _ {Z} X, Y \rangle + \langle X, \nabla _ {Z} Y \rangle + \theta ( Z) \langle X, Y\rangle , $$ where $ \nabla _ {Z} X $, the [[Covariant derivative|covariant derivative]] of $ X $ with respect to $ Z $, is defined by the formula $$ \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, $$ i.e. any torsion-free affine connection whose [[Holonomy group|holonomy group]] is the group of similitudes or one of its subgroups is a Weyl connection for some Riemannian metric on $ M $. If in (1) $ \omega ^ {i} = dx ^ {i} $, then for a Weyl connection $$ \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 [1].

References

[1]	H. Weyl, "Reine Infinitesimalgeometrie" Math. Z. , 2 (1918) pp. 384–411
[2]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[3]	G.B. Folland, "Weyl manifolds" J. Differential Geom. , 4 (1970) pp. 145–153