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

\begin{equation} \label{eq1} \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 \end{equation}

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

\begin{equation} \label{eq2} dg _ {ij} = g _ {kj} \omega _ {i} ^ {k} + g _ {ik} \omega _ {j} ^ {k} + \theta g _ {ij} . \end{equation}

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 \eqref{eq1} $ \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) Zbl 0925.53007
[3]	G.B. Folland, "Weyl manifolds" J. Differential Geom. , 4 (1970) pp. 145–153