  where the gauge function is \(\lambda(x) = e^{\theta(x)}\). Thus, a conformal structure plus length connection or gauge field \(A_{j}(x)\) determines a Weyl geometry equipped with a unique Weyl connection. Therefore, the  fundamental principle of infinitesimal geometry also holds in a Weyl  geometry; that is, the metric structure of a Weyl geometry determines a unique affine connection, namely, the Weyl connection. 