A generalized Weyl structure with arbitrary non-metricity

A Weyl structure is usually defined by an equivalence class of pairs $$(\mathbf{g}, {\varvec{\omega }})$$ related by Weyl transformations, which preserve the relation $$\nabla \mathbf{g}={\varvec{\omega }}\otimes \mathbf{g}$$, where $$\mathbf{g}$$ and $${\varvec{\omega }}$$ denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection $$\Gamma _{({\varvec{\omega }})}$$, which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity. This second view allows us to present two different generalizations of standard Weyl structures. The first one relies on conformal symmetry while allowing for a general non-metricity tensor, and the other comes from extending the symmetry to arbitrary (disformal) transformations of the metric.