Weyl Connections on Surfaces Complete Curvature Expansion and Dimensional Collapse

We present a complete and explicit derivation of the curvature tensors associated with torsion-free Weyl connections on Riemannian manifolds. Starting from the defining pure-trace non-metricity condition, we compute the connection coefficients, Riemann tensor, and Ricci tensor with a full separation of linear and quadratic contributions, as well as symmetric and antisymmetric sectors. All computations are carried out line by line, with fixed curvature conventions and without omission. In dimension two, we establish a structural collapse phenomenon: the symmetric part of the Weyl–Ricci tensor is forced to be pure trace, independently of gauge choices or field equations. This rigidity leads to a simplified scalar curvature formula and to a modified Gauss–Bonnet identity, where the -norm of the Weyl covector provides a global obstruction to conformal reduction. The results are purely geometric and clarify the special role of surfaces as a maximal rigidity regime within non-metric affine geometry.