The Ricci Tensor of Torsion-Free Weyl Connections in Dimension Two: Conformal Collapse, Pure-Trace Structure, and Gauss–Bonnet Implications

We investigate torsion-free Weyl connections with pure-trace non-metricity and analyze the structure of their Ricci tensor in arbitrary dimension. A general decomposition of the Ricci curvature is derived, separating Levi–Civita, linear, and quadratic contributions associated with the Weyl 1-form. We show that the quadratic anisotropic term is universally governed by a dimensional factor (n−2)(n-2)(n−2). As a consequence, in dimension two this term vanishes identically, forcing the symmetric part of the Ricci tensor to collapse to a pure-trace form. This phenomenon is shown to be structural rather than accidental, and is traced back to representation-theoretic constraints. In dimension two, we further establish an explicit formula for the Weyl scalar curvature and analyze its global implications. In particular, we show that the Gauss–Bonnet identity is modified by a non-topological contribution proportional to the L2L^2L2-norm of the Weyl 1-form, while all anisotropic curvature information disappears locally. Exact Weyl structures are shown to reduce to classical conformal geometry, recovering the standard conformal transformation law of scalar curvature on surfaces. Several explicit examples illustrate the collapse mechanism and its geometric consequences. These results clarify the special and rigid role of dimension two in Weyl geometry and provide a clean geometric baseline for the study of non-metric connections, independently of any variational or physical interpretation.