M\"obius-Type Structures in Non-Orientable Singular Semi-Riemannian Manifolds

Our objective is to illuminate the global structure of non-orientable manifolds with signature-changing metrics, with particular emphasis on global topological obstructions. Using explicit geometric constructions based on the topology of the M\"{o}bius strip, we produce examples of crosscap manifolds where the gluing junction coincides with the locus of signature change. Our main result shows that on non-orientable compact surfaces, the radical of such metrics cannot be everywhere transverse along the hypersurface of signature change. In particular, metrics arising from the transformation prescription $\tilde{g}=g+fV^{\flat}\otimes V^{\flat}$, with $g$ a Lorentzian metric and $f$ a smooth interpolation function, necessarily fail to satisfy the transversality condition. This obstruction is of purely global origin and is closely related to topological invariants such as the Euler characteristic and the non-existence of nowhere-vanishing vector fields. These results demonstrate that non-orientability imposes intrinsic limitations on the class of admissible signature-type changing metrics.