Our objective is to illuminate the global structure of non-orientable manifolds with signature-changing metrics. Using explicit constructions based on the topology of the M\"{ o}bius strip, we produce examples of crosscap manifolds where the gluing junction serves as the locus of signature change. In another set of examples, we convert the M\"{ o}bius strip into a singular signature-type changing manifold. For these resulting manifolds, we test whether the metric can be expressed as $\tilde{ g}=g+fV^{ \flat}\otimes V^{ \flat}$, with $g$ a Lorentzian metric and $f$ a smooth interpolation function between the Lorentzian and Riemannian regions, separated by the signature change hypersurface $\mathcal{ H}$. Our analysis reveals that the radical of the metric can transition from transverse to tangent at $\mathcal{ H}$, pseudo-space orientability is obstructed by the Euler characteristic, and pseudo-time orientability may still hold. These examples illustrate subtle obstructions to applying standard transformation prescriptions for signature change and highlight novel phenomena in compact, non-orientable semi-Riemannian manifolds.
Categories: math.DG gr-qc math-ph math.GT math.MP