dorsal/arxiv
View SchemaM\"obius-Type Structures in Non-Orientable Singular Semi-Riemannian Manifolds
| Authors | Nathalie E. Rieger |
|---|---|
| Categories | |
| ArXiv ID | 2601.10009vv1 |
| URL | https://arxiv.org/abs/2601.10009 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
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.
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"abstract": "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.",
"arxiv_id": "2601.10009",
"authors": [
"Nathalie E. Rieger"
],
"categories": [
"math.DG",
"gr-qc",
"math-ph",
"math.GT",
"math.MP"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "M\\\"obius-Type Structures in Non-Orientable Singular Semi-Riemannian Manifolds",
"url": "https://arxiv.org/abs/2601.10009",
"version": "v1"
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