Embedding Signature-Changing Manifolds: A Braneworld and Kaluza-Klein Perspective

TL;DR

This paper explores the geometry of signature-changing manifolds relevant to cosmology, proving the existence of global embeddings into higher-dimensional spaces and providing explicit constructions for specific models.

Contribution

It establishes the existence of global isometric embeddings for signature-changing manifolds and constructs explicit embeddings into Minkowski and Misner spaces for canonical models.

Findings

Proved global isometric embeddings into higher-dimensional pseudo-Euclidean spaces.

Constructed explicit embeddings into Minkowski and Misner spaces for specific models.

Provided new geometric tools for studying signature change and braneworlds.

Abstract

We investigate a class of semi-Riemannian manifolds characterized by smooth metric signature changes with a transverse radical. This class includes spacetimes relevant to cosmological models such as the Hartle-Hawking "no boundary" proposal, where a Riemannian manifold transitions smoothly into a Lorentzian spacetime without boundaries or singularities. For this class, we prove the existence of global isometric embeddings into higher-dimensional pseudo-Euclidean spaces. We then strengthen this result by demonstrating that a specific type of global isometric embedding, which we term an $\mathcal{H}$-global embedding, also exists into both Minkowski space and Misner space. For the canonical $n$-dimensional signature-changing model, we explicitly construct a full global isometric embedding into $(n+1)$-dimensional Minkowski and Misner spaces, a significantly stronger result than an $\mathcal{H}$-global embedding for this specific case. This embedding framework provides new geometric tools for studying signature change and braneworlds through the geometry of submanifolds embedded in a bulk, thus presenting a mathematically well-defined approach to these phenomena.