We provide a new theorem which asserts that any analytic n-dimensional pseudo-Riemannian manifold can be locally and isometrically embedded into (n+2)-dimensional pseudo-Euclidean spacetimes with at least two possible signatures \(\mathbb {E}^{n+1,1}\) and \(\mathbb {E}^{n,2}\) . Hence, such manifolds are at most of embedding class two. This theorem may be viewed as a direct consequence of the Dahia–Romero embedding theorem for embeddings into (n+1)-dimension pseudo-Riemannian space, in the context of vacuum and constant non-zero curvature. As consequence of this embedding theorem, we note that it resolves the open problem concerning the embedding class of the Gödel metric. We recapitulate the known Euclidean embedding results for FLRW geometries into pseudo-Euclidean spaces, and make some corrections to the possible signature. We also provide an explicit example demonstrating this.
