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## Main Question or Discussion Point

In Bernard Schutz's 'A first course in General Relativity', p325 (1st edition) he says

" [the constant-time hypersurface of a FLRW spacetime with k=-1 (hyperbolic)] is not realisable as a three-dimensional hypersurface in a four- or higher-dimensional Euclidean space."

On the face of it, this appears to contradict the Strong Whitney Embedding Theorem, which states that :

"any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean 2m-space, if m>0. "

So it should be possible to embed the constant-time hypersurface in Euclidean space of six dimensions, if not less.

Have I misunderstood either Schutz or Whitney here, or is Schutz just wrong?

" [the constant-time hypersurface of a FLRW spacetime with k=-1 (hyperbolic)] is not realisable as a three-dimensional hypersurface in a four- or higher-dimensional Euclidean space."

On the face of it, this appears to contradict the Strong Whitney Embedding Theorem, which states that :

"any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean 2m-space, if m>0. "

So it should be possible to embed the constant-time hypersurface in Euclidean space of six dimensions, if not less.

Have I misunderstood either Schutz or Whitney here, or is Schutz just wrong?