 Quote by HallsofIvy
That depends upon the manifold. You cannot, for example, embed the surface of the Klein bottle, a two dimensional surface, in Euclidean three space. But you can embed it in Euclidean four dimensional space and there exist a three dimensional subspace of that containing the surface of the Klein bottle.
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- There is no three dimensional subspace of R^4 that contains the Klein bottle.
- Another surface that can be embedded in R^4 but not in R^3 is the Projective plane.
- Try proving that a closed surface in R^3 must be orientable. It follows from this that neither the Klein bottle nor the Projective plane can be embedded in R^3.
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