Existence of non-orientable surface parametrized by periodic variables

[nonequilibrium]

Hi!

I was wondering: is it possible to have a non-orientable surface in 3D which is parametrized by u and v, with u and v periodic (i.e. is it possible to map the torus continuously into a non-orientable surface in 3D?)

If so, does anyone have any explicit examples?

 

[fzero]

Assuming that by 3D, you mean $\mathbb{R}^3$, then it is not possible to embed any nonorientable manifold in $\mathbb{R}^3$. There are a number of directions in which to prove this, some of which are summarized at http://mathoverflow.net/questions/18987/why-cant-the-klein-bottle-embed-in-mathbbr3. It is possible to embed a nonorientable surface (2d) in $\mathbb{R}^4$. This classic example http://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07501-4/S0002-9904-1941-07501-4.pdf shows an embedding of the Klein bottle in $\mathbb{R}^4$ using periodic coordinates.

 

[nonequilibrium]

Thank you. What is the reason that the Klein bottle as shown in 3D is not a manifold? I mean, suppose we actually regard our mapping 2-torus to Klein bottle representation in 3D (so with intersection) as the object of interest, then the intersection doesn't seem to be very "important" in the sense that they happen for very different coordinates, so if you use your coordinates on the torus as a sense of "locality/distance", then whether or not the image intersects doesn't seem to be an issue.

(To be clear, I'm in a physical situation where I would interested in the mapping from the torus to the Klein bottle, rather than just the object in 3D/4D on its own right.)

 

[lavinia]

Thank you. What is the reason that the Klein bottle as shown in 3D is not a manifold? I mean, suppose we actually regard our mapping 2-torus to Klein bottle representation in 3D (so with intersection) as the object of interest, then the intersection doesn't seem to be very "important" in the sense that they happen for very different coordinates, so if you use your coordinates on the torus as a sense of "locality/distance", then whether or not the image intersects doesn't seem to be an issue.

(To be clear, I'm in a physical situation where I would interested in the mapping from the torus to the Klein bottle, rather than just the object in 3D/4D on its own right.)

The self intersection of the Klein bottle in 3 space means that it is not an embedding. It is an immersion - a local embedding.

The torus covers the Klein bottle by the involution which rotates by 180 degrees along one axis circle and reflects along the other. The pairs of points that are the orbits of this involution map to the same point on the Klein bottle.

 

[nonequilibrium]

Thank you!