Flat Klein bottle in 6D

Higher-dimensional geometry (previously "Polyshapes").

Flat Klein bottle in 6D

Postby mr_e_man » Thu Sep 27, 2018 6:48 pm

The Klein bottle is usually depicted as a curved surface in 3D intersecting itself. It may be put in 4D so that the tube goes "around" or "over" rather than "through itself". But it's still curved.

When a topological Klein bottle is given uniform intrinsic geometry, it turns out to be flat Euclidean geometry. (That's related to the Euler characteristic being 0.)

This paper by C. Tompkins

http://www.ams.org/journals/bull/1941-4 ... 7501-4.pdf

provides an immersion of the Klein bottle in 4D with Gaussian curvature = 0. The position vector is

x(u,v) = (e1cos u + e2sin u) cos v + 2 (e3cos u/2 + e4sin u/2) sin v.

"Immersion", as opposed to "embedding", means it intersects itself. This happens along a circle x(u,0) = x(u+pi,pi).

We can remove the self-intersection by adding an orthogonal circle in u (with radius b), again making it go "around itself":

x(u,v) = (e1cos u + e2sin u) cos v + 2 (e3cos u/2 + e4sin u/2) sin v + b (e5cos u + e6sin u).

The metric for this surface is

ds2 = (1 + b2) du2 + (sin2v + 4 cos2v) dv2

which has zero curvature, for any b.

Is there an embedding of a flat Klein bottle in 4D or 5D?

This brings to mind the Hevea torus, an embedding of a flat torus in 3D. It's not quite smooth; I believe that requires 4D.
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Re: Flat Klein bottle in 6D

Postby quickfur » Tue Oct 30, 2018 5:57 pm

The Klein bottle is a bit of a misnomer, because in 4D, it cannot hold any liquid, being only a 2D manifold (4D requires a 3D manifold in order to confine a fluid). It's essentially (one of) the 4D equivalent(s) of the Möbius strip, and would exist as a curved loop of "3D paper strip" with a "twist" that joined one side of the strip with the other. (Strips in 4D have 3D extents, and a "flat" 3D object in 4D has two such surfaces if it's orientable. The Klein bottle has a twist that joins one surface to the other, so it effectively has only one surface and is thus non-orientable.) You wouldn't be able to flatten this without also damaging/removing the twist, since the twist is essential for the Klein bottle to be what it is, and such a twist requires at least 4 spatial dimensions to be (faithfully) embedded. To be flat in 4D, it would have to fit within a 3D hyperplane.

In 5D, though, you could certainly and easily embed a Klein bottle onto a flat 5D surface -- you just lift the 4D embedding into 5D and it's flat by default. :lol: But then in 5D, the Klein bottle has become the analogue of a rubber band, and no longer even occupies the space of a strip in 5D, but is as thin as a string, relatively speaking. Furthermore, this 5D embedding would have two 5D "sides" (think of it as a subset of a 4D hyperplane embedded in 5D), and would thus be orientable in the 5D sense, while at the same time still remain non-orientable in the 4D sense.
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Re: Flat Klein bottle in 6D

Postby mr_e_man » Mon Nov 26, 2018 12:15 pm

I meant "flat" in terms of intrinsic curvature, not embedding into a "flat" 3D or 4D subspace. A flat Klein bottle can be thought of as a rectangular piece of paper, rolled into a cylinder, and folded with a twist to connect the circular edges, without stretching the paper, only bending. (Also, sharp folds / creases are disallowed; it must be smooth.)

According to this, there is an embedding of any surface in 5D, including the intrinsically flat Klein bottle. I'm looking for a parametrization of such an embedding.
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