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) = (e

_{1}cos u + e

_{2}sin u) cos v + 2 (e

_{3}cos u/2 + e

_{4}sin 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) = (e

_{1}cos u + e

_{2}sin u) cos v + 2 (e

_{3}cos u/2 + e

_{4}sin u/2) sin v + b (e

_{5}cos u + e

_{6}sin u).

The metric for this surface is

ds

^{2}= (1 + b

^{2}) du

^{2}+ (sin

^{2}v + 4 cos

^{2}v) dv

^{2}

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.