Given a 3-manifold with coordinates at each point $(x,y,\theta)$. Where $x$ and $y$ are from $\mathbb{R}$ and $\theta$ is from $0..2\pi$.
I would like it to have these (and only) these symmetries:
1) Translation $\hat{T}_{a,b}(x,y,\theta)\rightarrow (x+a,y+b,\theta)$
2) Rotation $\hat{R}_\phi(x,y,\theta) \rightarrow (\cos(\phi)x+\sin(\phi)y, \cos(\phi)y-\sin(\phi)x, \phi+\theta)$
As you can see when it does a rotation around any point, it also translates along the compact direction. (Kind of like rotating a 2D vector field also rotates all the vectors).
Now this seems like the only manifold is $\mathbb{R}^2 \times S_1$. But this would also have the additional symmetries of a rotation without affecting the compact direction and separately a translation in the compact direction.
Is there by chance some of manifold that satisfies this? I can't really imagine one. On the other hand, does the symmetry define the manifold? In which case what topology does this define? (Maybe a fibre bundle?)
Edit I have an idea. If $x$ and $y$ are very large but not infinite. Then if we take the $x, y$ plane at $\theta=0$ and match it with the $x, y$ plane at $\theta=2\pi$ with a 360 degree twist. Then match the other sides to make a 3-torus. Perhaps that would work? Well it would work for the middle point, seeing as when you translate down the $\theta$ axis you are doing a 360 degree rotation. Don't think it would work for the rest of the points.