3d analogue of an open 2d cylinder geometry

 The natural 3D analogue of an open cylinder depends on which property of the cylinder you want to preserve.

For the 2D cylinder, we started with a plane (\mathbb{R}^2) and wrapped one dimension into a circle:

[ \mathbb{R}^2 ;\to; S^1\times\mathbb{R}. ]

One dimension is periodic, one remains infinite.

The direct 3D analogue is therefore

[ S^1\times\mathbb{R}^2. ]

You can think of it as ordinary 3-space where one coordinate wraps around.

Using coordinates ((\theta,x,y)),

[ \theta\in[0,2\pi),\qquad x,y\in\mathbb{R}, ]

with metric

[ ds^2=r^2d\theta^2+dx^2+dy^2. ]

Just as the 2D cylinder is locally identical to the Euclidean plane, this space is locally identical to ordinary Euclidean 3-space. Its curvature is zero everywhere.

Embedded picture

If you want to visualize it inside ordinary (\mathbb{R}^4), it is

[ x_1^2+x_2^2=r^2, ]

with two additional unconstrained coordinates (x_3,x_4).

This is a hypersurface in 4 dimensions having topology

[ S^1\times\mathbb{R}^2. ]

The hierarchy

You can see a pattern:

Dimension	Euclidean Space	One direction wrapped
1	(\mathbb{R})	(S^1) (circle)
2	(\mathbb{R}^2)	(S^1\times\mathbb{R}) (cylinder)
3	(\mathbb{R}^3)	(S^1\times\mathbb{R}^2)
4	(\mathbb{R}^4)	(S^1\times\mathbb{R}^3)

So the "3D cylinder space" is not usually called a cylinder. Topologists would simply call it (S^1\times\mathbb{R}^2): a flat three-dimensional manifold with one compact circular dimension and two infinite dimensions.

This is actually very close to an idea that appears in cosmology and relativity: a universe could be locally Euclidean yet have one spatial dimension that loops around on itself. If you traveled far enough in that direction, you would return to your starting position, just as walking around the circumference of a cylinder brings you back where you began. The geometry is locally flat, but the topology is nontrivial.