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.

2d plane that is an open cylinder with radius r

 A 2D open cylinder of radius (r) is usually defined as the surface

[ S={(x,y,z)\in\mathbb{R}^3 : x^2+y^2=r^2}. ]

The word open here means it extends infinitely along its axis (say the (z)-axis), not that it has boundary circles at the ends.

From the intrinsic viewpoint, this surface is a 2-dimensional manifold. A particularly useful coordinate system is

[ (\theta,z), ]

where

[ x=r\cos\theta,\qquad y=r\sin\theta,\qquad z=z, ]

with

[ \theta\in S^1,\qquad z\in\mathbb{R}. ]

So topologically the cylinder is

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

Metric

The Euclidean metric inherited from (\mathbb{R}^3) is

[ ds^2 = r^2,d\theta^2 + dz^2. ]

This completely defines the geometry.

Flatness

An interesting fact is that this geometry is intrinsically flat:

[ K=0 ]

everywhere, where (K) is the Gaussian curvature.

You can see this by introducing

[ u=r\theta. ]

Then the metric becomes

[ ds^2 = du^2 + dz^2, ]

which is exactly the Euclidean plane metric.

Locally, the cylinder and plane have the same geometry. A triangle drawn on the cylinder has angle sum (180^\circ), parallel lines behave as in the plane, etc.

Difference from the plane

The difference is global rather than local.

The plane is

[ \mathbb{R}^2, ]

while the cylinder is

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

One direction wraps around and is periodic:

[ (\theta,z)\sim(\theta+2\pi,z). ]

Equivalently, you can define the cylinder as the quotient

[ \mathbb{R}^2 /\bigl(u,z\bigr)\sim\bigl(u+2\pi r,z\bigr). ]

This is often the cleanest mathematical definition: take a flat plane and identify points separated by (2\pi r) in one direction.

Given your recent questions about spacetime and dimensions, it's worth noticing that an open cylinder is a nice example of a space that is locally Euclidean but globally different from Euclidean space, much like how some spacetime topologies in relativity can be locally Minkowskian while having nontrivial global structure.

General relativity and the block universe and pseudo-Riemanian spacetime

 In relativity, the answer is not exactly. What is negative is the squared spacetime interval under one common sign convention.

In ordinary Euclidean geometry, the distance between two nearby points is

[ ds^2 = dx^2 + dy^2 + dz^2 ]

which is always non-negative.

In flat Minkowski spacetime, one common convention is

[ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2. ]

Notice the minus sign. Because of it, (ds^2) can be:

• Negative (timelike separation)
  
• Zero (lightlike separation)
  
• Positive (spacelike separation)
  

For two events occurring at the same spatial location but at different times,

[ dx = dy = dz = 0 ]

so

[ ds^2 = -c^2dt^2. ]

This is negative. That's what you were noticing in our earlier discussion.

However, physicists often define the proper time (d\tau) along a timelike worldline as

[ d\tau^2 = -\frac{ds^2}{c^2} ]

(using the sign convention above).

Since (ds^2<0) for timelike motion,

[ d\tau^2 >0, ]

and the elapsed proper time (\tau) is a perfectly ordinary positive quantity.

What about General Relativity?

General relativity replaces the flat Minkowski metric with a curved metric:

[ ds^2 = g_{\mu\nu}dx^\mu dx^\nu. ]

Locally, at every point, spacetime still has the same Lorentzian signature as Minkowski space. The usual signatures are either

[ (-,+,+,+) ]

or

[ (+,-,-,-). ]

The choice is merely convention.

• With ((- + + +)), timelike intervals have (ds^2<0).
  
• With ((+ - - -)), timelike intervals have (ds^2>0).
  

Both describe exactly the same physics.

From a geometric perspective

Riemannian geometry uses a metric that is positive definite. Distances are always positive.

Spacetime in general relativity is actually a pseudo-Riemannian (or Lorentzian) manifold, not a purely Riemannian one. The metric is not positive definite, which is why intervals can be positive, negative, or zero.

So in the block universe picture of general relativity:

• The spacetime interval (ds^2) between timelike-separated events is often negative (depending on convention).
  
• Proper time along an observer's worldline is positive.
  
• The "negative interval" is not a negative amount of time. It is a consequence of the Lorentzian geometry that distinguishes the time dimension from the spatial dimensions.
  

An elegant way to think about it is that Euclidean geometry treats all directions alike, while Lorentzian geometry gives the time direction a different sign. That sign difference is precisely what creates light cones, causality, and the distinction between past, future, and elsewhere.

The number line and a circle as one dimensional geometry

 This is a subtle question because the answer gets at the difference between dimension and topology.

Both a straight number line and a circle are one-dimensional spaces. At any point on either one, you can move in only two directions: "forward" or "backward" along the space. So dimension alone cannot distinguish them.

The difference is in their global structure.

The Number Line

The ordinary number line (\mathbb{R}):

• extends infinitely in both directions,
  
• has no loops,
  
• has no beginning or end, but it is unbounded,
  
• can be cut at any point and still remain connected on each side.
  

Visually:

<------------------------>

Every point corresponds to exactly one real number.

The Circle

A circle (S^1):

• is finite in extent,
  
• loops back onto itself,
  
• has no endpoints,
  
• is bounded but has no boundary.
  

Visually:

   _____

 /       \

|         |

 \_______/

If you travel far enough in one direction, you eventually return to where you started.

Why a Circle Isn't Just a Curved Line

You can bend a line into all sorts of shapes without changing its fundamental topology. For example,

~~~~~~~~~~

or

/\/\/\/\/\

are still topologically equivalent to a line.

But when you connect the ends together, something fundamentally changes. You create a closed loop.

One way mathematicians detect this is by asking:

Can the space be continuously deformed into another without cutting or gluing?

A line cannot be deformed into a circle because forming the circle requires gluing two distinct points together.

Coordinates Reveal the Difference

On the number line, positions are labeled by real numbers:

[ x \in (-\infty,\infty) ]

On a circle, positions are often labeled by an angle:

[ \theta \in [0,2\pi) ]

but with the identification

[ 0 \equiv 2\pi. ]

This identification is the key. Two different coordinate values represent the same point.

In a sense, a circle can be viewed as

[ \mathbb{R}/\mathbb{Z}, ]

which means taking the real line and declaring every integer shift to be equivalent:

[ x \sim x+n. ]

The line becomes periodic.

Connecting This to Your Earlier Question

Notice how dimension alone is not enough to characterize a space:

• A point is 0-dimensional.
  
• A line is 1-dimensional.
  
• A circle is also 1-dimensional.
  

The line and circle have the same local dimensionality, but different global topology.

Similarly, in relativity:

• Minkowski spacetime and a curved spacetime can both be 4-dimensional.
  
• What differs is the geometric and topological structure, not the dimension count.
  

So the shortest answer is:

A circle and a number line are both one-dimensional because locally they look like a line. The difference is that the number line is open and infinite, while the circle is closed and periodic—the ends are identified with each other, giving the space a different topology.