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[Il Palazzo]

Now I'm lost. How do you define curvature then?

Well, for example, you look at whether parallel lines ever intersect, or whether angles in a triangle add up to more or less than 180 degrees.

[Sheb]

Okay, but I still can't imagine a space curved without another dimension for it to curve into. Oh, well, that's what you get from evolving for hunting antelopes rather than doing physics. 

[Evil Knievel]

I am quite sure that a smooth curvature throughout 3D space (=local property of space) alone, without any other constraints on it, would not guarantee a "closed" (=global property of space) 3D space embedded in 4D.

The curvature may also be changing (like it does on the surface of a potato) which again means, that there is no center to be defined.

[Sheb]

Of course, local curvature wouldn't be enough to guarantee a "closed" space. But I still cannot imagine an "open" space with curvature without a n+1th dimension to curve into.

[palsch]

Is there anything inherently different between a flat space and a curved space? In terms of having to be embedded in a higher dimension manifold that is.

[Sheb]

I guess not, it just goes against all my life experience of living in a space that at least look flat so I've got trouble visualizing it.

[scrdest]

I think it does, actually. Unlike a flat plane, curved plane is limited - the difference between it curving 'downwards' or 'upwards' relative to our perspective is whether we're on the inside or the outside of the sphere, but the sphere would nonetheless have finite volume in a given amount of time, whereas a flat plane does not have finite volume - and thus could not be contained in a finite 4-dimensional space.

[Sheb]

You could totally have an illimited, curved plane. Think of a parabolic surface for example.

[scrdest]

You could totally have an illimited, curved plane. Think of a parabolic surface for example.

IIRC, the thing that determines (or represents, can't remember) the curvature of our space is the interaction between gravity and universe expansion - if they reach the point where they are equal, it's a flat plane, otherwise it's curved in either direction, and they are consistent around the entire universe as far as we know, so any curved surface would eventually need to loop around.

[Darvi]

You could totally have an illimited, curved plane. Think of a parabolic surface for example.

Don't be silly. Spacetime is obviously hyperbolic.

[Il Palazzo]

if they reach the point where they are equal, it's a flat plane, otherwise it's curved in either direction, and they are consistent around the entire universe as far as we know, so any curved surface would eventually need to loop around.

No! Only if the curvature is globally positive the universe can be closed. Negative global curvature leads to open universe(a saddle-like shape of e.g., a Pringles crisp is a 2d analogue).
The difference between negative and positive curvature can be understood in terms of the angles of a triangle drawn in that space. If they add to more than 180 degree, it's positive(and it's the same result whichever side of the sphere you draw it in), if less than 180, it's negative.

[scrdest]

if they reach the point where they are equal, it's a flat plane, otherwise it's curved in either direction, and they are consistent around the entire universe as far as we know, so any curved surface would eventually need to loop around.

No! Only if the curvature is globally positive the universe can be closed. Negative global curvature leads to open universe(a saddle-like shape of e.g., a Pringles crisp is a 2d analogue).
The difference between negative and positive curvature can be understood in terms of the angles of a triangle drawn in that space. If they add to more than 180 degree, it's positive(and it's the same result whichever side of the sphere you draw it in), if less than 180, it's negative.

Yeah, I get the second part, I wasn't clear on the first one before.

[Bauglir]

Oh. Right. Yeah, okay, there's only a center if the universe's geometry is closed, and I seem to recall the evidence suggesting it ain't.

[Putnam]

Read flatland >_<

[Reelya]

It sounds like we live inside a sphere that is everywhere at the same time, allowing everywhere to be the centre! But we don't because a sphere is 3D and not... 4D or however many dimensions this universe has.

Lower dimension analogies are just about the best we can do to understand the limits of our perspective for higher-dimension stuff. The border of a 4D hypersphere is this 3D thing, but we know it's not really a 'sphere'. A sphere has flat space inside it, whereas the 3D-brane is a curved space.

The analogy in 2D would be sphere-surface living flatlanders. They know that they live on a sphere, but none of them can visualize it, so the best most of them can come up with is "it's like living inside a circle".