+,- and 0 curvature

DaveC426913

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In Brian Greene's 'Fabric of the Cosmos', he describes three possible curvatures that space may have: positive curvature (like a ball or torus), negative curvature (like a saddle) or zero curvature (like an infinite flat tabletop, or like a Pacman video game screen).

In his analogy to a video game screen, he demonstrates how, as in Pacman, if you exit side of the screen, you reappear at the other side, same with top/bottom, so it is with a flat universe - if you continue in one direction long enough in a zero-curvature universe, you will eventually wrap around, and arrive back where you started. He says that, mathematically, this is called a "2 dimensional torus".

???

Colour me hogtied, but I thought that was the quintessential closed, curved universe (be it spherical or toroidal). i.e.: the way you get 3 dimensional space to loop back on itself is to bend it in the 4th dimension so that it is a 4D sphere or torus.

A zero-curvature universe would very definitely NOT loop around like a video game screen, it would continue on forever.

Am I misunderstanding?
 

Hurkyl

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The great thing about differential geometry is it frees you of the notion that "space" has to be embedded in some (possibly higher-dimensional) Euclidean space.


Curvature is a purely local and intrinsic property. Looking at just a small piece of the pac-man gaming surface, it is indistinguishable from an ordinary plane. Thus, flat.

For another example, the surface of a cylinder is also flat!

Incidentally, the surface of a torus in 3-space is not flat -- you need 4 dimensions (more?) to get a torus with zero curvature.
 

DaveC426913

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Hurkyl said:
The great thing about differential geometry is it frees you of the notion that "space" has to be embedded in some (possibly higher-dimensional) Euclidean space.


Curvature is a purely local and intrinsic property. Looking at just a small piece of the pac-man gaming surface, it is indistinguishable from an ordinary plane. Thus, flat.

For another example, the surface of a cylinder is also flat!

Incidentally, the surface of a torus in 3-space is not flat -- you need 4 dimensions (more?) to get a torus with zero curvature.
I'm not sure how this answers my question.

If the Pacman video game wraps around behind to join up to the other side then, while locally, part of it might be flat, it's overall curvature is curved and closed. It is topologically identical to a sphere.

A cylinder, while is has flat portions, is still a closed, positively curved shape. It is topologically identical to a sphere.

Green is saying that a flat universe with zero curvature is toplogically unlike a curved, closed shape. Yet he claims that, somehow, it wraps around and rejoins without being topo-equivalent.
 

Hurkyl

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I'm pointing out the difference between your use of the word "curvature" and its actual meaning.


There's something called "extrinsic curvature" that, I believe, coincides with your interpretation of "curvature", but the "intrinsic curvature" is the only thing that matters. (At least as long as you stay within space!)
 
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DaveC426913 said:
In his analogy to a video game screen, he demonstrates how, as in Pacman, if you exit side of the screen, you reappear at the other side, same with top/bottom, so it is with a flat universe - if you continue in one direction long enough in a zero-curvature universe, you will eventually wrap around, and arrive back where you started. He says that, mathematically, this is called a "2 dimensional torus".
If the universe were spatially closed, and there was an initial even distribution of matter, then wouldn't we see matter entering the universe since it will have traveled to the point of return? Could this be the ZPE of the cosmological constant?
 

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