No, actually a hyperbolic space CAN be wrapped, provided it isn’t “simply connected”. The 2-D analog is the difference between a flat infinite plane and the flat but edges-wrap video game universe. I’m told the hyperbolic version would yield a universe that appeared to have an infinitely-repeating hall-of-mirrors effect with octagonal symmetry.
Not really. There’s are two differences between the video game worlds of which you speak and the surface of a torus. Let’s say moving horizontally across the video game world is like moving around the hole of the torus, and moving vertically is like moving around from the inside to the outside and back. If you measured the torus horizontally on the inside part, it would be narrower than on the outside part. So the world would be wider at certain latitudes than others. In order to draw a flat map of a torus, you’d have to either make the sides of the map curved, or make the map stretched at certain latitudes like on a cartisian map of the Earth, except it would be stretched a finite amount.
The other problem is that it would necessarily take a longer distance to travel around the torus horizontally (as I defined it above), at least around the outside part, than it would to travel around it vertically. In fact, it would necessarily be at least twice as wide, or the hole in the middle would have negative size which is impossible. All the video game worlds I’ve seen are pretty close to square.
I’m pretty sure there is no three-dimensional object of which the surface can be compared to video game worlds. It’s kind of like moving along the surface of a cylinder, except that when you reach the top of it, you magically appear at the bottom. When you connect the top of a cylinder to the bottom of a cylinder though, you get a torus, and I just explained the problems there. I guess it could be compared to a hollow cylinder with infinitely thin walls. You move up the top of the cylinder and then go back down on the other side and end up where you started. I think that’s the only thing that could possibly fit the bill, but since it’s infinitely thin, the inside wall is right on the outside wall, so it’s as if some latitudes are in the same space as others.
When I mentioned M-Theory, I was thinking of the other dimensions it predicts, and how they’re curved on a tiny scale, but those dimensions aren’t a part of our observable universe and are irrelevant to what I was responding to now that I think about it. So forget I mentioned it.
Wouldn’t a circle that wraps such that when you cross the edge at one point, you appear at the edge 180 degrees from it, be similar to the surface of a sphere? If not, what would it be? I’ll probably be thinking about that one for a while. This makes me want to go back to college and become a mathematician.
Its my understanding that the topological features of a surface do not depend on ratios of distances found within that surface. Waht makes two surfaces topologically equivalent is, very roughly speaking, if I’m remembering correctly, the number of sets of circles on that surface which can not be transformed into each other. So, for example, take the torus. Draw a circle around the hole. No matter how you smoothly stretch that circle, you will never be able to get it to go around the torus “cylindrically” so to speak. (By “cylindrically” I mean to refer to the way a torus can be thought of as a cylinder with its ends glued together. A circle around the torus’s hole can’t be transformed into a circle circumscribing the torus’s “cylinder.”
Chronos is arguing that while it may not be physically possible to realize a “flat” torus, still a mathematical representation of a “flat” torus can be constructed. This “flat” torus is topologically equivalent to the surface of a bagel, even though the ratios of various distances over the two surfaces differ.
-FrL-
I think it would be, but I don’t think it would be flat. On a sphere, a straight line is a circumfrance of that sphere. In the two-D surface you’ve described, then, one straight line would be the circle centered on the center of the larger circle, with a radius (I think) one half the radius of the larger circle. That circle corresponds to the “equator” of the sphere the 2-D surface is supposed to be equivalent to. But since this circle is a “straight line” on this 2-D surface, it follows (I think) that the surface is not a flat one.
I’m losing track of what “flat” means, though, so maybe I’m wrong. I thought it meant that on the surface in question, all triangles’ angles add up to 180. I’m not sure if that’s right, though.
-FrL-
Basically what you do is take little vectors at some point and move them around, trying to keep them as parallel to themselves as possible as you move them. The amount they turn as you move them around various curves and back to the starting point tells you about intrinsic curvature. As an example of how parallel transport works on a curved surface, imagine walking from the north pole to the equator down the prime meridian, around the earth 90 degrees, and back up to the north pole, pointing south the whole time. You see that you’re now pointing in a different direction than when you started. That’s curvature.
You can get a torus by taking a sheet of paper and sewing the left edge to the right one, then the top edge to the bottom one. Since the paper was flat (you can slide vectors around without changing them) the torus is too.
And it’s not a 3-sphere anyway. It’s a 3-ball. The sphere is the surface.
Think of the torus as being made from a rolled-up piece of paper like I just posted a little earlier (somewhere up there ^).
Now, the curves on the torus that are the image of straight lines on the sheet of paper are straight on the torus.
No, it can’t. There’s a really wonderful theorem called the “Gauss-Bonnet Theorem” which basically says that if you take a certain measure of the local curvature of a surface (the scalar curvature at a point) and integrate that function over the entire surface you get 2pi times the Euler characteristic of the surface, which is 2-2g for a genus g surface.
That is, for a sphere you will always get 4pi no matter how you deform the sphere. If it were flat you’d be integrating 0 over the surface so you’d have to get 0.
No, the surface you’d get is the projective plane, which is what happens when you identify antipodal points on the 2-sphere.
Is Chronos right that a torus can be flat?
How is it that a torus can be flat but not a sphere?
-FrL-
I gave the “rolling up a sheet of paper” description before, and Chronos gave the “Asteroids” description. A torus can be flat because its genus is 1, so its Euler characteristic is zero, which means the Gauss-Bonnet integral is zero. On the other hand, a surface of genus 2 (a “torus with two holes”) has Euler characteristic -2, so its Gauss-Bonnet integral must be -4pi. On average it’s negatively curved.
[speculation]So… do I understand correctly that even if measurements showed that the universe is flat overall, it could still be for all we know that there is a maximum distance between objects?[/speculation]
-FrL-
Didn’t some accountant’s Special Theories of Disaster Area Tax returns prove that the whole fabric of the space-time continuum is not merely curved, but is in fact totally bent?
In case Mathochist’s answer, which was correct, communicated very little, you might want to think about the difference between topology and geometry. Topologies differ as Frylock described. If you can draw curves that can’t be shrunk indefinitely, then you can think of the space as having holes in it. A torus is thus topologically equivalent to a coffee mug. If you made them out of a sufficiently flexible material, you could deform one into the other, without tearing anything. Topology is about connectedness and boundaries.
Geometry is about the distance between two points, straight lines, volume, and differentiation. The distance between two points is determined by a function called a metric. When you talk about curved spaces, you are talking geometry. A flat geometry is a Euclidean geometry.
You can place different geometries on the same topology. So, a torus can be flat, if you use the asteroid game geometry that Chronos described. Or, it can be curved. If you were to measure distances on a doughnut, using the metric implied by our 3-D Euclidean metric, the doughnut would appear to be curved.
Geometry and topology are related, however. You can’t place just any geometry on any topology. It is impossible to to put a flat geometry on a 2-sphere, basically because the curvature is constant. (You prove it as mathochist describes.) General Relativity primarily determines geometry, not topology, because you can think of it as a set of equations for determining a metric.
Isn’t the projective plane a regular plane together with a single “point at infinity”? I thought that was equivalent to a regular (non-identified) sphere. Or do you have a whole 1-d set of points at infinity? And while we’re on the subject, what’s the genus of the antipodally-identified sphere, 1/2? I thought that genus had to be an integer (you can’t have half a hole), but perhaps I’m integrating over the wrong limits?
That is correct. If the Universe is uniformly positively curved, there must be a maximum distance between points. If the Universe is flat or uniformly negatively curved, then there may or may not be a maximum distance between points, depending on the topology of the Universe. All we can say right now (and presumably, all we’ll ever be able to say) is that if there is such a maximum distance, it’s larger than the size of the observable Universe.
Adding one point at infinity – the “one-point compactification” – to the plane is a sphere. The projective plane adds a line at infinity, with one point for each class of parallel lines in the plane. That is, take the circle (the interior of which is diffeomorphic to the plane) and identify the boundary points if they’re opposite each other, which means they’re on the same line through the origin.
Well genera are only really defined for orientable compact surfaces. It’s the rank of the first homology module, and for unorientable surfaces you start getting Z[sub]2[/sub] terms showing up there so it’s no longer free.
There is something called the “non-orientable genus” k which is the number of projective planes in a connected-sum decomposition of the surface. For unorientable surfaces the Euler characteristic is 2-k.