By use of maps, the surface of the Earth can be definitively defined as a sphere. However, if one only specifies that going in any direction on the Earth will eventually return one to his or her starting point, then many different manifolds qualify. The Earth could have just as easily been a torus, or any other finite manifold without an edge. Only by confirming the curvature within several different "patches" of a manifold can one uniquely determine it.
The same problem exists with the Universe. However, it is a rather more difficult one.
A common misconception concerning the shape of the Universe is that if it is finite, it has an edge or boundary. This is not true. On a 3-sphere, (recall that the surface of the Earth is a 2-sphere) one could go indefinitely through 3-dimensional space in one direction and never reach an edge, although he or she may return to his or her starting point. In fact, the Universe's lack of an edge is mostly agreed upon, as there would be disagreements in density and isotropy that make an edge unlikely.
Also, the curvature of the Universe can be determined to a fairly accurate degree by measuring its density. The density of matter in the Universe determines how fast it continues to expand. This is because the amount of gravity counteracting the expansion of the Universe is dependent on the amount of matter and energy present, and this in turn, determines whether the Universe is expanding, and at what rate. Since the presence of matter also determines how space is bent, the local curvature of the Universe around the aforementioned matter can be calculated as spherical, Euclidean, or hyperbolic.
Throughout all of the Universe that we can see (the observable Universe) the matter seems evenly distributed at a sufficiently high scale. Obviously, on (relatively) small scales, there is a large difference in density between stars and the interstellar medium, and between galaxies and the intergalactic void, but when one considers density on the scale of billions of light-years, the density is remarkably uniform. From this, one can assume that the curvature of the Universe is constant.
Additionally, observations up to the present have suggested that the density of the Universe is very close to the so called "critical density", at which the curvature of the Universe would be exactly 0. The constant Ω, known as the density parameter, representing the ratio of the actual density to the critical density, would then be 1 for a flat Universe, greater than 1 for a spherical Universe, and less than 1 for a hyperbolic one.
As a caveat to assumptions about the finite or infinite nature of the Universe, note that the curvature of the Universe does not determine its global structure topologically. Many people assume that if the Universe has flat geometry, it must be infinite, just as the flat Euclidean plane is. However, it was noted in previous posts that the torus has Euclidean geometry as well.
In fact, when one considers 3-manifolds, there are 10 possibilities for finite Euclidean 3-manifolds alone, the most simple of which is the 3-torus.
The fundamental cube for the 3-torus. Imagine that the upper right end face of the cube is the front. Each face is connected with the opposite one (front to back, left to right, top to bottom) while preserving the orientation of faces. This means that the edge C1 is attached (facing up) to the corresponding edge also labeled C1 on the opposite C face.
Many telltale signs would exist if the Universe was a 3-torus. Using the 2-torus (previously known as simply the torus) as an analog, one can explore the remarkable phenomena of a toroidal Universe. To do this, consider life forms occupying the surface of a 2-torus, and then consider the 3-dimensional extension.
The first of these is the closed and finite nature of this Universe. An inhabitant of the 2-torus would see another copy of himself looking around the top of the donut! (see below)
Inhabitants of this Universe would see an image of themselves by looking in the direction indicated by the arrow. The light itself actually leaves the other side of the blue oval, and travels along the orange path, reaching the blue oval again after one revolution. One might argue that the curvature of the torus would obscure the view, but this is not true, as the Universe is the surface of the torus, and light can only travel along this surface. Therefore, light would traverse a closed circle in certain directions. In fact, the number of these directions is infinite!
By looking around the "ring" of the torus, inhabitants of the blue oval might also see themselves. The same goes for an observer looking diagonally, where the light would circle around the bottom as it goes around, any number of times! Infinite images of the same blue oval would exist in their "sky"!
Extending this system to our Universe, the 3-torus shape could easily be identified by images of our galaxy, the Milky Way, in the night sky, right? Unfortunately, it isn't that easy. First, the sheer size of the Universe may be so large that even the closest image point may be many billions of light years away-beyond the scope of the observable Universe. And even if there was an image point within our view, we couldn't easily identify it, as it would be an image of the galaxy from billions of years ago! Although we may not know it, a galaxy looked upon by the Hubble could possibly be our own, simply at a different stage of evolution!
How then, does a donut Universe leave its mark? The signs may be more subtle, but they are there. For example, consider a point in space emitting light uniformly in all directions. Assuming that the intensity of the light dies away over distance, one would expect that the brightness at a constant distance would remain constant. This is not the case.
Returning to our above example, light rays would converge on the opposite side of the torus, and in multiple places in between. By observing from each and every point on the torus, one could construct a contour diagram of the brightness compared to what would otherwise be expected. Judging from their distance from the source, some points receive more light than they "should" in a generic flat, infinite Universe. The two points that deviate most are the diametrically opposite point on the torus, and the point on the bottom of the ring. Other points would be intermediately shaded. These "hot spots" are indicative of a donut Universe, but there are too many light sources for the specific example above to take effect.
Finally, perhaps the most important phenomenon is the discrepancies in gravity that would occur. Treating gravity waves similar to the light example above, one can clearly see that an object, which exerts gravity on its surroundings uniformly, would exert more gravity on some point in the toroidal Universe than others. The only difference here is that direction matters. For an object on the exact opposite side of the torus Universe, the gravity waves converge from all directions in a symmetrical way, adding up to a net zero force. For an object in the vicinity of the original object, the convergence of gravitational waves adds more attraction to the object than one would normally expect given the distance for an infinite Universe. These results are summarized in the figure below with the two-dimensional analogs of both the flat and the toroidal Universe. (click to enlarge)
In both figures, the dotted lines represent gravitational attraction. It is assumed for simplicity that the blue objects are the only bodies exerting gravity, and that the remainder of the objects are of negligible mass. In the flat Universe, only one line connects two points, but on the torus, multiple lines between two points are a result of the finite cyclic nature of the manifold; a line going around the manifold will come back to the vicinity of its original position.
In the next post of the Manifolds Series, other theories are considered.
Sources: http://en.wikipedia.org/wiki/Shape_of_the_Universe, The Road to Reality by Roger Penrose, The Poincare Conjecture by Donal O'Shea, http://www.astronomy.ohio-state.edu/~ryden/ast162_9/notes40.html, http://www.math.brown.edu/~banchoff/STG/ma8/papers/leckstein/Cosmo/torus.html
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