Topology and manifolds, part three

Continuing!

So, what do spherical, Euclidean, and hyperbolic geometry have to do with manifolds? Well, it turns out that all two-dimensional manifolds have a natural geometric structure.

That statement will require some unpacking.

Let’s start with the torus, i.e. the surface of a donut, which is commonly described as follows: take a rectangle (imagine it to be made of some thin, flexible material) and then glue two opposite edges together to form a tube. Then bend the tube, and glue the ends of the tube together to get a torus. This is an example of a Euclidean geometric structure on a torus, which is a fancy way of saying that we’ve constructed a torus out of pieces of the Euclidean plane…namely, the rectangle…in such a way that all of the edges and corners glue together neatly.

This “gluing together neatly” part is important. Specifically, it’s very important that the corners come together neatly. In the case of the torus above, the rectangle had four corners, each of which was a right angle, that is, π/2 radians or 90 degrees. Those four corners all come together to form a single point after all of that gluing, and when they do those four angles add up to 2π radians or 360 degrees, i.e. a full circle. That’s exactly what we want to happen: the single “corner” of the resulting torus lies flat.

Let’s take another example. Instead of a single rectangle, let’s take six squares and glue them together to form the surface of a cube. Topologically, a cube is the same as a sphere, but geometrically we have a problem: the corners don’t lie flat. At each corner of the cube there are three right angles coming together, and that only adds up to 3π/2 radians, less than a full circle. Gluing six squares together in this way does not result in a Euclidean geometric structure on a sphere.

But not surprisingly, spherical geometry steps in here. In spherical geometry, the angles of every triangle, and indeed every polygon, add up to more than they would in Euclidean geometry. It’s possible to construct a square in spherical geometry whose angles are each 2π/3 radians, or 120 degrees…all four of them. If you take six of those squares and glue them together, you’ll get a spherical “cube” where the corners do lie flat: each corner is made up of three angles of 2π/3 radians, which adds up to a full circle. A sphere has, not surprisingly, a spherical geometric structure: we can construct it neatly out of pieces of the sphere.

One last example. This is the fun one, but you might want to draw some pictures as you go along.

Imagine two pairs of pants. Stitch all the holes together: waist to waist, leg to leg, and other leg to other leg. What you’ll end up with is a two-holed torus. Now, cut the two-holed torus back into two pairs of pants. Then, cut each pair of pants into two pieces with three cuts, as follows: from the left edge of the waist down the outside of the left leg to the bottom, from the right edge of the waist down the outside of the right leg to the bottom, and finally along the inside of both legs, from the left hem up to the crotch and down to the right hem. (Basically, cut the pants into the front half and the back half.) You’ve now cut your two-holed torus into four pieces in total. Each piece is a hexagon, made of three cuts and three half-hems (if you consider the waist to be a hem). When you stitch the hexagons back together to make the two-holed torus, the corners of the hexagons will come together in sets of four. If we want those corners to lie flat, the simplest way would be to ensure that each corner of each hexagon is a right angle; four of those will add up to a full circle, just like in the torus example at the beginning.

Hexagons with six right angles? Try drawing that. It’s just crazy talk…in Euclidean geometry. But we can construct right-angled hexagons in hyperbolic geometry. Thus, the two-holed torus has a hyperbolic geometric structure.

Whew! Let’s cut to the chase: every two-dimensional manifold has some kind of geometric structure, either spherical, Euclidean, or hyperbolic. Talking about the compact manifolds I described in part two, the sphere and the projective plane have spherical geometric structures, the torus and the Klein bottle have Euclidean geometric structures, and everything else has a hyperbolic geometric structure.

But I said in the interlude that all of this geometry had something to do with three-dimensional manifolds, didn’t I? Yes, yes I did…

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