Curvature

What's the difference between a plane and a sphere? Well, for one, a sphere is curved while a plane is flat. Another example of a flat space is the cylinder, since we can just unroll it and it becomes a plane. A sphere can't simply be unrolled, though. It would need to be stretched to flatten it out. Unrolling can be done without doing any stretching. Now in Riemannian geometry we have a way of making this general concept into a mathematical concept. This is called curvature. We say a sphere is positively curved. In fact, it has constant curvature which means the curvature at every point is the same and positive (whatever that means). Similarly a plane is flat, so the curvature is zero (whatever that means) at every point. You might ask what negative curvature means. Negative curvature is exemplified by a saddle shape, which is the shape of a Pringles potato chip. If the curvature is negative and the same at every point, we have what is called Hyperbolic space. This was one of the first examples of what is called "Non-Euclidean geometry," sacrilege to classicists! Curvature, it turns out, is a very complicated concept, especially in higher dimensions. It is not completely understood. We do know that curvature determines the metric and places restrictions on the topology. Thus if we want to understand our manifold, it is often easiest to try to understand the curvature. Easier said than done, though, huh? So let's give a couple of ideas of what curvature means in two dimensions. If we draw a triangle in the plane, we find that the sum of the interior angles is 180 degrees (which is, incidentally $\pi$ radians and can even be thought of as the definition of $\pi$, some number that happens to be approximately 3.141592653589793). No matter which triangle we draw, the sum of the angles is 180 degrees. But what happens if we draw triangle on the sphere? Let's say the sphere is really big (like the size of the Earth) and we draw a little triangle on it, say on the your driveway. What will the sum of the angles be? They would essentially be 180 degrees, right? But now let's say we have a small globe and draw a triangle with a right angle (90 degrees) at the north pole and whose third side is on the equator. (Recall that a straight line on the sphere consists of great circles, so the equator is a straight line and the other two straight lines are two longitudes meeting at right angles.) Thus we have a triangle which is essentially one-fourth of the northern hemisphere or one-eighth of the sphere. What is the sum of the angles? If you look closely, you see that there are three right angles! Hence the sum of the angles is $90+90+90 = 270$ degrees. It's bigger than 180 degrees! This is what positive curvature means. If you have a triangle in positive curvature, the sum of the angles of a triangle is bigger than 180 degrees. Negative curvature, similarly, means the sum of the angles is less than 180 degrees. You might think about what this means on a Pringles potato chip! In the standard model of negative curvature, you can even have triangles which have a sum of angles almost 0! Another way to think about curvature is in terms of area of disks with the same perimeter. Let's say we have a circle. What is a circle? It is the set of points the same distance away from a single point, say distance 1. (This is rather arbitrary, so we just choose 1 because it makes the formulas come out better. Hooray!) This makes sense for any surface with a distance on it. Now, in flat curvature (the plane), the circle bounds a disk with area $\pi$ (this is another place where we could have defined $\pi$). Now, suppose the disk was made out of clay and you wanted to stretch that disk to fit on a sphere? You'd have to stretch it! Thus the enclosed area is larger than $\pi$. Now what if you wanted to put it on a Pringles potato chip? It would be too big and you would have to make the area smaller. This is another difference between positive and negative curvature. And the actual curvature relates to how much you need to stretch or shrink the disk at each point to fit on the surface. Now curvature takes place at a point, so we can have positive, negative, and flat curvature on the same surface. Picture, for instance, the surface of a doughnut (called a torus). If you think about it, the outside surface has a positive curvature and the part inside (the "hole") has negative curvature. In between, there is some zero (flat) curvature. (Can you guess where? If it's flat, you should be able to sit a plane on it so that it touches on a line or anything else with dimension more than just a point.)