DG, Part 5: I Think It's Touching

[stoutfellow]

I'm not going to give anything like a definition of "surface", as the word is used by differential geometers. There are simply too many steps in the construction to make it worthwhile in a forum like this. Instead, I'll give a few examples and general rubrics before delving into some constructions.

The first thing to emphasize is that surfaces are two-dimensional objects; that is, to locate a point on a given surface requires you to specify two numbers. The usual coordinates on the plane are an example. On the surface of a sphere, think of latitude and longitude. (When I use the word "sphere", take it for granted that I mean the surface; if I mean the interior as well, I use the word "ball" - but I probably won't have to, in this series of posts.)

Here are a few more examples.

• The cylinder. Here, I mean only the rounded side of a cylinder, not the flat (circular) ends. Points on an upright cylinder can be identified by saying how high up they are and how far around from some specified vertical line (analogous to the Greenwich meridian).


• The torus. Again, this means the surface of a doughnut-shaped object. If you think of it as lying horizontally, you could place a compass in the center; to identify a point, you'd say what angle on the compass points to it and how far around from, say, the innermost circle of the torus it is.


• The Möbius strip, obtained by taking a rectangular strip of paper, giving it a half-twist, and joining the ends together. (Without the half-twist, you'd get a cylinder.) This is an important example, as it has some unusual properties; most famously, it has only one side. (The importance of this will become apparent a bit later.)

In general, we want two-dimensionality, and there are certain smoothness conditions. From the point of view of differential geometry, a cube, for instance, is not a surface, as it has sharp edges. (Topologically, that's not a problem, but it gets in the way if you want to take derivatives.)

Now, we need to describe tangent planes. "Tangent" is derived from a Latin word meaning "touching", and when we're talking about the tangent line to a curve, that's a reasonable word for it. When applied to surfaces, however, it's just a little misleading. (Not so much so as to warrant abandoning the word, but a little.) A couple of examples might clarify the problem.

First, consider a sphere. The tangent plane to the sphere at, say, the "North Pole" is simply the horizontal plane resting on top. More generally, if you pick any point of the sphere and draw the radius from that point to the center, the tangent plane at that point is the plane perpendicular to the radius. So far, so good: the tangent plane does, indeed, just touch the surface.

But consider a torus instead, and look at a point on the innermost circle - inside the hole, so to speak. There's no way to pass a plane through that point that "just touches" the torus; any plane will slice into it. So how do we construct the tangent plane?

Let's try something else. We can draw (smooth) curves on the surface, running through the given point. Each of those curves will have a tangent line at that point, and those tangent lines together sweep out a plane. That's the tangent plane.

We don't really need to look at all of those curves; it's enough to pick two, running in different directions. (Two lines through a point determine a plane.) So, for our torus, it works like this. Think of it as lying horizontally. There's a vertical circle through the point, with its center in the interior of the torus (not the hole; where the doughnut itself would be). There's also a horizontal circle, with its center in the hole. The tangents to those circles at the given point mark out the tangent plane.

Similarly with the cylinder: again, think of it as standing upright. There's a vertical line through the point (and that line is its own tangent line), and there's a horizontal circle as well, providing you with a second tangent line. (The tangent plane, in this case, runs parallel to the central axis of the cylinder, and touches the cylinder all along that vertical line.)

There's one last construction I want to mention. A "normal vector" at a point of a surface is a length-one vector perpendicular to the tangent plane. At any given point, there are exactly two normal vectors, pointing in opposite directions. (For the sphere, for example, there's one pointing in towards the center, and another pointing outward.) If you choose a small neighborhood of a point, you can assign a normal vector to every point in the neighborhood, in a continuous way. (For the sphere, pick the outward vector every time, or the inward vector every time.) For some surfaces - the sphere, the cylinder, the torus - you can pick a normal vector at every point, in a continuous way. You can't do that with the Möbius strip. (Think about it. That's what it means to be "one-sided"; picking a normal vector amounts to choosing a side.) Normal vectors will be important in the next bit; fortunately, it'll be enough to assign them "locally" - i.e., in the neighborhood of the point we're examining.

This post has been rather dry, I'm afraid, but the payoff, in the next post, will be worth it, I think. Bear with me.