Riemannian Geometry

We have described what we are looking at topologically, but we are also interested in geometry. Riemannian geometry is one way of looking at distances on manifolds. This seems an easy enough concept when you first think of it, but after further though we realize it is not so easy. Sure we know how to measure distances on a plane. The shortest distance between two points is a straight line, right? So just draw the line and measure the distance (first we set what unit measure is, for instance 1 meter, and then compare the distance we want to measure to our set standard unit distance, say the meter stick). But on a sphere how do we measure distance? Or on a torus (the surface of a doughnut)? A sphere we can think of as living in Euclidean 3-space and then just say that the distance between 2 points on the sphere is just the distance between those points in 3-space, as described by the Pythagorean theorem, which says that for points $(x_1,y_1)$ and $(x_2,y_2)$ the distance between them is

$$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$

which is the fact that the length of the hypotenuse (longest side) of a right triangle is equal the square root of the sum of the squares of the other two sides.

In addition, we can change this slightly by deforming the sphere in 3-space so that it is still topologically a sphere but not geometrically a perfect sphere. This is a perfectly legitimate way to define distances. It has one disadvantage, however. It requires that the sphere be "embedded" in 3-space. In other words, we can think of the sphere as living in 3-space, but maybe our geometric sphere cannot live in 3-space. This concept is a little weird in the case of the sphere, so let's look at this problem one dimension down. Consider the circle. Now, we can think of the circle as living in Euclidean 2-space (the plane). Now I claim that we can find topological circles that we cannot put in Euclidean 2-space! These are called knots and they are exactly what you consider to be knots. Take a circle, then disconnect it, then tie a knot in it, then reattach the ends. We now still have a topological circle. If you lived on the circle, your couldn't tell the difference than if there was no knot tied in it. But we can't crunch the knot down so it lives in the plane. Thus we need to have the circle live in 3-space and we are fine. So we need to first know which sized (dimension) Euclidean space we live in to use this method. Of course, there is no reason to believe that any manifold lives in some appropriately sized Euclidean space, although this fact is true (but hard to show). We are going to skirt these issues by taking another approach to distances. Suppose we could measure the length of curves. Then we could simply define the distance between two points to be the length of the shortest path, if one exists. Even if one does not exist, we could express the distance as the largest lower bound for lengths of paths between the two points (this is called the infimum). This isn't too important so let's assume that we can find a shortest path. So we could measure distances if we could measure the lengths of paths. This is where a Riemannian metric comes in. So if you are driving, how would you measure the distance you traveled? Well, one way is to look at your speedometer and remember how fast you are going at every point and then consider how long it took you to get where you were going. Since speed is just distance per unit time (like miles per hour) we just multiply the speed times the time and we get the distance. That's how far we traveled. The only problem is that our speed changes and our direction changes, so we actually need to take into account which direction we are traveling (this means using the velocity vector instead of just the speed, which is the length of the velocity, ignoring direction). Also, our formula only works if the velocity does not change. But our velocity changes. What do we do? We chop our time intervals up into smaller segments which have constant velocity. This doesn't quite work because there is no time when we are going at a constant velocity (most likely), but as we chop up the time interval more and more, our approximation is more and more accurate. In the limit, we end up with an integral and can calculate the actual length by computing that integral (this involves some basic calculus). For culture, let's look at how we would write this:

$$l(t) = \int_0^t \vert c '(s) \vert ds$$

which we read as the length $l(t)$ of the curve (path) $c$ is the sum (the integral sign $\int$ is a German 's' standing for sum) of the speed of the curve, denoted $\vert c '(s)\vert$, multiplied by a really small length of time, denoted $ds$. The the term $l(t)$ just means we measure the path from time 0 to time $t$. Anyway, the what we get out of this discussion is that only need to know the velocity at every point. The velocity at every point can be considered as a vector (arrow) indicating where we are going (direction) and at what speed. It is tangent to our path because it tells us which direction to move. And it is only defined at each point. We just need to be able to measure these velocities and then we can compute lengths of curves. Thus we need to be able to measure the length of vectors at every point. This is what a Riemannian metric does. It provides us with a way of measuring vectors at every point. Since these are local things and we understand what vectors at different points in Euclidean space are, we can do the same things on manifolds, since locally they look like Euclidean space. So the Riemannian metric is a function defined at every point that takes two vectors and gives a number. Technically, a Riemannian metric must be a symmetric bilinear form (otherwise it is called a Finsler metric). This condition isn't too important for us to understand at this point, but it is easy to state so why don't we do it anyway. We express the Riemannian metric as $g$ and we express vectors at a point $p$ in our manifold as $V_p,W_p,U_p$. then we can measure quantities like $g(V_p,W_p)$. The fact that is symmetric means:

$$g(V_p,W_p) = g(W_p,V_p)$$

and bilinearity (when coupled with symmetry) is the following condition:

$$g(V_p + W_p,U_p) = g(V_p,U_p) + g(W_p,U_p)$$

But you probably don't have to understand this too much. I just get carried away sometimes. So the important thing is that the Riemannian metric gives us a way to measure lengths of vectors at each point in the manifold, and also gives us a way of measuring lengths on the manifold.