In 1851, Bernhard Riemann got his PhD in mathematics. In the German university system at the time, in order to become a professor^{1}, you had write an extra thesis, which Riemann did on what we would now call Fourier series. In addition, you had to give a probationary first lecture.
To make it worse, you didn’t even get to choose the topic of your talk. You were to propose three different subjects, and then the faculty, which at that time was headed by Friedrich Gauss, chose between them. Two of the subjects Riemann proposed were ones he had already worked extensively in. The third was “On the foundations of geometry.”
Unfortunately for Riemann, Gauss was a troll. Against all tradition, he picked the third topic.
Fortunately for mathematicians, Riemann was a mathematical boss.
Of course, Riemann was also researching in mathematical physics and working as what we would probably call a TA (teaching assistant). The strain of another major research project on top of all that led to a temporary mental breakdown. Fortunately, he quickly recovered, and a few months later, gave his talk.
In his probationary lecture, Riemann invented the idea of a manifold, as we talked about in previous posts. He also introduced the important idea of removing manifolds from some space like . And, finally, he came up with a way of measuring lengths on these manifolds, without having to refer to .
That way of measuring distances is the topic of this post.
We’re going to slowly build up ideas in order understand Riemann’s idea.
How do we measure lengths?
Let’s start in the plane, . Suppose we want to measure the distance between two points, like these:
In this case, we can just use the Pythagorean Theorem, . Here, is the horizontal distance and is the vertical distance between the points.
Now, that’s fine and dandy for , but it doesn’t really work on other shapes. After all, what’s a “straight line” on a torus?^{2}
We need a way to measure the length of a path between the two points, not just the straight one.
How can we do that?
The key idea actually comes from a car. When we drive, our speedometer tells us our speed. If we traveled at the same speed for an entire trip, the length of our travel would just be our speed times the time we traveled. Now, of course you don’t travel at a constant speed. Sometimes you go slower, sometimes faster. But our odometer figures out our total distance traveled just the same.
One way to think about what the odometer is doing is it is adding up the distance traveled for each second. If you’re going faster for a second, the odometer ticks up a bit more; if slower, a bit less. At any given second, you’re almost going a constant speed, and so this is a good approximation of your distance.
Now, of course, your speed isn’t constant during that second, but the odometer is essentially adding up the distances you travel over each infinitesimal interval of time.
Not to scare you, but it will be convenient to introduce some calculus here.
Don’t worry, we won’t actually do any calculations. We just need the basic ideas of the derivative and the integral of a function. We’ll illustrate with a basic example.
A function takes some input and gives you some output. For our example, let’s look at the function . The input, , is the time since leaving your house, while the output, measures the distance you are away from your house.
The derivative of a function measures the rate of change of the output. So, the derivative of , which we’ll write as (read “s prime of t”), is the rate of change of position, i.e., the speed.
The integral is essentially the opposite of the derivative. The integral of speed (i.e., of ) from to adds up the rates of change in order to calculate the total change in position from time 0 to time 10. The notation here is , but we shouldn’t need to use that here.
While we won’t be worrying about formulas or calculating these things, it is important to note that, given a function, it’s a relatively simple thing to find derivatives and integrals of it.^{3}
In terms of our car description, the speedometer measures the derivative of your position, which is your speed, while the odometer integrates your speed in order to calculate your total distance traveled.
Riemann’s idea is to measure lengths the same way.
How can we calculate speed along a path?
If we have a path in, for instance, , we can think of a person driving along that path. Her current position is given by a point on the path, which we can label as . So, gives her position in the direction seconds after beginning, and similarly for .
With a derivative, it’s easy to say how much her position is changing. The function , by definition, is the rate of change of her position in the direction, and similarly for . If we put these together as , this tangent vector tells us what direction the driver is going, and its length tells us how quickly her position is changing.
To calculate speed, we need to calculate the length of a vector in . Fortunately, that’s as simple as using the Pythagorean theorem. Since the tangent vector is wide and tall (i.e., represented by the vector ), the Pythagorean theorem says that the length of the vector (and thus the speed) is given by , and so .
Let’s write that rule a bit differently. Let’s write that as . You can think of as the derivative of the position, while is the derivative of the position, or, in other words, the speed.
This way of measuring vectors, , is called a metric.^{4}
So, to review, to find the length of a path, we first find its tangent vector (via a derivative). We calculate the length of these vectors using the metric, which tells us the speed. Then we add up the speeds (via an integral) to get the length of the path.
If we had a path in space (i.e., in ) instead, the only difference is that we have three directions (x, y and z) instead of two (x and y). We can find the length of the tangent vector in the same way, only using . The term is the same as the others, just for the extra direction.
How does this help us with manifolds?
Let’s look at a specific case first. Think of the sphere sitting in space . The first thing to notice is that a path on the sphere is also a path in space!
This means that we can use the metric in space to find the lengths of the tangent vectors, and thus the speed. The integration works the same for any path, so that gives us what we need.
But we want to be able to define distances without thinking of the manifold in space.
To get around that, recall that the definition of a manifold tells us that, for a 2-dimensional manifold, we always have two coordinates. For the sphere, we can choose the two coordinates to be latitude and longitude. Longitude is measured as an angle from some fixed north-south line.^{5} Latitude is measured as an angle from the north pole.^{6}
A path on the surface of the sphere in could be written like , but we could also write it in terms of latitude and longitude, as . It’s the same path, just written in terms of the coordinates for the sphere.
Similarly, instead of writing the tangent vector as , we can write it in terms of the new coordinates, . Again, these are the same vectors, just with different names, if you will.
And if they’re the same vectors, they should have the same lengths. So, we want to figure out what the metric should look like in terms of the new coordinates. After a calculation I’ll explain in a footnote^{7}, we find that the metric on a sphere of radius 1 is . In this, represents the change in the coordinate (i.e., ), while represents the value of the coordinate (i.e., ).
This is the same metric as before, but it looks very different because we’re using different coordinates. But the same vector will have the same length.
The important thing to note here is that this new form of the metric makes no reference to . It only refers to the coordinates on the manifold!
This immediately gives us how to measure lengths on any manifold.
On any manifold, we can choose some coordinates, say . Then we can write down a metric using those coordinates. It might be ugly, like , but that’s okay. There are a few things we generally want to worry about (like “smoothness”), but we won’t here. We can then use that metric to measure the lengths of tangent vectors, then integrate those lengths (i.e., speeds) to find the total length of the path.
Yay! We now understand Riemann’s idea for measuring lengths on a manifold!
How did Gauss react to Riemann’s lecture? Recall that Gauss was the foremost mathematician of his time. In 1827, he had proven a theorem about geometry so awesome that, to this day, it is officially called “The Awesome Theorem.”^{8} Gauss’s influence would decide whether Riemann got the job.
Riemann blew Gauss away.
Dedekind, who later replaced Gauss as head of the faculty, said that Gauss sat at the lecture “which surpassed all his expectations, in the greatest astonishment, and on the way back from the faculty meeting he spoke to Wilhelm Weber, with the greatest appreciation, and with an excitement rare for him, about the depth of the ideas presented by Riemann.”^{9}
Needless to say, Riemann got the job.
There is a lot more we could talk about metrics and manifolds. And we will. After all, I do want to tell you about “The Awesome Theorem.” But, before that, I want to talk about the mathematics of something important that Riemann’s ideas allowed to be invented: Einstein’s theory of special relativity.
- The position (Privatdocent) is a bit different than how we think of a professor. You did not have a regular salary, but were simply forwarded fees paid by any students who chose to attend your lectures. Riemann did eventually get a regular salary, but not for a few years. ↩
- There definitely is an idea of a straight line, called a geodesic, and they’re super important. We’ll talk about them at some point, but not in this post. ↩
- Well, at least, mathematically, it’s simple to say that the derivative or integral exists, though writing out an explicit formula is sometimes hard or impossible. Fortunately, that doesn’t stop us from using the concepts. ↩
- There is another concept called a metric space. A manifold with a metric is a metric space, though a metric space is more general. The “metric” on a metric space defines the distances between points, rather than giving a way of measuring speed. Clearly related, but not quite the same. ↩
- On Earth, this line is the north-south line through Greenwich, England. ↩
- Latitudes on Earth usually measure from the equator, rather than the north pole, but mathematicians will do as they want. Also, following a master plan to confuse everyone, physicists reverse the names of these coordinates, switching and . We’ll use the mathematical names. ↩
- This calculation is actually fairly simple. The coordinates are related by, for instance, . By taking the (exterior) derivative of both sides, we see that . You do this for each coordinate, then substitute what you get in and simplify what you get. ↩
- Okay, technically, it’s called “Theorema Egregium.” But that’s Latin for “The Totally Awesome Theorem,” so I stand by my claim. We’ll definitely be talking about this theorem when we eventually talk about curvature. ↩
- My story about Riemann comes from Spivak’s Introduction to Differential Geometry, Volume 2, which includes the translated lecture in chapter 4. Though probably only a few in the audience were able to appreciate the lecture at the time, it was the pivot point in the history of geometry. ↩