The Birth of Metrics

In 1851, Bernhard Riemann got his PhD in mathematics. In the German university system at the time, in order to become a professor1, 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 $\mathbb{R}^n$. And, finally, he came up with a way of measuring lengths on these manifolds, without having to refer to $\mathbb{R}^n$.

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, $\mathbb{R}^2$. Suppose we want to measure the distance between two points, like these:

In this case, we can just use the Pythagorean Theorem, $a^2 + b^2 = c^2$. Here, $a$ is the horizontal distance and $b$ is the vertical distance between the points.

Now, that’s fine and dandy for $\mathbb{R}^2$, 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 $s(t)$. The input, $t$, is the time since leaving your house, while the output, $s(t)$ 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 $s(t)$, which we’ll write as $s'(t)$ (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 $s'(t)$) from $t=0$ to $t=10$ 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 $\int_0^{10} s'(t) dt$, 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, $\mathbb{R}^2$, 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 $(x(t), y(t))$. So, $x(t)$ gives her position in the $x$ direction $t$ seconds after beginning, and similarly for $y(t)$.

With a derivative, it’s easy to say how much her position is changing. The function $x'(t)$, by definition, is the rate of change of her position in the $x$ direction, and similarly for $y'(t)$. If we put these together as $(x'(t),y'(t))$, 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 $\mathbb{R}^2$. Fortunately, that’s as simple as using the Pythagorean theorem. Since the tangent vector is $x'$ wide and $y'$ tall (i.e., represented by the vector $(x',y')$), the Pythagorean theorem says that the length $s'$ of the vector (and thus the speed) is given by $s'^2 = x'^2 + y'^2$, and so $s' = \sqrt{x'^2 + y'^2}$.

Let’s write that rule a bit differently. Let’s write that as $ds^2 = dx^2 + dy^2$. You can think of $dx$ as the derivative of the $x$ position, while $ds$ is the derivative of the position, or, in other words, the speed.

This way of measuring vectors, $ds^2 = dx^2 + dy^2$, 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 $\mathbb{R}^3$) 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 $ds^2 = dx^2 + dy^2 + dz^2$. The term $dz$ is the same as the others, just for the extra $z$ direction.

How does this help us with manifolds?

Let’s look at a specific case first. Think of the sphere $\mathbb{S}^2$ sitting in space $\mathbb{R}^3$. 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 $\theta$ from some fixed north-south line.5 Latitude is measured as an angle $\phi$ from the north pole.6

A path on the surface of the sphere in $\mathbb{R}^3$ could be written like $(x(t), y(t), z(t))$, but we could also write it in terms of latitude and longitude, as $(\theta(t), \phi(t))$. It’s the same path, just written in terms of the coordinates for the sphere.

Similarly, instead of writing the tangent vector as $(x'(t), y'(t), z'(t))$, we can write it in terms of the new coordinates, $(\theta'(t), \phi'(t))$. 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 footnote7, we find that the metric on a sphere of radius 1 is $ds^2 = \sin^2(\phi) d\theta^2 + d\phi^2$. In this, $d\phi$ represents the change in the $\phi$ coordinate (i.e., $\phi'(t)$), while $\phi$ represents the value of the $\phi$ coordinate (i.e., $\phi(t)$).

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 $\mathbb{R}^3$. 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 $(x,y,z)$. Then we can write down a metric using those coordinates. It might be ugly, like $ds^2 = dx^2 +\cos^4(z) dydx + dy^2 +e^{xy}dz^2$, 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.

1. 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.
2. 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.
3. 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.
4. 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.
5. On Earth, this line is the north-south line through Greenwich, England.
6. 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 $\phi$ and $\theta$. We’ll use the mathematical names.
7. This calculation is actually fairly simple. The coordinates are related by, for instance, $z = \cos(\phi)$. By taking the (exterior) derivative of both sides, we see that $dz = -\sin(\phi) d\phi$. You do this for each coordinate, then substitute what you get in $ds^2 = dx^2 + dy^2 + dz^2$ and simplify what you get.
8. 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.
9. 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.

Manifold Menagerie

In the last post, we introduced the idea of manifolds. A two-dimensional manifold is essentially a shape on which we can go in two directions, say up-down and left-right, but not forward-backward as well. We talked about some simple examples, like a sphere or a torus, but there are so many more wonderful manifolds than that!

Let me give you a brief tour of the manifold menagerie.

The first manifold I want to show you is an interesting variation of the torus. Last time, we showed how you can make a torus by considering a rectangle, and identifying opposite sides.

Now, notice how the arrows are pointing in the same direction. What would happen if we changed the direction of one of the arrows? You’d get what we call a Klein bottle.1

What this means is that the left edge is still identified with the right edge, but with, for instance, points near the bottom left being identified with points near the top right edge.

Another way to visualize this is that if you played Asteroids on a Klein bottle, and you flew through the left side facing up, you would come out the right side, facing down!

What does this Klein bottle thing even look like, anyway?

Well, first we can roll up the rectangle, like we did with the torus. But we can’t bend the two ends together quite like before because the orientation (i.e., the direction of the arrows) doesn’t match up. Instead, we have to do something funny.

Take the end of the tube through the wall near the other end, then bend it up through the end. Then, flange over that end to match up the two ends. Okay… that description needs a picture.

Weird looking, right?

Now, you may want to raise an objection. Dr. Dilts, you may say, isn’t that not a manifold? After all, where the tube intersects itself, it doesn’t look like $\mathbb{R}^2$!

You’d be right to mention that. In fact, that’s one of the interesting things about this example.

If you look back at the rectangle representation of the Klein bottle, there’s nothing that stops it from being a manifold. There’s no self-intersection there. The problem is not in the Klein bottle itself, but in how we chose to visualize it in three-dimensional space, $\mathbb{R}^3$!

As an anology, consider a circle.2 We can think of it as a loop of string. In that standard configuration, it looks like a manifold, since if you zoom in, it looks like a line, i.e., a one-dimensional space. But we could pick up the end and pull it over itself like such:

It no longer looks like a manifold, but, again if we think of it as a string, the places where the loop overlaps itself are not the same points; they just look like it because they overlap. The crossing lines don’t “see” each other.

Again, the problem is not with our loop of string, which certainly is a manifold. The problem is with how we are visualizing it in the plane, $\mathbb{R}^2$.

So, the Klein bottle is a manifold.

This raises a question. Can we visualize the Klein bottle so that it doesn’t intersect itself?3

The answer is yes, but, surprisingly, you can’t do it in $\mathbb{R}^3$. You need a fourth dimension to do it!

The idea is something like this.

If we go back to the loop of string example, where the string overlaps itself, the visualization intersects itself. However, it’s easy to resolve this problem. Simply pick up the string near where the intersection is; in other words, pull it from two dimensions into the third.

In order to visualize the Klein bottle without an intersection, we have to do the same thing, only with a fourth dimension. You grab the Klein bottle near the intersection, and pull it from three dimensions into the fourth.

Weird, huh?

Now, this worked for the Klein bottle, this trick of pulling it into a higher dimension. But you can imagine there are other manifolds that have this same problem of self-intersection. (In fact, we’ll even get to another one in this post!)

If you pick any manifold, is there always a visualization that doesn’t have self-intersections?

The answer is yes, though it’s hardly obvious how to prove such a statement. The Whitney embedding theorem says that any manifold can be visualized without self-intersection, but it may require a high dimension to do so. Precisely, it may take as many as twice as many dimensions as the manifold has.

If the manifold has $n$ dimensions, this visualization may require as high as $2n$ dimensions. Our Klein bottle (with $n=2$) is an example of this, as it requires $4$ dimensions. A six-dimensional manifold might take twelve dimensions!

Okay, back to the funky Klein bottle thing.

Guess what? It doesn’t have an inside or an outside! Weird, yes? This fun fact about the Klein bottle is exactly like how a Möbius strip has no front or back.4

With a sphere, if you think of yourself walking on the outside surface, you can’t “walk” to the inside of the surface; you’d have to punch a hole through. But look closely at the Klein bottle, if you started walking on the “outside,” and followed along just like you know you should, you can end up underneath where you began.

Trippy.

Okay, the Klein bottle is weird.5

Let’s talk about some more pedestrian6 examples.

In the last post, we talked about the sphere, by which I mean a shape like the surface of the Earth. We call this the two-dimensional sphere, $\mathbb{S}^2$ (read “S 2”), since there are two directions you can go.

Along with that, you have the one-dimensional sphere, $\mathbb{S}^1$. This has one direction you can go… in other words, it’s a circle!

We can go higher in dimensions as well. What is $\mathbb{S}^3$, the three-dimensional sphere?

Well, it’s not the inside of the sphere, which is what your mind might go to first. Instead, it will help to use analogy.

If you look at $\mathbb{S}^2$, horizontal slices of it are just circles, $\mathbb{S}^1$.

So, as we take slices of $\mathbb{S}^2$, we have small circles that get larger till we pass the equator, then get smaller again till they disappear.

We can visualize $\mathbb{S}^3$ the same way. The problem is, it’s naturally in $\mathbb{R}^4$ instead of $\mathbb{R}^3$, so it’s harder to see. To help us, let’s think of $\mathbb{R}^4$ as the three normal dimensions, plus time.

So, if we take $\mathbb{S}^3$, and we look at what we can see at any instant in time (which is equivalent to looking at horizontal slices in the previous example), we can see a sphere, $\mathbb{S}^2$, which starts small, grows till it reaches the “equator,” then shrinks back down till it disappears.

Yeah, four dimensions is hard.

MORE manifolds!

The torus we talked about before is just one example of a torus. Since it’s a two-dimensional manifold, we call it $\mathbb{T}^2$ (read “T 2”). Using the same idea, it’s not hard to make higher dimensional versions.

If we take a cube (instead of a rectangle), we can identify opposite sides of the cube. So, if you are playing (3-d!) Asteroids in this, and you fly out the top, you’d fly in from the bottom. This gives us a new, higher dimensional torus, $\mathbb{T}^3$.

Pretty cool. Of course, it’s much harder to visualize as sitting inside of $\mathbb{R}^4$. Good thing we don’t have to!

There’s another variation we can do.

Instead of taking a square, take an octagon. Now, identify the sides in this pattern.

Now, it’s much harder to see what’s going on without a hint. What is this shape?

It’s a two-holed torus.

You can actually do this for any number of holes. You just need 4 sides for each hole you want your torus to have.

Let’s see, what else weird can we do?

Oh, I know!

Okay. Take the sphere ($\mathbb{S}^2$). Now, for each point, identify it with the point on the exact opposite end of the sphere.

Another way to think of this is to think just of the upper hemisphere of the sphere. Then, just on the equator, again identify the points on the opposite side.

In other words, if you fly south, and go past the equator, you end up coming north out of the equator on the other side!

At first consideration, this really seems to just be a sphere, since on a sphere, you can fly through the south pole just the same.  But it really is different. It’s called $\mathbb{RP}^2$ (read “R P 2”), where the $\mathbb{RP}$ stand for “real projective” space.

And it’s really weird.

To explain it, we’re going to need one of those funny mathematical objects; an infinitely stretchable and infinitely compressible rubber band.

Look at the sphere. If you put a rubber band on it through the north and south poles, you can always have the rubber band shrink down, like this:

However, if we took the equivalent path on $\mathbb{RP}^2$, it can’t shrink down! See, if you try to move the parts that hit the equator closer to each other, you can’t, since they, by the definition of the space, have to stay on the opposite sides of the hemisphere! Essentially, you can only rotate the rubber band, but never have it come off.

But, if we wrap the rubber band around twice, all of a sudden we can shrink it down.

Wait, what?

The trick is to run one part down each side of the hemisphere. Thus we can have two small half circles getting smaller, and finally shrink to nothing like on a sphere.

So, in $\mathbb{RP}^2$, one closed path is stuck. But if you wrap it around twice, all of a sudden you can shrink it down to nothing!7

So this new space is not a sphere!

Again, we call this space $\mathbb{RP}^2$. As you can see, it turns out to be about as interesting as the Klein bottle. In fact, we can even use a similar rectangle to define it, like this!

Like the Klein bottle, $\mathbb{RP}^2$ cannot be visualized in three dimensions without self-intersections. Here’s one way to look at it, with those opposite parts of the equator matched up.

$\mathbb{RP}^2$ also doesn’t have an inside or an outside. Which is really weird, since it seems like if you take a sphere, anything you get from it should still have an inside… But, you know… strange but true statements like this are the heart of mathematics.

There are plenty of other weird manifolds out there, but that’s a good introduction to some interesting and important ones.

Next, we want to come up with a way to measure distances on a manifold, without visualizing it in a space like $\mathbb{R}^3$.

1. Apparently, it was originally called Kleinsche Fläche (Klein surface), but, perhaps due to its appearance, it was misinterpreted as Kleinsche Flasche (Klein bottle), which may have led to the use of this term in German as well.
2. Remember, a circle is the outer edge, not the filled-in shape.
3. A visualization of a manifold (in $\mathbb{R}^n$, for instance, and with a few other conditions) is called an immersion. If the immersion doesn’t intersect itself, it’s called an embedding. So, our current visualization of the Klein bottle is an immersion, and the question is whether we can find an embedding.
4. In fact, a Klein bottle is two Möbius strips with their edges glued together.
5. Klein bottles are so cool that you can get blown-glass Klein bottles. The man who runs that site, Cliff Stoll, looks like a mad scientist. And he’s obsessed… Check out this interview of Cliff Stoll on Numberphile. He also shows off his Klein bottle hats and Möbius scarfs!
6. Pun! Ba-dum-ching!
7. This idea of drawing paths and asking whether or not they can be shrunk to nothing is an important one in topology, called the fundamental group of the manifold. I was about to link to the Wikipedia page, but on second glance it’d be a horrible introduction…

Asteroids on a Donut

Have you ever played the game Asteroids?

In this game, you fly a little triangular spaceship and try to  destroy all of the asteroids floating around you while avoiding collision.1 One interesting feature of this game is that if you fly through the right side of the screen, you come out the left side. If you fly through the top of the screen, you come out the bottom.

In other words, the game takes place on the surface of a donut.

This visualization is hard the first time you think of it, so let’s work through this.

We can think of the game area as the surface of a sheet of paper. However, the top edge and the bottom edge have been identified2. We’ll draw arrows on the sides to help us remember which edges have been identified.

Since the top and bottom are the same, we could take this piece of paper and roll it up to match up those edges. This leaves us with a tube of paper.

The two ends of the tube are the original left and right end of the screen, and so they are identified. Unfortunately, paper isn’t stretchy, but if you could bend it again, we could bring those two ends together to form the surface of a donut!

A mathematician would call the shape a torus.

When we are looking at the shape like this, we are thinking of a two-dimensional torus sitting inside of a three-dimensional space. But looking back at the game, there’s nothing about it that requires three dimensions. Even if the torus were sitting in three-dimensional space, the spaceship and the asteroids would have no way of realizing it.3 They can go up and down, left and right, but they have no way of interacting with the third dimension we visualized them in.

This turns out to be a key observation, and the basis for the mathematical idea of a manifold.

A manifold (with, say, 2 dimensions) is a shape where you can always, and only, go in two directions and the directions between those. You can go, say, up and down and left and right, as well as diagonally, but there is not even a conception of forward and backward through the two-dimensional screen. This is exactly what the game area for Asteroids looks like.

Another way to say this is that a 2-D manifold is a shape that locally looks like the standard 2-D space: a plane, like a piece of paper. So, in Asteroids, if we zoom in on an area just around where the ship is, the area around the ship also looks like a small piece of everyday two-dimensional space, the plane.

Yet another way to say this is that a manifold is a shape that locally can be described using as many coordinates as the manifold has dimensions. So, in Asteroids, the coordinates could be the $(x,y)$ coordinates on the screen. And a 2 by 2 matrix is a four-dimensional manifold, since there are four numbers.

These three different ideas of a manifold all mean the same thing, though it may take some thinking to see why. The technical definition, while important when actually working with these things, doesn’t really add much to our intuition, and so we won’t worry about it here.4

The idea of purposely not thinking of these shapes inside of some high-dimensional space is surprisingly important and useful. (For example, thinking of our donut-torus as a two-dimensional surface rather than in three-dimensions or above.) But, before we get into that, let’s go through a few more examples of shapes that are (or are not) manifolds.

The most basic example is just n-dimensional space itself. You look closely at it and, unsurprisingly, it still looks like n-dimensional space. (n here is a number like 2 or 3 or 6.) Because this space is so important, I’ll finally give it it’s usual name, $\mathbb{R}^n$ (read “R n.”) This is the prototypical manifold, and the model on which the rest are built.

Another example is the surface of a sphere. As on the surface of the Earth, there are two coordinates: latitude and longitude. We have two directions we can go: north/south and east/west. If you look closely, or, in other words, if the sphere is large enough, it looks like $\mathbb{R}^2$.

A 1-dimensional manifold has only one direction to travel, and so it has to (locally) look like a line. It turns out that the only examples of this are either a line or a circle5.

Let’s make sure we understand the term circle. If you’re imagining this from your Sesame Street days:

Sorry, to break it to you, but a mathematician would not call this a circle. A mathematician would call that a disc… or a two-dimensional ball. You know, ‘cuz it’s filled in.

Now THIS is what a mathematician calls a circle. Just a line closed in on itself. NOT filled in. The space inside has nothing to do with the circle.

Of course, they might also call this a one-dimensional sphere…

Anyway.

What about an infinity figure? Is that a one-dimensional manifold?

If you focus at most points of this infinity, it looks like a line. So far so good.

However, if you focus on the point where the two parts cross, the shape instead looks like a cross, which is a problem if you were hoping to call it a manifold. At that point, there two directions you can go, but you can’t go in directions between those two directions. So it isn’t quite one-dimensional there, nor is it quite two-dimensional…

The infinity figure is not a manifold.

Most simple examples of shapes that aren’t quite manifolds are of this kind; most of the shape is like a manifold, but there are some places where these parts of manifolds come together is crosses. Another example would be two spheres just barely touching.

Now, back to the question: Why do we try so hard to think of manifolds as not sitting inside of some space like $\mathbb{R}^n$?

There are two main reasons, as I see it. First, it lets us understand some objects that aren’t obviously “shapes” in a new way. For instance, the set of 2 by 2 matrices with determinant one6 (sometimes called $SL(2,\mathbb{R})$) turns out to be a three-dimensional manifold, not a four dimensional one, since the matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ can be written as $\begin{bmatrix}a&b\\c&\frac{1+bc}{a}\end{bmatrix}$ and so only has three coordinates. If we can understand manifolds better, it will tell us something about matrices!

The second main reason for not wanting to think of manifolds in, say, three-dimensional space, is that how you draw your shape implies things about your manifold that are not intrinsic to the manifold.

For instance, look at a flat sheet of paper. If I were a two-dimensional being living on that paper, and you picked up the paper and bent it a bit, nothing that I could perceive from the two dimensional paper would let me know you bent it.

By thinking of the paper in three dimensional space, we can see the curving of the paper. And there are lots of cases where we might care about that, but we are also interested in what we can say about a shape independent of how we put it into $\mathbb{R}^n$.7

Another piece of information we actually want to ignore that would be forced on us if we put our shapes in $\mathbb{R}^n$ is distance (or length). As far as the definition of manifold is concerned, a circle (circle, not a disc!) or a horseshoe shape should be the same. And ignoring lengths, they are. (If you have trouble accepting that, just pretend these are made of yarn. You could easily rearrange the horseshoe to be a circle.)

It might seem that there isn’t much we can tell about a manifold without distances, but it turns out there is a lot that can be said. For instance, if you draw a circle (again, circle, not a disc!) on the surface of a sphere, there is a clearly defined inside and outside.

But if you draw the right circle on a torus, there is no inside or outside.

The study of shapes ignoring things like distances and angles is a whole field of study, called (manifold) topology.

But we like distances. So, in a coming post, we’ll talk about a way to define distances on manifolds, without thinking of the shapes in $\mathbb{R}^n$.

First, though, to satisfy my love of manifolds, we’ll have to have a post exhibiting a menagerie of weird and exotic manifolds.

1. The odds of success are approximately three thousand seven hundred and twenty to one!
2. Identified is the technical term, meaning we treat a point on the top edge as the same point as the corresponding point on the bottom edge.
3. Unless, they could “look up,” if you will, and see into the other dimension.
4. The technical definition is that a manifold is a “topological space,” which means that you can say what “close” means, that is locally Euclidean (as per our second intuitive idea), second countable and (usually) Hausdorff. Second countable and Hausdorff are technical conditions that don’t come up that often. For instance, Hausdorff means that two separate points can’t be “infinitely close,” which only happens in kind of weird situations anyway.
5. The technical term for this is simple closed path. The simple means it doesn’t cross itself.
6. In case you aren’t familiar or have forgotten, a matrix is just a rectangle of numbers, in this case 2 by 2, so 4 total numbers. The determinant of a matrix is calculated from these numbers and tells you something important about the matrix (if it is invertible), though what the determinant is precisely doesn’t matter here.
7. The technical term for how we think of the shape inside of $\mathbb{R}^n$ is the embedding of the shape. You can think of embedding as a way to draw a shape into space such that the shape doesn’t intersect itself. An immersion is drawing of the shape that lets it intersect itself. So, an immersion of a circle could be the infinity sign.