The Gauss-Bonnet Theorem

Topology is the study of shapes and, in particular, what doesn’t change when you bend and squish them.

This is kind of weird, since you wouldn’t normally think of a donut and a coffee mug as the same shape. But to a topologist, they are the same.

Again, these are the same “shape.” To get from one to the other, we didn’t have to cut or break or glue the shape, just kind of squish it.1

Another way to say this is that topologists don’t care about distances, areas or angles, but are looking for some intrinsic “shape-ness” of a shape. Though topologists study more shapes than just manifolds, in the case of manifolds, this means they study what you can say about a manifold without a metric.

Now, at first, it may seem that there isn’t all that much you can say about a shape without distances or areas, but it turns out there’s quite a lot.

For instance, does the manifold have an inside and an outside? The Klein bottle was an example of a shape without an inside or an outside. This is called orientability. A sphere is orientable; the Klein bottle is nonorientable.

kleinwalk

Another example is the number of holes. Both a donut and a mug have one hole. The number of holes a surface has is called its genus,2 so both are genus 1.

The last one I’ll mention here is more complicated, but is vital for this post. It is the Euler characteristic of a surface.

The Euler characteristic only makes sense for 2-manifolds, i.e., surfaces like the surface of a donut or a sphere. In order to calculate it, first you have to cut up the surface into a bunch of polygons.3 For instance, we can cut up the sphere into its octants:

octants

Or, we can take the torus (surface of a donut) and cut it out into a single rectangle. (This is the reverse of how we constructed the torus back in the Asteroids post.)

torus

Then, we count vertices (i.e., corners), edges and faces (i.e., polygons.) The way we cut up the sphere has 6 vertices, 12 edges and 8 faces. The torus had 1 vertex, 2 edges and 1 face.

The Euler characteristic is then defined to be \chi = V-E+F. So, the Euler characteristic for the sphere is 6-12+8=2, while the Euler characteristic for the torus is zero.

It’s kind of weird that this doesn’t depend on how you choose your polygons, but it is true. For instance, we could have split the sphere into two “squares,” which meet at the equator.4

squares

Now, we have 4 vertices, 4 edges and 2 faces. But \chi is still 2! No matter how we cut up the surface, the Euler characteristic will be the same.5

Something that’s easier to see is that it also won’t change if we stretch the surface. After all, without a metric to tell us what straight lines (i.e., geodesics) are, there’s nothing stopping us from using squiggles for our polygon edges.

squiggles

Similarly, squishing our torus into a coffee cup isn’t going to change the connectedness we measured with the Euler characteristic.

We’ve now gone through all this trouble to talk about topology, and how there are wonderful things you can measure without a metric. It may seem odd, but we’re going to go back to metrics and curvature.

See, it turns out that topology has a deep and strong relationship with curvature.

This seems weird. I mean, if you take a torus and stretch it into a coffee cup, obviously the metric and curvature are changing a lot. But it turns out that, despite all that, topology and curvature are linked.

This is the heart of the Gauss-Bonnet theorem.6

The Gauss-Bonnet theorem says that, for a closed7 manifold M, \displaystyle\int_M K dA = 2\pi \chi. The left hand side is the integral of the Gaussian curvature K over the manifold M. Integrals add up what’s inside them, so this integral represents the total amount of curvature of the manifold. The right hand side is some constant times the Euler characteristic.

Let’s look at an example.

For a standard sphere S^2, Gaussian curvature is a constant, 1/r^2, where r is the radius of the sphere. That means that \displaystyle\int_{S^2} KdA is just the area times the curvature, so 4 \pi r^2 \cdot 1/r^2 = 4\pi. Since the Euler characteristic of a sphere is 2, this lines up nicely with the Gauss-Bonnet theorem.

Now, none of this seems to weird until you remember that the right hand side, 2\pi \chi is the same for any metric on the sphere. In other words, it doesn’t change no matter how much we bend or squish the sphere. But when we do that, the Gaussian curvature changes, and so it seems like the left hand integral should change.

But that’s exactly why this theorem is so interesting.

Somehow, no matter how you squish or bend or deform a surface, the positive curvature you may add is exactly canceled out by negative curvature added somewhere else.

And not only that! The theorem says that the topology of a surface restricts what kinds of curvature you have.

For instance, consider the torus.

The torus has a Euler characteristic of zero. That means that no matter how we bend it around, the total curvature has to be zero. The standard way you think of a torus, like a donut, has positive curvature on the outside edge, and negative on the inside edge, that exactly cancel out in the integral.

positivenegative

The torus of Asteroids, which is a flat rectangle with edges identified, has zero curvature everywhere.

flat

But what you can’t do is somehow make a torus that has positive curvature everywhere. If you get close, there’s always going to be a tiny bit of the torus with some negative curvature exactly canceling out your best efforts.

finagleabagel

To finish, let’s look back at those weird triangles, with angles adding up to 270 or zero degrees.

weirdtriangles

While the version of the Gauss-Bonnet theorem I originally gave only works for closed manifolds, there is another version of it that allows for shapes with boundaries, and thus for triangles.

If the edges of the triangle are “straight,” i.e., are geodesics, then the Gauss-Bonnet theorem changes slightly to \displaystyle\int_M K\,dV = 2\pi \chi - \kappa_{turning}, where \kappa_{turning} is the sum of all the “turning” angles. The turning angle is \pi minus the standard interior angle.

turningangle

Since a triangle has 3 corners, 3 sides and 1 face, its Euler characteristic is 1. Combining all that, we see that \kappa_{int} = \pi + \displaystyle\int_M K\,dV. In words, the sum of the angles of a triangle add up to \pi (i.e., 180 degrees), plus the total curvature inside the triangle.

This is exactly one of the interpretations of curvature I gave earlier; positive curvature makes triangles have angles add up to more than 180 degrees, while negative curvature does the opposite.

Next time, we’re going to put all this geometry to work and introduce more general curvatures and general relativity!

 


  1. This kind of a transformation is a homeomorphism. Two shapes are the same, to a topologist, if they are homeomorphic. 
  2. Technically, the genus of an orientable manifold is the number of times you can cut along a closed curve but leave the manifold in one piece, but intuitively, that’s the number of holes. There’s a definition for nonorientable manifolds as well. 
  3. You can just use triangles if you want. One theorem says that every 2-manifold has a triangulation, so you can always do this. 
  4. Another way to think of this is to take two squares of rubber, and then glue them around their edges. If you inflated this up, you’d get a ball. You could do the same with triangles or pentagons or whatever. 
  5. In fact, for a closed, orientable surface, the Euler characteristic works out to \chi = 2-2g, where g is the genus. 
  6. Gauss really has his name all over everything, doesn’t he? Oh, and Bonnet is pronounced BOH-neh (French), though often said Boh-NAY. 
  7. A closed manifold is one that closes up on itself, like a torus or a sphere. This is in contrast to a filled in circle, which has a boundary, or the plane, where you can run off towards infinity. 
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