I can do the impossible.
I can draw a triangle with three right angles!
How do I accomplish this impossible feat?
I draw the triangle on a sphere.
This is actually easy enough. Start from the north pole and draw a line down to the equator. On a sphere, this “great circle” route is a “straight line.”^{1} Then make a turn and travel along the equator till you’ve gone a fourth of the way around. Finally, turn north and travel back to the north pole.
See?
Three right angles.
So why did we learn in grade school that the angles in a triangle add up to 180 degrees?
The angles of a triangle do add up to 180 degrees… but only if we’re drawing it on a flat sheet of paper. Curvature messes things up.^{2}
Notice that this triangle weirdness is something intrinsic to the surface. If I were an ant living on the surface of a sphere, even if I was unaware of a third dimension above and below me, I could physically walk along those paths and measure that I turned more than 180 degrees. I could tell there was some sort of curvature, completely independent of whether or not the third dimension existed!
This is an important kind of curvature. Just like we wanted the idea of a manifold and a metric to be independent of how we visualized a shape in space, there is some sort of intrinsic curvature of a surface that does not depend on how the shape is visualized.^{3}
Some curvature is not intrinsic to the surface, but is rather based on how the surface is visualized in space. This kind of curvature is important, but it’s also important to differentiate the two.
For example, take a flat sheet of paper.
Certainly a sheet of paper is flat. Any triangle you draw on it will have angles adding up to 180 degrees.
But you can always roll up that sheet of paper into a cylinder to make a toy spyglass.
When we roll it up this way, clearly some sort of curvature changes. But notice that the triangles drawn on the paper are still the same. They still have 180 degrees.
The cylinder, intrinsically, is still flat!
To be clear, something living on the surface of that cylinder could walk around the cylinder and realize it got back to where it started. But that’s repetition, and not really curvature.^{4}
So, to start trying to understand curvature, let’s start with the simplest case: the curvature of a curved line.
We want to define a quantity that is big when the path curves a lot, and small when it only curves a little.
The key idea is to draw the osculating circle.
Yeah, okay. I’ll admit that “osculating circle” isn’t exactly a common term. But the idea is pretty simple.
A tangent line, as you may know, is a line that just barely touches a curve, like this:
The osculating circle is essentially a tangent circle.
In order for this circle to be as good a fit as possible for the path, it has to have just the right radius. Too big or too small, and it will barely touch the curve, but it won’t bend like the curve bends.
If the perfect radius for the osculating circle is , we define the curvature of the path to be . A tightly curved path will have a small osculating circle, and so a large curvature, while a slowly meandering path will have a large osculating circle, and so a small curvature. A straight line will have an infinitely big osculating circle, and so will have zero curvature.
Of course, a curve could bend left or right, which are slightly different. Sometimes this doesn’t matter, but, if it does, you can just say the curvature is positive if it bends one way, and negative if it bends the other way.
As far as paths are concerned, this is the only kind of curvature there is. Unfortunately, this curvature is not intrinsic to the path. If something lived on the line, it could only detect the length it traveled, forward and backward. If it had no awareness of other dimensions, there would be no way to tell the line was curved.
Darn.
How can we define curvature for a two-dimensional surface?
One idea would be to try an “osculating sphere,” i.e., a sphere that just barely touches the surface. The problem is that surfaces might bend up and down at the same time, and so no one sphere really fits just right.
Instead, we continue to use osculating circles, but now there are lots of them.
First, let’s pick a normal vector. “Normal,” here, is used in the sense of “perpendicular.” A normal vector is an arrow that points perpendicularly away from the surface.
Next, if we focus on a point on our surface, and slice the surface with a (perpendicular) plane, we get a path through that point for each different direction.
Each path is in a plane (the plane we sliced the surface with), and so we can find the osculating circle for that path. Again, one over the radius gives the curvature of the path. The curvature is positive if the circle is on the same side of the surface as the chosen normal vector, and negative if it’s on the opposite side.
Of course, there are infinitely many of these curvatures, but it turns out that there’s really only two that matter. The principal curvatures are the biggest and smallest (most negative, if there is a negative curvature) of these curvatures.
For example, on a cylinder, the slices that give the principal curvatures are the slice along the cylinder and the slice across the cylinder.
That means the principal curvatures for the cylinder are zero (for the slice along the cylinder) and , if is the radius of the cylinder.^{5}
How about another example. On a sphere, each of the slices will give an equivalent path, and so the principal curvatures are both , where, again, is the radius of the sphere.
One last example. Consider a saddle point, a point on a surface which partly bends up and partly down.
For a saddle point, one principal curvature is positive and the other is negative, though the exact values depend on precisely which surface we draw.
You may have noticed that the principal curvatures in the examples we drew come from slices that are perpendicular to each other. It’s not immediately obvious, but it turns out that the principal curvatures always come from perpendicular slices like this.^{6}
So, now that we have these nice principal curvatures, do they fulfill our goal of being something intrinsic to the surface? Something detectable by someone living on the surface, and not just an artifact of how the surface was visualized in space, ?
Unfortunately, no.
Let’s go back to the cylinder example.
For a sheet of paper, the principal curvatures are both zero, since it’s not curving at all.
But once we roll it up, a cylinder has principal curvatures zero and .
So, the curvatures are different.
That is somewhat confusing. We know there should be some sort of intrinsic curvature, since the sphere has some. However, if the principal curvatures aren’t intrinsic to the surface, it’s hard to imagine some other way of measuring curvature.
The surprising fact is that you can combine the principal curvatures in order to make some sort of intrinsic curvature! And this is the theorem that’s so awesome, it’s called the awesome theorem.
But that’ll have to wait till next time.
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- Technically, a geodesic. We’ll get there eventually. ↩
- In particular, it causes the parallel postulate of Euclid to fail. Euclid’s proof that the angles of a triangle add up to 180 degrees depends upon this assumption. ↩
- In this case, “does not depend on how the shape is visualized” means “depends only on the manifold and metric.” I mention this since the (implied or induced) metric is different depending on how the manifold is visualized in space. But there is more than one way to visualize the same surface with the same (induced) metric. ↩
- Like in the game Asteroids, the screen is flat, but when you go out the right side you come in the left side. Again, it’s flat, but still closed in on itself. Now, this is the flat torus. The standard donut shape you visualize is not flat. ↩
- Here, we took the inward pointing normal vector, so that the curvature was positive. In the end, it doesn’t really matter, as long as you are consistent with your choice. ↩
- (Warning: math ahead!) One way to prove this is to note that the actual calculation for the principal curvatures don’t use more than the second derivatives of the surface’s equation. Thus, if we approximate the surface with a 2nd-order approximation (like a Taylor series, but for multiple dimensions), the principal curvatures have to be the same, since the two surfaces have the same second derivatives. For that kind of approximate surface (which has equation very much like or similar), it is “easy” to see that the principle curvatures should come from slices perpendicular to each other. ↩