What is (Gaussian) curvature?
Tags: howtos, research
A previous article already introduced manifolds and some of their properties. Along the way, I briefly mentioned curvature but never got around to explain it properly. This article aims to fill in some of the gaps in a very visual way that is accessible to many people.
What we talk about when we talk about curvature
Unfortunately, curvature is one of the concepts that crops up in very different contexts in mathematics. This can cause quite a lot of confusion! Let us thus start with an intuitive view first: intuitively, curvature measures to what extent an object, such as a surface or a solid, deviates from being a ‘flat’ plane1. Moreover, to be very precise, this article deals with intrinsic curvature, i.e. curvature that does not change when we change the way an object is embedded in space. This is relevant because it means that the curvature of such objects can be determined by measuring angles, for example. Thus, we never have to ’leave’ the object2 to compute it; the disadvantage is that we will not be able to tell some objects apart based on this purely intrinsic property.
Since most of the initial discoveries of this form of curvature are due to Carl Friedrich Gauss, mathematician extraordinaire, we should credit him appropriately and use the term Gaussian curvature throughout this article. However, in the interest of conciseness, I shall only use the term curvature. Many of the concepts that are illustrated here also apply to more complicated variants of curvature, but those will be covered in another article.
Let us dive into curvature by thinking about triangles. Triangles are great: everyone learns about them at some point in their life and everyone can visualise them in their mind. One of the core concepts in Euclidean geometry, i.e. the type of geometry you typically learn in high school, is that the sum of all angles in a triangle is $180^\circ$ or $\pi$, if you like your angles in radians3. Here are some examples of triangles, with their respective angles indicate as dotted lines. Try as you might, all of them will have an angle sum of $\pi$.
Let us call all objects that satisfy the same angle sum property planar or flat objects because their geometry is the same as that of the plane. Are there other objects out there with different angle sums? There sure are! We already encountered a triangle with three right angles on the sphere4:
We call spaces or objects with angle sums larger than $\pi$ spherical spaces. This is reminiscent of the fact that the sphere, as shown here, is our ‘prototypical’ space. Sometimes, you will also see the term positive curvature thrown around—this is just another way of expressing the fact that triangles are somehow ‘more curved’ than on the plane (which is considered to be a space of zero curvature).
Now the question we have all been waiting for: if there are spaces with angle sums exactly $\pi$, and angle sums larger than $\pi$, are there also space with angle sums less than $\pi$? As it turns out, such spaces exist and are even relatively natural5. Take a saddle as it is used when riding a horse, for example. We can inscribe a triangle on this object6 without any problems:
When doing so, however, we notice two interesting things:
- The ’triangle’ has curved edges because we must not ’leave’ the saddle when drawing it.
- Given a sufficiently ‘curved’ saddle, we can make the interior angles as small as we want. In mathematical terms, this means that the infimum of the angle sum is zero.
Thus, we conclude that spaces or objects with an angle sum of less than $\pi$ exist. They are commonly referred to as hyperbolic spaces or, equivalently, spaces of negative curvature. The name comes from the class of hyperboloids, which are objects that you obtain by rotating a hyperbola around one of its axes. Here is an example7:
You can also think of this object as ‘pinched’ cylinder—which thus needs to have a negative curvature (because of the pinching).
But can we express it?
We have seen that there are three types of spaces, each one being characterised by the sign of its curvature. But there is even more information available—for some of the spaces, closed-form expressions of curvature exist. For example, a plane has a curvature of $K = 0$. A sphere of radius $r$, by contrast, has a curvature of $K = 1/r^2$ at every point. As an (algebraic) topologist, this sort of expression makes me slightly nervous, because it means that a property depends on the size of a space. Or, to put it differently, the larger the sphere becomes, the less curved it will appear to be8. An expression of curvature for hyperboloids also exists, but its is even more complicated and depends on its parameters. Look it up on MathWorld if you are interested.
Interestingly, curvature can also change along an object; it is a local property. While a plane and a sphere have the same curvature everywhere, a torus, i.e. a ‘donut’ (albeit an unfilled one) has regions of varying curvature:
Here, red indicates negative curvature, while blue indicates positive curvature. Green corresponds to zero curvature, i.e. local ‘flatness’. Thus, as we move from the outside of the torus to the inside, the sign of curvature flips. In light of what we have seen above, this makes perfect sense: the outside of the torus locally looks like a sphere, while the inside is bent like a saddle. This tells us that curvature cannot be constant here. Equivalently, we can say that there are regions of the torus that are locally convex (positive curvature) or locally concave (negative curvature).
The remarkable thing about curvature
We are now able to prescribe certain numbers to mathematical objects to measure how curved they are. This seems to roughly coincide with our intuition—a plane is flat, but a sphere is not.
The real remarkable fact about curvature, though, is that we can determine it from length measurements alone—at least, this holds for surfaces, i.e. two-dimensional manifolds. Even more surprisingly, Gaussian curvature itself does not depend on how such a manifold is located in its ‘ambient’ space: if we change the position of a sphere of radius $r$, its curvature does not change. In fact, an even stronger theorem holds: the Gaussian curvature is invariant9 under isometries10!
This is not a trivial thing to see or to prove; Carl Friedrich Gauss himself—the first one to discover, and prove it—dubbed it Theorema Egregium, which is Latin for ‘remarkable theorem’ or ‘outstanding theorem’. This theorem has an exciting application that we can all appreciate: since a plane has a curvature of $K = 0$, and a sphere has non-zero curvature, there is no way to ‘unroll’ a sphere into a plane without distorting lengths. In other words: it is impossible to generate a perfect map of our Earth; every map suffers at least from some distortion. This also works in the other direction: there is no way to generate a sphere from a (flat) piece of paper without crumpling it, i.e. without distorting some lengths or angles.
The limits of Gaussian curvature
So far, Gaussian curvature proved to be very helpful: we were able to distinguish three different types of manifolds from each other. There are limits to its capabilities, however. Take a cylinder, for example. Since we can obtain a cylinder from a plane by ‘rolling it up’, which is an isometry, its curvature is $K = 0$. As a small bug living on such a cylinder, we would thus not be able to detect that our space is different from that of plane—at least, we would not be able to detect this difference based on purely intrinsic measurements, i.e. measurements performed on the surface.
Not surprisingly, this obvious deficit led to the creation of even more variants of curvature in the mathematics community. Mentioning them all would go well beyond the scope of this article; suffice it to say that the mean curvature is non-zero for the cylinder, but zero for the plane.
This once again demonstrates how mathematics progresses: first, relatively simple properties are investigated until their limits are known. Afterwards, mathematicians try to update them in order to encompass a greater class of objects.
Curvature continues to play a big role in modern mathematics. A more complex and more generic definition of curvature called Ricci curvature was instrumental in solving the Poincaré conjecture, one of the Millennium Prize Problems. Solving one of these problems guarantees honour, fame, and 1 million USD, which seems like a good deal. Obtaining closed-form expressions of certain curvature formulations is thus still an active research topic. For example, the Ricci curvature of graph-based data was formally introduced by Lin, Liu, and Yau in their 2011 article Ricci curvature of graphs. For mathematicians, this practically happened yesterday11. The future of curvature may thus not only be bright, it will surely also be bendy.
Until next time, stay curvy!
Nonetheless, there will be some surprises in this article that do not necessarily fit our idea of a space being curved. ↩︎
Think of it like this: the small bug introduced in the previous article does not need a spaceship in order to measure certain properties of his home world. ↩︎
As you should. Radians sounds so much more professional than degrees. Moreover, it will not be confused with a temperature. ↩︎
You can construct the same triangle on our earth. Please see the previous article for a brief recipe. ↩︎
For a given value of natural. ↩︎
This figure is a modified variant of a figure from Wikipedia, which has been placed in the public domain. Thanks to the author ‘LucasVB’ for that! ↩︎
This figure is a modified variant of another figure from Wikipedia. Thanks to the original author ‘Nicoguaro’ for that! ↩︎
Insert your favourite ‘flat earth’ joke here. ↩︎
A mathematical term meaning that it does not change. Mathematicians like invariants because they enable us to classify spaces regardless of the form they appear in. ↩︎
Another mathematical term referring to transformations of a space that preserve lengths. The term comes has its roots in the Greek language, and literally means ‘same measure’. ↩︎
Whereas in the machine learning community, anything from that time is considered to come straight from the Jurassic period. ↩︎