# 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’ plane^{1}. 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 object^{2} 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.

# Triangles

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
radians^{3}. 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 sphere^{4}:

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 natural^{5}. Take a *saddle* as it
is used when riding a horse, for example. We can inscribe a triangle on
this object^{6} 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 example^{7}:

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 be^{8}.
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
*invariant*^{9} under *isometries*^{10}!

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 now

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 yesterday^{11}. 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.
^{↩}