The study of differential geometry begins innocuously with the notion of the manifold. Sounds ominous! Not really. In fact, you have been working with manifolds your entire life but you just didn’t know it. Every interior angle of a triangle you’ve had the pleasure of measuring, every parallel line, every circle you’ve ever tried to draw; this all took place on a manifold. Very early on, you would have called it a piece of paper. A little later you would have called it a plane. A little later still perhaps a Euclidean plane. Without realising it you’ve actually worked with manifolds – the very centerpiece of modern differential geometry, and it’s something they don’t teach you until you’ve made it through three years of math at university.
So Why Keep it a Secret?
Simply because manifolds, on their own, are very basic objects. They’re so basic in fact that we can’t even be sure that they’re flat. If they’re not flat then they’re curved and that means you can’t do classical geometry on them. It also means you can’t measure simple stuff like distance or angle. Not even area.
So it’s not really that they’re kept a secret, it’s just that we really don’t need something so general so early on in our mathematical lives. So why do we need them in post-grad math? This is because a lot of post-grad math isn’t your usual primary school math. Us post-grads don’t need to know the angle-sum of triangles, we don’t need flat space (besides, what’s flat in this universe anyway?). The point is, post-grad math often requires us to drop structure, or to relax constraints, in order to either study a general framework (topology and topological spaces, for example); or to then rebuild it along a different path of assumptions to arrive at something completely different (Riemannian geometry for example).
What is a Manifold?
A manifold is a very relaxed, basic, simplified mathematical object. It doesn’t really do much but sit there staring back at you much like a blank canvas waiting to be painted. Bringing a manifold to life is an easy task. You definitely don’t have to do much to claim that you have one and that you are about to work with it; all it takes is for someone to say that they have this object that appears kind of flat when you look closely enough at it. That is certainly a vague statement, after all, pretty much every single object I’ve ever seen looks pretty flat when I look really closely at it (some may require the use of a microscope). The Earth is a prime example of a manifold, it certainly isn’t flat but we thought it was until Socrates suggested otherwise. Geometrically speaking, forcing a mathematical object to look kind of flat when you look close enough is just about the most basic thing you can do. But as soon as you do this, you immediately have a manifold.
So why is it so important to have this ‘thing’ that looks kind of flat but not really? Well, it’s the flat part that is important. When one says that some object is flat what they really mean is that the object is Euclidean. But what is Euclidean? An object is Euclidean if the distance between any two points on (or in) the object can be found using the Euclidean metric – or in other words, using a ruler. Which means that manifolds are objects that kind of look flat and we can measure distances between points using rulers when we look close enough. Now, we are getting somewhere! Keep in mind that a manifold need not be Euclidean everywhere, only when you look close enough – mathematicians say locally instead of ‘close enough’ so we will too from now: manifolds are locally Euclidean.
You might be wondering why bother making something look locally Euclidean? It’s pretty simple and it has to do with the fact that for centuries we’ve done everything based on flat geometries. We’ve developed sophisticated tools that work extremely well on flat spaces. Due to this, a manifold is a cheat way to study non-flat geometries but still keep a toe in flat space.
Before the concept of a manifold existed in mathematics there were only several important results that were not linked. Non-Euclidean geometry (not flat) considered spaces where Euclid’s parallel postulate fails. Giovanni Saccheri (1667 – 1733) first began to conceptualise the idea of a manifold in 1733 but it was not for another century when Bernhard Riemann (1826 – 1866), extending on the work that Nikolai Lobachevsky (1792 – 1856) had done, first defined the modern idea of a manifold. Both Riemann and Lobachevsky’s research had uncovered two types of spaces whose geometry differed completely from that of Euclidean space; these two were in fact hyperbolic and elliptic geometry.
Whatever a manifold is, it is at least a topological space that can be covered with flat, Euclidean-like patches. By a topological space we mean that there is an understanding of what an open set is, indeed the topology (which is structure) defines precisely what is and what is not an open set; thus if we discard all the non-open sets and just take the manifold and the open sets then the pair is called a topological space. And by covered with flat patches we mean that each point of the manifold, , is the center of some open subset (also called a neighbourhood) of the manifold, which is isomorphic (there exists an isomorphism, essentially they’re identical) to flat Euclidean space . For example, a sphere could be regarded as a manifold as every point on a sphere looks like flat Euclidean space, in fact many people on Earth once believed that it was flat. To draw the analogy even further, the flat patches are actually called charts, and the collection of all the charts covering a manifold is called an atlas! Let’s look at these two guys a little further.
We begin with a definition: A chart for a topological manifold is a pair consisting of an open subset (we can talk about open sets because manifolds come equipped with topologies), and these subsets are defined about a point . There is also a homeomorphism , from the open subset to an open subset of Euclidean space. In symbols:
Charts can work backward too, mapping the real numbers on to an open subset of a manifold essentially giving each point on a manifold a coordinate! It is this way that we may define a local parametrisation as the pair .
Another definition. An atlas for a manifold is a collection of charts such that the union of all the charts is precisely equal to the entire manifold, i.e. .
Up Next: 2. Manifolds – Adding Structure