Manifolds, Proper

Here’s the idea behind manifolds. We want manifolds to be like a general structureless space, with none of those troublesome ’special points’ like the origin and irrational points and whatnot. But how do we characterise such a space?

We are most used to working in flat space — . The thing about is that it can be nicely expressed in independent coordinates: each of x, y, and z in a position vector (x, y, z) are like independent properties of the point being referred to, and we can perform operations on the vector as though each coordinate is independent of the rest (unless specified otherwise).

It is hence natural to try to frame manifolds in the language of Euclidean space. And we do this by the idea of transformations. We set up a system such that we can transform any given point in a manifold to any other point in a corresponding Euclidean space. But what kind of transformations? It would not be of much use to have rigid transformations that leave curved and twisted surfaces essentially curved and twisted, and straight objects essentially straight. We could not characterise a meaningful array of non-Euclidean spaces with that. We also want maximal transfer of information about the manifold to the corresponding Euclidean structures. That is, we don’t want the ‘translation’ to Euclidean space (or Euclidean space-like, as we shall see) to butcher the original manifolds so much that we wouldn’t be handling the original manifolds in our operations on the Euclidean structures. After all, the whole point of translation was to be able to handle manifolds in convenient Euclidean notation.

The natural thing to do might be to map an entire manifold to a slice of Euclidean space. For example, we can map the whole of a spherical surface, minus a pole, to the infinite real plane , as follows:

You place the sphere on the plane. Call the point on the sphere furthest from the plane the ‘north pole’. Draw a line from the north pole of the sphere towards any point on the plane. It will intersect the sphere at exactly one point. Thus each point on the plane maps to a unique point on the sphere, and every point on the sphere except the north pole maps to a unique point on the plane. So the spherical surface sans north pole is a manifold which maps whole and nicely to a whole slice of Euclidean space.

But this is not always the case. That is, we want to deal with manifolds that do not map so nicely. We want to deal with the whole sphere, for instance, north pole and all. We don’t want to stop ourselves from calling funny twisty structures manifolds just because we cannot press them flat (metaphorically) into a continuous slice of Euclidean space. So we relax our demands and try instead to cover any given manifold with a patchwork of maps to separate slices of Euclidean space. The patchwork can be made from arbitrarily small (but still non-zero) patches. Now that we can have many patches, we can characterise even structures that are ‘very far’ from flatness as manifolds. So long as we find the right combination of patches.

Take the surface of a 3D sphere, for example. We can now cover it with six overlapping patches and not miss a point.

Going by the colour scheme above, we use the following hemispherical patches: blue+pink, pink+green, green+brown, brown+blue, the hemisphere above the plane, the hemisphere below the plane. They cover every point of the sphere. We can use a simple function to translate every point in a given hemisphere to a corresponding point in Euclidean space. Finally, they also overlap with at least one other patch. This will turn out to be important.

The manifold has no seams. It is supposed to be a structureless surface. So when we divide it into patches to translate into Euclidean space, we don’t want to make special conditions for the translation of points that happen to fall on the edges of the patches. So we insist on overlaps. So there aren’t any ’special’ points that necessarily end up on the edge of the corresponding slice its patch is translated to. But we have to go further, for still we can see where the overlaps are. We insist that every possible combination of overlapping patches is a valid translation of the manifold. So no particular cutting up is favoured over another, as long as they preserve the essential, structureless characteristics of the manifold. This is why the manifold isn’t just one particular translation of it into patches of Euclidean space. It must be all possible qualified translations, for favouring any one over the others destroys its structureless nature. That is not to say that we do not favour one over others for pragmatic reasons, for ease of computation. But we recognise that the others are mathematically as representative of the manifold as whichever one we choose to use. At this point, a term that my professor used to describe the different possible patchworks pops into my head: a democracy. One translation, one vote. None more important than the others.

I could, and perhaps should, mention what are the mathematical formulations of ‘overlaps’, ‘maps’ and so on. In particular, there are niggling but important details about the kind of maps (differentiable) that are allowed. But I’m lazy, I figure that anyone who has the mathematical background probably knows what I’m talking about or can apply my intuitive description to help her learn from a dry axiomatized textbook description, and that anyone without the mathematical background wouldn’t benefit from reading technical details anyway.