is nice. By nice I mean that it is flat, that it contains (by definition) our intuitive notions of distance in flat spaces, that everything that one intuitively suspects is true turns out to be true in it, and so on.
Unfortunately, 
often does not suffice as an arena for many physical phenomena, which can take place in spaces with different structures. In fact, one can argue that 
simply cannot be a true description of physical phenomena, because we don’t think the coordinate system one chooses is anything more than a tool for the physics — it is certainly not part of the physical world. By this I mean that we don’t think there are literally points in the spaces objects move in, or in the abstract spaces that represent their velocities and positions and various other properties, that have the properties of having irrational coordinates. Or that it is important that a particular point is labelled as (1, 2) rather than (2.59, 0.98). We also don’t always need the idea of a distance as defined in . And when we throw in complex spaces, it becomes less clear that is adequate for everything.
Manifolds are a way of describing all continuous spaces. Manifolds are essentially the most structureless possible spaces. And they are everywhere in physics. (83% of physics is manifolds, as my professor quipped.) It is, from an aesthetic and philosophical point of view, more appropriate to think of physics in terms of manifolds than in terms of or the complex plane or any of those structured things, because physics doesn’t really care what coordinates you put on it. Physics may involve distances, but you can easily do that in manifolds as well. Ultimately, physics involves things moving around in the most general possible spaces. Although these may often be translated into things moving about in , the part is strictly ornamental.
Hence manifolds. The nub, though, and the part about manifolds that makes them so general and yet at bottom so easy to deal with, is that they are on small scales. One can think of this as saying that they are locally flat.
In the next post, I will try to explain how all this is manifested in the usual rigorous definitions of manifolds.
