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A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds).

  1. What is the special feature of Euclidean spaces that makes them interesting?
  2. Is there a field that studies spaces that are neither globally nor locally Euclidean spaces?
  3. If such a field exists, are there any practical uses for it (as in physically existing models)?
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Well, the universe is (approximately) locally Euclidean... I think that's a good enough reason, yeah? –  Qiaochu Yuan Nov 3 '10 at 21:54
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@Qiaochu: I was going to put that as part of the question but I left it for later (it probably warrants a separate one): Is the universe locally Euclidean, or are we just inclined to model it that way? –  Muhammad Alkarouri Nov 3 '10 at 22:00
    
locally Euclidean spaces are interesting regardless of the answer to that question (because the real question is whether the model works, and the answer, I think you have to agree, is a resounding yes). –  Qiaochu Yuan Nov 3 '10 at 22:11
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@Qiaochu: Of course I agree. The reason I excluded that part is that it seemed to belong to physics or metaphysics. That being said, saying that the universe is (approximately) locally Euclidean becomes no longer a valid reason for our interest. –  Muhammad Alkarouri Nov 3 '10 at 22:21
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we have pretty compelling evidence (in particular, the success of Newtonian dynamics) to believe that the universe is locally Euclidean to very close approximation at least; if there were any substantial intrinsic curvature then we'd have detected it. –  Steven Stadnicki Dec 9 '10 at 0:27

5 Answers 5

up vote 13 down vote accepted

One of the main reasons for interest is that manifolds are homogeneous. So from a geometric point of view, we study manifolds for much the same reason we study groups.

Put in less abstract ways, there is the "classical problem" of whether or not the earth is flat. The "flat earth problem" deals with the possibility that on small scales something may look linear, but macroscopically it need not be. Manifolds are the abstract manifestation of the flat earth problem.

People do study large families of objects that are locally modelled on spaces that aren't Euclidean spaces. There's infinite-dimensional manifolds, fractal manifolds, orbifolds (locally modelled on euclidean space mod a finite group action), etc... Fibre bundles are maps that are locally projection maps. This idea resurfaces in mathematics in many different forms and many different ways.

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Can you please explain what does homogeneous mean here? –  Muhammad Alkarouri Nov 3 '10 at 22:23
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If $M$ is a connected manifold, given any two points $p,q \in M$ there is a homeo/diffeomorphism $f : M \to M$ such that $f(p)=q$. So I'm using homogeneous in the sense that $Homeo(M)$ or $Diffeo(M)$ acts transitively on $M$, depending on if you're in the topological or smooth category of manifolds. It's also true in the PL category. –  Ryan Budney Nov 3 '10 at 22:27

In my opinion (because it really is an opinion), the main reason to care about euclidean spaces is that they come equipped with a lot of particularly nice structures. In particular, they are finite-dimensional Hilbert spaces. This means that they are:

  • Finite-dimensional vector spaces (so we can talk about addition and scaling)
  • Complete metric spaces (so we can talk about distances and limits, and limits behave as we would like them to)
  • Inner product spaces (so we can talk about the notion of "angle," and thereby do all sorts of geometric things.)

And really, there's just so much geometric intuition that comes along with these ideas -- not to mention an entire calculus apparatus. After all, there are plenty of topological spaces where the above notions are not defined or otherwise fail to be true.

The reason (at least to me) to study locally euclidean spaces is that we want to study spaces that are more general than euclidean spaces, yet still retain many of their nice features. In particular, we want a place where calculus makes sense.

Areas of math which study more abstract spaces include topology and algebraic geometry. Admittedly, I'm not very well-versed in either just yet, but I'm sure practical uses (and physical models) have been found in both.

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Sorry if this is all stuff you already knew, but this is how I interpreted your question. –  Jesse Madnick Nov 4 '10 at 4:24
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To add to your comment below the bullet points, another reason to study manifolds is that generic level-sets of smooth functions are manifolds by Sard's theorem. So if you care about smooth functions, you have to care at least a little about manifolds. –  Ryan Budney Nov 4 '10 at 4:40
    
No need to apologise, this is a very good answer. The reason I asked this question is to see the forest for the trees, and your answer weaves individual concepts that I know into a beautiful viewpoint. –  Muhammad Alkarouri Nov 4 '10 at 9:33

I'm interested in differential topology, so that baises my answer. $\mathbb{R}$, and Euclidean spaces generally, are interesting because you can do calculus on them. Manifolds are interesting because, not only as OrbiculaR says, you can do standard analysis in a chart, but you can sensibly extend the concepts of analysis to the entire manifold.

You can also do analysis in the p-adics, so you might wonder why aren't studying p-adic manifolds. (I don't actually know.)

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Dear Cotton, Some of use are (studying $p$-adic manifolds)! Best wishes, –  Matt E Dec 8 '10 at 7:11

Well, my answer to 1. is that you can convienently do standard analysis in a chart. Did you ever try working on more complicated (singular) objects? It's a pain in the neck! (2. and 3. have already been answered)

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In topology people study intensely some naturally occurring abut quite abstract and not "locally Euclidean" spaces. Examples are path spaces(space of paths in a given topological space or manifold) and some natural $H$-spaces in the homotopy category.

Anyway there a whole lot of new kind of "geometry" in the field of algebraic geometry, which is not at all Euclidean.

But your point is good, that the Euclidean space is quite specially useful in physics. With this in mind, some people are actually doing adelic physics these days.

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