Problems that differential geometry solves

Recently, I've been studying a course in differential geometry.

Some keywords include (differentiable) manifold, atlas, (co)tangent space, vector field, integral curve, lie derivative, lie bracket, connections, riemannian geometry, symplectic geometry.

However, I've been wondering what problems in pure mathematics that are obvious and interesting can be solved with tools from differential geometry. In other words what questions could one ask that will motivate the study of differential geometry for someone who's interested in pure mathematics mainly.

Please don't limit yourself to merely stating a problem, but include a hint, full solution, or reference on how exactly differential geometry becomes useful.

Here are some possible answers with my own comment that may inspire you:

• It's a great language to formulate problems in physics. That may be true, but unfortunately I don't care about pysics (enough).
• It's a language to talk about differential equations, which are "naturally interesting". But honestly, I don't think I care about applications to differential equations, knowing that what it comes down to in the end is that the equations are rammed into a computer and brutally 'solved' by numerical methods anyway, no matter how fancy you formulate them.
• Perelman's solution of the Poincaré conjecture, which may be considered a topic in pure mathematics, uses differential geometry. Apparently it does, but isn't that a bit like using Wiles' solution to FLT as a motivation for a course in commutative algebra?
• It provides deeper insights in the whole of mathematics. Well, I just made that up. But maybe it provides a wealth of examples in, for instance, topology, or maybe techniques can be borrowed from differential geometry and used in other branches. But then again, doesn't it make more sense to study them where they become interesting?

As a final example, a simple question that I consider motivating for exploring groups beyond their definition would be: "how many groups of order 35 are there?": it's an easy question, only refering to one easy definition with a somwhat surprising answer where the surprise vanishes once you've developed a sufficient amount of theory.

ps - Since there is no best answer to my question maybe it should be community wiki. I'm sure some moderator will do what's appropriate.

pps - In reply to Thomas Rot's answer I must apologise for the tone when I'm talking about differential equations. Actually I'm a person who obtained a degree in applied physics before turning over to "pure" (in a sense that I don't mind if it's hard to find applications in real life situations) math. I've seen how these people solve differential equations -- I've seen how I used to do it Myself, actually. No cute mathematical theory, just discretize everything and put it into a computer. If it doesn't work out, let's try a finer grid, let's leave out a term or two, until it works. Surprisingly they don't use cotangent spaces to even state the problem, still it appears sufficient to calculate the heat distribution in nuclear reactors, or calculate the electron density around a deuteron. Because I've seen all this and didn't think it is pretty, I've turned away from it. But feel free to change my mind on the subject.

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Could you clarify your definition of pure mathematics? I for one consider differential geometry a purely pure branch of mathematics –  t.b. Apr 18 '11 at 14:52
I think your statements about physics and differential equations illustrates more your personal ignorance than anything else. Since you've managed to rule out the two disciplines that gave birth to much of modern mathematics, you may also want to provide a definition of what you consider "pure mathematics". –  Willie Wong Apr 18 '11 at 14:56
@Theo Buehler: I think you misunderstood, since I didn't mean to say differential geometry isn't a branch of "pure" mathematics (whatever that's supposed to mean). I'm just looking for a motivation for studying an introductory course, if you're not planning to use them in formulations of physical theories or differential equations. By the way, (@Willie Wong) I don't mind if you reproach me ignorance, but if you make no attempt to correct that ignorance, it feels more like an insult. –  Myself Apr 18 '11 at 15:03
Good question., –  quanta Apr 18 '11 at 15:23
The tone of your question and of your comments completely demotivates me to make the effort to write an answer, point you to whatever it might take your surprise you with the beauty and deepness of the subject, or anything else, really. –  Mariano Suárez-Alvarez Apr 18 '11 at 17:22

Maybe you aren't, but Gauss was interested in inherent geometry of surfaces. In fact, so interested that he proved a remarkable theorem.

But ok, say you don't care about inherent geometry. Then surely you care about ambient geometry. Which is fine, because you know, it's pretty damn interesting.

What's that? You just don't like questions about spaces? Well, fie on you. But it doesn't matter. Because, you know lots of algebra has its backbone in geometry.

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The Feit-Thompson theorem? Please explain, since actually I have read that wikipedia entry a few times already in the past and I don't see the connection with differential geometry. I'm quite fond of applications of geometry in algebra and vice versa Myself btw, so don't think that I don't like geometry or 'questions about spaces' (for instance, coxeter-presentations versus hyperbolic tessalations). You just don't need the all the diff-geo-equipment for those things. –  Myself Apr 18 '11 at 17:30
@Myself: What if your primary interest is measuring curvature? (Disclosure: that's certainly my main interest.) I actually don't know of any subject other than differential geometry which lets you do that.... –  Jesse Madnick Apr 19 '11 at 7:21
@Jesse: Measuring curvature? You mean like in the Lisa project lisa.nasa.gov ? Or simply to provide information about the curvature of an arbitrary surface? –  Myself Apr 19 '11 at 7:40
@Myself: I mean the latter. That is, how can we make sense of the concept of "curving," and what theoretical consequences might curving have on other properties of the space. To my mind, that's differential geometry. (Anyway, I don't mean to hijack the thread, which should probably focus on BBischof's excellent answer.) –  Jesse Madnick Apr 19 '11 at 7:48

If you find the question: "How many groups are there of order 35?" motivating, why don't you find the question: "How many differentiable manifolds of dimension 2 are there?", motivating as well?

There are millions of applications of manifolds in pure mathematics. Lie groups (continuous symmetries) are a beautiful example.

As an aside, your question got downvoted (not by me), because the tone is somewhat arrogant. For example your statement:

"It's a language to talk about differential equations, which are "naturally interesting". But honestly, I don't think I care about applications to differential equations, knowing that what it comes down to in the end is that the equations are rammed into a computer and brutally 'solved' by numerical methods anyway, no matter how fancy you formulate them."

is an incredibly ill informed view of the subject. The theory of differential equations is extremely rich (both from a pure and applied viewpoint).

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thank you for pointing out that people may feel insulted by my comment on differential equations. The question that you state, yes I do find it motivating actually! But how does the machinery of tangent-spaces, Lie derivatives,... and all that help to answer that question? –  Myself Apr 18 '11 at 15:24
@Myself: it is not that they may feel insulted---this is the internet, after all, and no one really will take your evaluation of their favourite subject that seriously anyways... It is that such statements in 99.99% of the cases only display the ignorance of those making them. –  Mariano Suárez-Alvarez Apr 18 '11 at 17:28
@Myself: How does group theory explain you how many groups of order 35 are there? You could just write down all multiplication tables of 35 elements right? And then check which ones are the same. More machinery makes this takes way easier though –  Thomas Rot Apr 21 '11 at 11:50

I must admit, I'm not sure I fully understand the question. If you're interested in pure mathematics, and consider differential geometry to be a part of that, then certainly you must be interested in differential geometry for its own sake. The types of motivating problems might be, say, differential geometric ones.

So since I don't fully understand what motivates you, perhaps I can at least explain what motivates me.

(1) Say you want to talk about integrating vector fields over curves and surfaces. Sure, maybe you saw that in multivariable calculus, but perhaps you didn't find the presentation very convincing. If you were like me, you probably thought that the whole thing involved too much hand-waving and not enough rigor or attention to detail.

Well, the language of differential forms solves that problem. But wait: technically speaking we're only really integrating covector fields and not vector fields. If we want to integrate vector fields, one approach might be to introduce a metric structure, and then take advantage of the induced isomorphism between tangent and cotangent spaces.

(2) Say you're interested in topology. Suppose you want easily-checked sufficient conditions for a space to be orientable or simply-connected. Synge's Theorem does that. Or say you want sufficient conditions for a space to be homeomorphic to $\mathbb{R}^n$ or the $n$-sphere $\mathbb{S}^n$. There are theorems (Cartan-Hadamard) (Sphere Theorem) which do that, too. The list goes on, the most famous example being the Gauss-Bonnet Theorem.

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In the study of elliptic curves you can make lots of use of differential geometry. I don't have any examples but you can find them if you study elliptic curves.

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Probably this is just my own ignorance, but I thought the theory of elliptic curves in the smooth setting is rather trivial? –  Aaron Mazel-Gee Apr 18 '13 at 16:15