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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 mathematics2011-04-18
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    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".2011-04-18
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    @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.2011-04-18
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    Good question.,2011-04-18
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    For starters: by claiming that "what [differential equations] comes down to in the end is that the equations are rammed into a computer and brutally 'solved' by numerical methods anyway" you are in one sentence dismissing the subject that featured prominently in the work of at least 6 of the past 18 Fields medalists (most recent 5 ICMs), and at least 3 of the first 8 (first 5 ICMs), and many in between.2011-04-18
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    @Willi Wong: It appears you feel insulted by my point of view. I can't change the way I feel about the subject but I certainly didn't mean to insult anyone, also see my last paragraph. But maybe you could just refer me to an elementary exposition of the problems in the theory of differential equations that these that these fields medalists and others solved with the tools from differential geometry, and surprise me with the beauty and deepness of the subject.2011-04-18
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    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.2011-04-18
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    @Mariano: Same.2011-04-18
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    @Mariano Suárez-Alvarez @Didier Piau: I'm really sorry you feel that way. Apparently you think I'll reject any thoughts on the subject because I don't find them beautiful of interesting enough, but the contrary is true: I'll read whatever thoughts you have on the subject with eager enthousiasm. Please don't read my question as criticism on the study of *any* branch of mathematics at all, I just want to see the machinery of differential geometry at work somewhere...2011-04-18
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    You can also edit the question...2011-04-18
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    @Mariano I didn't think of that. Thank you. It still doesn't close the useless discussion though.2011-04-18
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    You mentioned your *disap(p)ointment* at least twice, you invented that some people felt *insulted* and you provided endless precisions about your views on (what you see as) such and such branches of mathematics and about your decision to close your question (to vote for this, really). Instead, you could have acknowledged the substantial amount of *expert* pieces of advice you received here. For example, @BBishop's answer contains *eight links*, did you read them? I can assure you that, despite the light tone of BB's answer, each of these links is *highly* relevant to the question you asked.2011-04-18
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    @Didier Piau: I mentioned my disappointment once in a now removed comment and once more because I editted my post. (For you information: I read all wikipedia-entries BBishop provided, still don't see the link with the Feit-Thompson theorem, and I am on page 5 of the pdf-document by Misha Gromov.) Maybe I feel insulted myself then by the tone of your comments. Anyway I just want my question closed or deleted.2011-04-18
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    @Myself: I've rolled back to the last revision of the question. I consider it very impolite towards all the people who took their time discussing and answering the issue that you decide to tear their responses out of context. If you want to delete the thread, flag it for moderator attention (I can't because I already have for turning it into CW mode).2011-04-18
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    @Theo Buehler: ok thank you. Maybe it's better that way (I was merely following Mariano's suggestion) Please note that I do appreciate the answers that are given (as opposed to fruitless discussion in the comment boxes that isn't informing me about math but about the fact that I know very little about it).2011-04-18
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    @Myself: I can understand your reaction and the discussion in the comment thread was indeed fruitless and dominated by an accusatory tone. Maybe you could consider formulating a less opinionated version of your question? e.g. What are your favourite results and applications of differential geometry? (I'd say the Sphere theorem; Cartan-Hadamard; Lie theory; everything that connects negative curvature, number theory and dynamical systems are each most fascinating topics).2011-04-18
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    @Myself: I did not suggest that you edit the text of the question away, but that you edit it into something which will encourage answers that are useful to you.2011-04-18
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    Just to be clear: there is one group of order 35: $\mathbb{Z}/35\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/7\mathbb{Z}$, right?2012-11-04

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