Problems that differential geometry solves

Question

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.

Answer 1

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).

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

Differential geometry is a basic tool in mathematical physics, in particular in mechanics. I strongly recommend the lecture of the book of Arnold "Mathematical Methods of the Classical mechanics", and the book of Penrose "The Road to Reality: A Complete Guide to the Laws of the Universe". Enstein theory of relativity (what is space-time) cannot even be formulated without the basic concepts of diff geometry ; symplectic geometry is now a standard tool in Hamiltonian mechanics etc etc...