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I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (equivalent to holomorphicity). There are many important differences indeed analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.

For example in the complex case we have Liouville's theorem (any bounded complex analytic function defined on the whole complex plane is constant). But in the real analytic case this is false, just take $f(x)=\frac{1}{1+x^2}$ Actually, this example gives to us much more differences then this simple one.

In the real case it's a standard fact that any real-analytic function is $C^\infty$ but the converse is not true because of the following example \begin{equation*} {\displaystyle f(x)={\begin{cases}e^{-{\frac {1}{x}}}&{\text{if }}x>0,\\0&{\text{if }}x\leq 0,\end{cases}}} \end{equation*}It is $C^\infty$ but it is not real-analytic at the origin. In light of this example I would like to ask this question

How does the category of real-analytic manifolds differs from that of $C^\infty$ ones? Or rather, are there many examples of manifolds that admit smooth but not real-analytic atlases?

EDIT

Here I'll try to resume all the comments below. Thanks everybody for the answers!

(1) Take a look at this question Can every manifold be given an analytic structure? with much more discussion

(2) The main resut is that: Any $C^1$ manifold admits a compatible real analytic structure (proved by Whitney)

John117
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    Any smooth (or even $C^1$) manifold admits a compatible real analytic structure. This is a result of Whitney (see, for instance, this question https://mathoverflow.net/questions/8789/can-every-manifold-be-given-an-analytic-structure with much discussion). A real analytic manifold is distinct from merely a smooth one in that there is a well defined notion of real analytic function on such a manifold, but a smooth structure does not suffice for this. – RBega2 Feb 07 '21 at 16:59
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    It is a result of Grauert, building on prior work of Whitney. Whitney proved that every $C^1$ admits a compatible $C^{\infty}$ structure, unique up to $C^1$ diffeomorphism. Grauert improved the result, using very different methods (Stein manifolds) to replace $C^{\infty}$ by real analytic. – Ben McKay Feb 07 '21 at 17:08
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    The forgetful functor from analytic to smooth manifolds is essentially surjective and faithful, but not full, as your example illustrates. In more detail, the fact that an analytic map between connected manifolds is completely determined by its restriction to an arbitrarily small open set means that analytic manifolds are "rigid" whereas smooth manifolds are "floppy". An actual analytic geometer could say more. (e.g. I don't know but I would assume in the presence of nontrivial topology that not every smooth map is homotopic to an analytic one.) – Tim Campion Feb 07 '21 at 17:09
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    I don't have an immediate duplicate in mind, but I'm sure I've seen some closely related variant of this question before …. It's such a good and natural question that it's hard to imagine we went over a decade without asking it here! – LSpice Feb 07 '21 at 17:15
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    @TimCampion You are incorrect, there are analytic approximation theorems. Everything is the same after passing to the homotopy category. A short chapter discussing analytic approximation theorems is in Hirsch's differential topology book, though I don't remember if he talks about relative analytic approximation. – mme Feb 07 '21 at 18:40
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    @MikeMiller Thanks, I'm glad to correct that misconception! And now I'm really interested in seeing somebody explain in an accurate way how rigidity vs. floppiness plays out here in the difference between the two categories. – Tim Campion Feb 07 '21 at 18:44
  • It looks like Hirsch doesn't talk about relative approximation (the "smoothly homotopic implies analytically homotopic" step), but I remember seeing this in print somewhere. Anyway, you might like this excerpt: https://imgur.com/a/6fwXy8v Naturally the different classes of differentiability are very important in dynamics or its cousin foliation theory, where you will often see that theorem statements make careful consideration of differentiability class. For an example in PDE consider the Cauchy–Kovalevskaya theorem. – mme Feb 07 '21 at 18:53
  • In the comments here nfdc23 seems to suggest that for these reasons, it may not be so useful to think of real analytic geometry as "rigid" and smooth geometry as "floppy" the way I said, in contrast to the comparison between complex-analytic and smooth geometry. Maybe to understand the rigidity of complex geometry, it's not enough to observe that functions are determined by germs, but you really do have to invoke the "elliptic PDE" angle. – Tim Campion Feb 07 '21 at 19:06
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    The homotopy type of the space of the space of automorphisms of $M$ in the $C^\infty$ category is the same as for the analytic category. But of course the behavior of a particular automorphism is severely constrained by analyticity. – Tom Goodwillie Feb 07 '21 at 20:06
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    This is probably tangential to what you are looking for but from the functional analytic point of view there is a substantial difference between the smooth case and the real analytic one, namely the corresponding function space is, as a topological vector space, considerably more complicated. In the $C^\infty$ case (as in the complex analytic one), it is a nuclear Fréchet space (which is about as good as it gets) whereas in the latter, one has to employ much more delicate structures. These have studied by Martineau and Vogt (amongst others). – bathalf15320 Feb 07 '21 at 20:38
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    I felt I should, there are a number of subtle distinctions between existence of real analytic versus smooth geometric structures. My personal favorite has to do with uniqueness of tangent maps for energy minimizing harmonic maps at an isolated singularity. If the target has a real analytic Riemannian metric, then Leon Simon showed the tangent maps are unique, but if the target only has a smooth metric, then Brian White gives an example with non-unique tangent map. This has to to with the Lojasiewicz inequality for real analytic functions. – RBega2 Feb 07 '21 at 22:20

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