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This question stems from a discussion with a friend about the apparent jump in rigidity of smooth geometry to complex geometry. The word rigidity here is somewhat vague; I mean this in the sense it imposes significantly more restrictions on behavior of these objects (as H. R. Miller once said in a lecture, "groups are rigid, but spaces are squishy").

In complex analysis, asking for a complex derivative is an incredibly restrictive condition. Not only are these functions everywhere analytic, but we get results such as Cauchy's theorem, Liouville's theorem, and Hadamard factorization as "basic" consequences. The theory of functions of a complex variable has a very different character to functions of a real variable.

If we're building a manifold, and require the charts to produce holomorphic transition maps, we get a similarly restrictive condition. These manifolds behave more like algebraic varieties over $\mathbb{C}$ than smooth manifolds. Complex manifolds are automatically smooth and canonically oriented. Most spheres fail to be complex manifolds (except $S^2$, the Riemann sphere), hinting that complex structure is a somehow "rare" phenomenon.

Ultimately, I want to know what it is about having derivatives on $\mathbb{C}$ that causes this apparent rigidity. Some random candidates that come to mind are ideas from the proof of Cauchy's theorem, algebraic closure of $\mathbb{C}$, and the fact that $\dim_{\mathbb{R}} \mathbb{C} = 2$. There really aren't any good analogies of this phenomenon that I know of, but I'd be interested in hearing about them if they exist.

As commenters have pointed out, real analytic geometry behaves similarly to complex geometry, though real analytic geometry can be done on a seemingly wider class of manifolds, while complex manifolds are somehow "rare". Presumably this is to do with the fact that holomorphic functions, in addition to being analytic, satisfy an elliptic PDE (the C-R equations). I have little background in PDEs, though, so the fundamental reason for this "rigidity" is still somewhat mysterious to me.

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    $\begingroup$ Having a $C$-derivative means an elliptic pde, while having an $R$-derivative is only a smoothness condition. Solutions of elliptic pde are analytic. What other explanation can you expect? $\endgroup$ – Alexandre Eremenko Oct 22 '17 at 15:39
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    $\begingroup$ Real-analytic functions and geometry are also 'rigid' as opposed to only smooth functions and manifolds. So, I think the key point is analyticity and not the complex numbers per se. In a sense, the ring $\mathbb{K}\{x_1,...,x_n\}$ is much closer to $\mathbb{K}[[x_1,...,x_n]]$ than it is to $\mathcal{C}^{\infty}(\mathbb{R}^n)$, so it's more "algebraic". Moreover, you don't have bump functions and such in the analytic setting. $\endgroup$ – M.G. Oct 22 '17 at 15:57
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    $\begingroup$ @July: Amazingly, real-analytic geometry inherits some "floppiness" from the smooth world: any (paracompact Hausdorff) smooth manifold admits a real-analytic structure unique up to diffeomorphism (Grauert-Morrey), any smooth vector bundle on a real-analytic manifold admits a real-analytic structure, any smooth section of a real-analytic vector bundle on a real-analytic manifold is well-approximated by real-analytic ones, and any smooth map between real-analytic manifolds is well-approximated by a real-analytic map. See Hirch's Differential Topology and 30.12 in the book of Kreigl & Michor. $\endgroup$ – nfdc23 Oct 22 '17 at 16:33
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    $\begingroup$ @July: Though in real-analytic geometry there are no bump functions, there is a (remarkable) close connection to Stein spaces, which are well-suited for globalizing "local" constructions in complex-analytic geometry. Together with considerations based on Weierstrass approximation, that provides tools for approximating smooth structures by real-analytic ones, unlike anything available in the complex-analytic setting. It is really disorienting at first, because of the expectations based on rigidity of power series that you mention. $\endgroup$ – nfdc23 Oct 22 '17 at 16:38
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    $\begingroup$ @July: The specific place I referred to within Kriegl (sorry about the typo) & Michor's book The Convenient Setting of Global Analysis (visible on Google Books, which perhaps these days makes it more "real" than an actual book?) provides some precise references. Also look at 5.7 and 5.8 (and references therein) in From Stein to Weinstein and Back by Cieliebak and Eliashberg available as a real book from the American Math Society, or go the virtual route: math.uni-augsburg.de/prof/geo/Dokumente/stein106.pdf $\endgroup$ – nfdc23 Oct 22 '17 at 17:21
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To expand my comment, let me mention the wonderful book by R. Penrose, The road to reality. A complete guide to the laws of universe, A.A. Knopf, NY 2006. He discusses this under the title "Magic of complex numbers". The magic in this particular case lies in the general facts (S. Bernstein's theorem; Weyl's lemma) that solutions of elliptic PDE are analytic.

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