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A map $f:X\to Y$ between Riemann surfaces is called holomorphic if for every complex charts $\varphi $ and $ \psi$ (of $X$ and $Y$ resp) we have that $\psi \circ f \circ \varphi^{-1}$ is holomorphic. I would like to know if there is a local definition of this i.e. I would like to know if given a map $f:X\to Y$ between Riemann surfaces and $ x\in X$ a point is there a notion of $f$ being holomorphic at $x$. Thank you all in advance

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    $\begingroup$ The notion of “holomorphic” isn’t applied to points, but instead “complex differentiable”. Then you can define a function of Riemann surfaces $f \colon X → Y$ to be differentiable at $x$, if for any charts $φ$ on $X$ and $ψ$ on $Y$, $ψ ∘ f ∘ φ^{-1}$ is complex differentiable at $φ(x)$. $\endgroup$
    – k.stm
    May 29, 2019 at 22:33
  • $\begingroup$ @k.stm: Typically "holomorphic at a point" is defined to mean "complex differentiable in a neighborhood of the point". $\endgroup$ May 29, 2019 at 22:35
  • $\begingroup$ @EricWofsey Ah, yeah. Makes sense. I like that. $\endgroup$
    – k.stm
    May 29, 2019 at 22:36
  • $\begingroup$ @EricWofsey If you say so. I'd restrict "holomorphic" to open sets; the standard meaning of "analytic on E" is "holomorphic in some nbd of E". $\endgroup$ May 29, 2019 at 23:11

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Yes, it's just the obvious thing: $f$ is holomorphic at $x$ if whenever $\varphi$ is a complex chart of $X$ defined at $x$ and $\psi$ is a complex chart of $Y$ defined at $f(x)$, $\psi\circ f\circ\varphi^{-1}$ is holomorphic at $\varphi(x)$.

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  • $\begingroup$ Have you got a reference handy for this usage? $\endgroup$ May 29, 2019 at 23:12
  • $\begingroup$ Nope. I'm not sure I've ever actually seen it used, since typically the only reason to be interested in a function holomorphic at a point is for studying local properties and so there's no reason to talk about Riemann surfaces rather than just $\mathbb{C}$. It's the obvious definition though and can be considered as a special case of the very general principle that any local property on Euclidean space gives rise to a corresponding local definition on manifolds. $\endgroup$ May 29, 2019 at 23:35
  • $\begingroup$ "Obvious" doesn't make it an actual standard definition. I really don't believe the word is used except for open sets - "analytic'' is there to fill the need for this use of "holomorphic'. (Really - if you've never seen it used how can you say that is the definition?) $\endgroup$ May 29, 2019 at 23:58
  • $\begingroup$ I have never seen "analytic" and "holomorphic" distinguished the way you say. I have absolutely seen holomorphic functions on non-open subsets of $\mathbb{C}$ though (I just don't specifically recall seeing it more generally on Riemann surfaces). For instance, Stein and Shakarchi's Complex Analysis defines holomorphic functions at a single point of $\mathbb{C}$, or more generally on closed subsets of $\mathbb{C}$. $\endgroup$ May 30, 2019 at 0:03
  • $\begingroup$ " the very general principle that any local property on Euclidean space gives rise to a corresponding local definition on manifolds" hmm, perhaps I haven't been clear. Yes of course if one did speak of "holomorphic at $x$" in a Euclidean context then the extension to Riemann surfaces would be exactly as you say. I'm asserting that there iis no "holomorphic at $x$" in $\Bbb C$. $\endgroup$ May 30, 2019 at 0:04

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