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Riemann surfaces(Riemannian surfaces) is one dimensional complex manifold. For questions about classical examples in complex analysis, complex geometry, surface topology.

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Is complex analytic extension of real-analytic diffeomorphism a diffeomorphism ?

Indeed, a real-analytic map can always be extended to some complex neighbourhood. The problem is, the neighbourhood may be very small. Consider, for example, the map $$f:I\to I,\, I=[-1,1],$$ define …
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