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The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multivalued functions such as the square root and other algebraic functions, or the logarithm.
Every Riemann surface is a twodimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions. A twodimensional real manifold can be turned into a Riemann surface (usually in several inequivalent ways) if and only if it is orientable and metrizable. So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not.
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