Riemann Surface

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In complex analysis, a Riemann surface is a one-dimensional complex manifold, a deformed version of the complex plane, and a two-dimensional real analytic manifold (i.e., a surface). A two-dimensional real manifold can be turned into a Riemann surface if and only if it is orientable and metrizable. A sphere and torus are classified as Riemann surfaces but not the Mobius strip, for example. The main interest in Riemann surfaces is that holomorphic functions, complex-valued functions of one or more complex values that is at every point in its domain complex differentiable in a neighborhood of the point, may be defined between them. These surfaces were first studied by and named after Bernard Riemann.