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In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. 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 multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional 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 two-dimensional 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. Geometrical facts about Riemann surfaces are as "nice" as possible, and they often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem is a prime example of this influence. == Definitions == There are several equivalent definitions of a Riemann surface. # A Riemann surface ''X'' is a complex manifold of complex dimension one. This means that ''X'' is a Hausdorff topological space endowed with an atlas: for every point ''x'' ∈ ''X'' there is a neighbourhood containing ''x'' homeomorphic to the unit disk of the complex plane. The map carrying the structure of the complex plane to the Riemann surface is called a ''chart''. Additionally, the transition maps between two overlapping charts are required to be holomorphic. # A Riemann surface is an oriented manifold of (real) dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point ''x'' of ''X'', the space is homeomorphic to a subset of the real plane. The supplement "Riemann" signifies that ''X'' is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same. Choosing an equivalence class of metrics on ''X'' is the additional datum of the conformal structure. A complex structure gives rise to a conformal structure by choosing the standard Euclidean metric given on the complex plane and transporting it to ''X'' by means of the charts. Showing that a conformal structure determines a complex structure is more difficult.〔See for the construction of a corresponding complex structure.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemann surface」の詳細全文を読む スポンサード リンク
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