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Teichmüller space
In mathematics, the Teichmüller space ''TX'' of a (real) topological surface ''X'', is a space that parameterizes complex structures on ''X'' up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in ''TX'' may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from ''X'' to ''X''.
The Teichmüller space is the universal covering orbifold of the Riemann moduli space.
The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it was introduced by .
==Complex structures and Riemann surfaces==
Each topological atlas for a (real) surface ''X'' consists of injective maps from open subsets of ''X'' into the Euclidean plane. Identify the Euclidean plane with the complex plane via . A topological atlas is a complex atlas for ''X'' if each transition map is a biholomorphism. Two complex atlases are equivalent provided their union is a complex atlas. An equivalence class of complex atlases is called a complex structure. A topological surface ''X'' equipped with a complex structure is called a Riemann surface. Among all atlases belonging to a complex structure, there is a maximal atlas which is the union of all complex atlases in the complex structure. One may identify each complex structure with this maximal atlas.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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