|
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus ''g'' > 1, stating that the number of such automorphisms cannot exceed 84(''g'' − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.〔Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms.〕 The theorem is named after Adolf Hurwitz, who proved it in . == Interpretation in terms of hyperbolicity == One of the fundamental themes in differential geometry is a trichotomy between the Riemannian manifolds of positive, zero, and negative curvature ''K''. It manifests itself in many diverse situations and on several levels. In the context of compact Riemann surfaces ''X'', via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies: * ''X'' a sphere, a compact Riemann surface of genus zero with ''K'' > 0; * ''X'' a flat torus, or an elliptic curve, a Riemann surface of genus one with ''K'' = 0; * and ''X'' a hyperbolic surface, which has genus greater than one and ''K'' < 0. While in the first two cases the surface ''X'' admits infinitely many conformal automorphisms (in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus), a hyperbolic Riemann surface only admits a discrete set of automorphisms. Hurwitz's theorem claims that in fact more is true: it provides a uniform bound on the order of the automorphism group as a function of the genus and characterizes those Riemann surfaces for which the bound is sharp. 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「Hurwitz's automorphisms theorem」の詳細全文を読む スポンサード リンク
|