|
In mathematics, a Klein surface is a dianalytic manifold of complex dimension 1. Klein surfaces may have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex numbers analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by Felix Klein in 1882.〔 A Klein surface is a surface (i.e., a differentiable manifold of real dimension 2) on which the notion of angle between two tangent vectors at a given point is well-defined, and so is the angle between two intersecting curves on the surface. These angles are in the range (); since the surface carries no notion of orientation, it is not possible to distinguish between the angles α and −α. (By contrast, on Riemann surfaces are oriented and angles in the range of (-π,π] can be meaningfully defined.) The length of curves, the area of submanifolds and the notion of geodesic are not defined on Klein surfaces. Two Klein surfaces ''X'' and ''Y'' are considered equivalent if there are conformal (i.e.: angle-preserving but not necessarily orientation-preserving) differentiable maps ''f'':''X''→''Y'' and ''g'':''Y''→''X'' that map boundary to boundary and satisfy ''fg'' = id''Y'' and ''gf'' = id''X''. ==Examples== Every Riemann surface (analytic manifold of complex dimension 1, without boundary) is a Klein surface. Examples include open subsets of the complex plane (non-compact), the Riemann sphere (compact), and tori (compact). Note that there are many different inequivalent Riemann surfaces with the same underlying torus as manifold. A closed disk in the complex plane is a Klein surface (compact, with boundary). All closed disks are equivalent as Klein surfaces. A closed annulus in the complex plane is a Klein surface (compact, with boundary). Not all annuli are equivalent as Klein surfaces: there is a one-parameter family of inequivalent Klein surfaces arising in this way from annuli. By removing a number of open disks from the Riemann sphere, we obtain another class of Klein surfaces (compact, with boundary). The real projective plane can be turned into a Klein surface (compact, without boundary), in essentially only one way. The Klein bottle can be turned into a Klein surface (compact, without boundary); there is a one-parameter family of inequivalent Klein surfaces structures defined on the Klein bottle. Similarly, there is a one-parameter family of inequivalent Klein surface structures (compact, with boundary) defined on the Möbius strip.〔 Every compact topological 2-manifold (possibly with boundary) can be turned into a Klein surface,〔 often in many different inequivalent ways. 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「Klein surface」の詳細全文を読む スポンサード リンク
|