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 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. 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 differentiable maps f:X→Y and g:Y→X that map boundary to boundary and satisfy fg = idY and gf = idX.
Examples
Every Riemann surface is a Klein surface. Examples include open subsets of the complex plane, the Riemann sphere, and tori. 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. All closed disks are equivalent as Klein surfaces. A closed annulus in the complex plane is a Klein surface. 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. The real projective plane can be turned into a Klein surface, in essentially only one way. The Klein bottle can be turned into a Klein surface ; 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 defined on the Möbius strip. Every compact topological 2-manifold can be turned into a Klein surface, often in many different inequivalent ways.
Properties
The boundary of a compact Klein surface consists of finitely manyconnected components, each of which being homeomorphic to a circle. These components are called the ovals of the Klein surface. Suppose Σ is a Riemann surface and τ:Σ→Σ is an anti-holomorphic involution. Then the quotient Σ/τ carries a natural Klein surface structure, and every Klein surface can be obtained in this manner in essentially only one way. The fixed points of τ correspond to the boundary points of Σ/τ. The surface Σ is called an "analytic double" of Σ/τ. The Klein surfaces form a category; a morphism from the Klein surface X to the Klein surface Y is a differentiable mapf:X→Y which on each coordinate patch is either holomorphic or the complex conjugate of a holomorphic map and furthermore maps the boundary of X to the boundary of Y. There is a one-to-one correspondence between smoothprojective algebraic curves over the reals and compact connected Klein surfaces. The real points of the curve correspond to the boundary points of the Klein surface. Indeed, there is an equivalence of categories between the category of smooth projective algebraic curves over R and the category of compact connected Klein surfaces. This is akin to the correspondence between smooth projective algebraic curves over the complex numbers and compact connected Riemann surfaces. There is also a one-to-one correspondence between compact connected Klein surfaces and algebraic function fields in one variable over R. This correspondence is akin to the one between compact connected Riemann surfaces and algebraic function fields over the complex numbers. If X is a Klein surface, a function f:X→Cu is called meromorphic if, on each coordinate patch, f or its complex conjugate is meromorphic in the ordinary sense, and if f takes only real values on the boundary of X. Given a connected Klein surface X, the set of meromorphic functions defined on X form a field M, an algebraic function field in one variable over R. M is a contravariant functor and yields a duality between the category of compact connected Klein surfaces and the category of function fields in one variable over the reals. One can classify the compact connected Klein surfaces X up to homeomorphism by specifying three numbers : the genusg of the analytic double Σ, the number k of connected components of the boundary of X, and the number a, defined by a=0 if X is orientable and a=1 otherwise. We always have k ≤ g+1. The Euler characteristic of X equals 1-g.
