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 number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s analytically, the former are used to study algebraic curves over the real numbers analytically. Klein surfaces were introduced by

Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...

in 1882. A Klein surface is a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

(i.e., a

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...

of real dimension 2) on which the notion of angle between two

tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ...

s 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 ,π 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 In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

(analytic manifold of complex dimension 1, without boundary) is a Klein surface. Examples include open subsets of the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

(non-compact), the

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...

(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 In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

can be turned into a Klein surface (compact, without boundary), in essentially only one way. The

Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...

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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...

. Every compact topological 2-manifold (possibly with boundary) can be turned into a Klein surface, often in many different inequivalent ways.

Properties

The boundary of a compact Klein surface consists of finitely many connected components, each of which being

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to a circle. These components are called the ''ovals'' of the Klein surface. Suppose Σ is a (not necessarily connected) Riemann surface and τ:Σ→Σ is an anti-holomorphic (orientation-reversing)

involution Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

. 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 Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

; a morphism from the Klein surface ''X'' to the Klein surface ''Y'' is a differentiable map ''f'':''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 smooth projective

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s over the reals (up to

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

) and compact connected Klein surfaces (up to equivalence). The real points of the curve correspond to the boundary points of the Klein surface. Indeed, there is an

equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two Category (mathematics), categories that establishes that these categories are "essentially the same". There are numerous examples of cate ...

between the category of smooth projective algebraic curves over R (with regular maps as morphisms) 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. (Note that the algebraic curves considered here are abstract curves:

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

, separated one-dimensional schemes of finite type over R. Such a curve need not have any R-rational points (like the curve ''X''²+''Y''²+1=0 over R), in which case its Klein surface will have empty boundary.) There is also a one-to-one correspondence between compact connected Klein surfaces (up to equivalence) and algebraic function fields in one variable over R (up to R-isomorphism). 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 (or ∞) on the boundary of ''X''. Given a connected Klein surface ''X'', the set of meromorphic functions defined on ''X'' form a field M(''X''), an algebraic function field in one variable over R. M is a

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

and yields a duality (contravariant equivalence) between the category of compact connected Klein surfaces (with non-constant morphisms) and the category of function fields in one variable over the reals. One can classify the compact connected Klein surfaces ''X'' up to

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

(not up to equivalence!) by specifying three numbers (''g'', ''k'', ''a''): the

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

''g'' 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

of ''X'' equals 1-''g''.

Examples

Properties

References

Further reading