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.
Similarly, there is a one-parameter family of inequivalent Klein surface structures (compact, with boundary) defined on the Möbius strip.
[3] Suppose Σ is a (not necessarily connected) Riemann surface and τ:Σ→Σ is an anti-holomorphic (orientation-reversing) involution.
There is a one-to-one correspondence between smooth projective algebraic curves over the reals (up to isomorphism) and compact connected Klein surfaces (up to equivalence).
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 (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.