Examples of non-Riemann surfaces? While studying Complex Analysis, I have come across Riemann Surfaces:
http://mathworld.wolfram.com/RiemannSurface.html
Can anyone please provide some examples of non-Riemannable surfaces? Thanks a lot!
 A: A Riemann surface is a $1$-dimensional complex manifold, i.e. a surface that admits a complex structure. The complex structure on a Riemann surface induces a canonical orientation. So, in particular, a nonorientable surface cannot be a Riemann surface. Examples of nonorientable surfaces are the real projective plane and the Klein bottle.
A: Your question doesn't really make a lot of sense.  I'll explain why.
"Riemann" isn't an adjective that's used to classify surfaces.  That is, there's not some classification of surfaces into "Riemann surfaces" and "non-Riemann surfaces".
Instead, a Riemann surface is a surface together with some extra structure.  In particular, a Riemann surface is a surface with a complex structure,  which lets you define things like holomorphic functions on the surface.
Asking for a surface that isn't a Riemann surface is a lot like asking for a set that isn't a group.  A group isn't a special kind of set -- it's a set that has been endowed with extra structure, namely a binary operation satisfying certain axioms.  Some sets can be a group in several different ways, possibly using several different binary operations.  Also, some sets (e.g. the empty set) can't be given the structure of a group.  Finally, there's lots of sets that don't have a "natural" or "obvious" group structure, but could be made into a group if you define an appropriate binary operation.
Typical Riemann surfaces include:


*

*The Riemann sphere

*Open subsets of the complex plane

*Covers of open subsets of the complex plane or other Riemann surfaces

*Quotients of the complex plane by lattices

*Hyperbolic surfaces, which can be described as quotients of the unit disk by groups of Möbius transformations.

*Nonsingular surfaces in $\mathbb{C}^n$ (or $\mathbb{CP}^n$) defined by polynomial equations (or more generally equations involving holomorphic functions).  For example, every complex elliptic curve is a Riemann surface.


In each case, the way that the surface is constructed gives it a natural complex structure.  Other ways of making surfaces (e.g. surfaces you find in $\mathbb{R}^n$) often don't come with a complex structure, so they aren't Riemann surfaces unless you endow them with one.  Moreover, some surfaces (such as a torus) can be endowed with a complex structure in several non-equivalent ways.
Finally, as Henry T. Horton points out, non-orientable surfaces cannot be given a complex structure, since holomorphic maps are always orientation-preserving.  Every compact orientable surface can be given a complex structure, though in some cases there are several possibilities which lead to different Riemann surfaces.
