Examples of non-Riemann surfaces?

Question

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!

Answer 1

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.

Answer 2

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.