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While studying Complex Analysis, I have come across Riemann Surfaces:

Can anyone please provide some examples of non-Riemannable surfaces? Thanks a lot!

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Do you mean just any surface or does it have to have special properties? – Gregor Bruns Jun 3 '12 at 23:24
@GregorBruns : yes, any! ^_^ – ltt Jun 3 '12 at 23:26
up vote 17 down vote accepted

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.

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In particular, it is useful to recall that complex conjugation of the complex plane is not holomorphic, and see how this relates to orientation. This of course relates to the condition that transition maps are biholomorphisms. – AnonymousCoward Jun 3 '12 at 23:35

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.

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Yep. If you wanted a name for the property "can be given the structure of a Riemann surface," I guess you could say "Riemannable surface." – Qiaochu Yuan Jun 4 '12 at 2:13
Thanks so much! – ltt Jun 5 '12 at 8:19

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