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 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.
