Any two dimensional orientable manifold is complex Let $M$ be a $2$-dimensional orientable manifold. Is $M$ a Riemann surface? If it is true, how can I show it? Thank you very much.
existence of complex structure in dimension 2 Does an orientable closed 2 manifold always admit a complex structure?﻿  I know that an almost complex doesn' t necessarily give rise to a complex structure(the almost complex structure should be integrable) I am studying 2 manifolds over R, and I wonder if an almost complex structure always exist? This almost structure may be chosen integrable?  I ask because there are certain known results on the different possible geometries in dimension 2.I don' t know exactly how these manifolds (closed and orientable) are classifyied, but I know there are three cases. May be I am confused, and the complex structure is irrelevant to this classification. Some help please!
When can you build up all representations from the fundamental and antifundamental ones? Under what conditions can you determine all representations of a Lie algebra from the fundamental and antifundamental ones using just the tensor product, direct sum and Clebsch-Gordan decomposition? I think this is true for $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{C})$, or at least that's what physics books lead me to believe!   Is it true in general for semisimple Lie algebras?