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!
May be this link to WIKIPEDIA covers entirely my question...
https://en.wikipedia.org/wiki/Surface_(topology)#Classification_of_closed_surfaces
Orientability should also be taken into account...