TITLE: Foliation with leaves which are and are not dense
QUESTION [7 upvotes]: Do there exist a foliation on a closed surface (i.e. real dimension 2) which has a dense leaf and also a leaf which is not dense?

REPLY [11 votes]: No, there doesn't exist such a foliation. The existence of any foliation would mean the Euler characteristic is zero, so the surface must be either a torus or a Klein bottle. Foliations for these surfaces are understood well enough to rule out having both dense and non-dense leaves.
Any foliation will contain a "Reeb component" (for which no leaf is dense) or will be a suspension of a homeomorphism of $S^1$. If a circle homeomorphism has one dense orbit, all orbits are dense, so the situation you describe can't exist.
If you allow surfaces with boundary, you need to consider a Mobius band or annulus, but it's still not possible to have a dense and non-dense leaf: the foliation will have pieces that are either Reeb components or suspensions of interval homeomorphisms, neither of which can have a dense leaf.
For the definition of a Reeb component and complete details, see this page, especially the "Foliations of Surfaces" section:
http://www.map.mpim-bonn.mpg.de/Foliations