Example of a foliation where leaves are not embedded

Question

In the definition of a foliation, the leaves are only required to be immersed and not embedded. I haven't quite found an example of a foliation where the leaves are NOT embedded, could someone please help?

Answer 1

A simple example is the foliation whose leaves are dense curves on the torus.

Viewing the torus $T^2\cong S^1\times S^1$ as the quotient $T^2=\mathbb{R}^2/\mathbb{Z}^2$, any constant vector field on $\mathbb{R}^2$ will descend to a vector field on the quotient. Corresponding to each such vector field (except zero) is a $1$-dimensional foliation whose leaves are tangent to it. If this vector field has rational slope, the leaves will be embedded and periodic (i.e. embedded copies of $S^1$). Otherwise, they will be dense and nnonperiodic (i.e. immersed, but not embedded copies of $\mathbb{R}$). This question discusses why these curves are not embedded.