= Dense immersed cylinder in a three-dimensional torus
{title2=$F(s,t)=(s,t,\alpha s)\bmod\mathbb Z^3,\quad\alpha\notin\mathbb Q$}
The map descends to an injective <immersion> from $\mathbb R\times(\mathbb R/\mathbb Z)$ into the <three-dimensional torus>. Injectivity follows because an integer difference in $s$ with an integer difference in $\alpha s$ must be zero. Its image is dense by an <irrational rotation of the circle>, and is tangent to the commuting independent fields $\partial_x+\alpha\partial_z$ and $\partial_y$. It is a <leaf of a regular foliation>, but cannot be an <embedded submanifold>: in an embedding chart the coordinate plane would be locally closed yet contain a dense ambient subset.
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