The irrationally sloped line in the torus is a smooth, simple, immersion of $\mathbb{R}$ into $T^2$. This line is even dense in $T^2$ (thus not all injective immersions are embeddings).
When I tried to concoct a curve with similar properties in $\mathbb{R}^2$ I found it much more difficult and eventually threw up my hands in defeat.
So I'm curious:
- Is there a smooth, simple, immersed, dense curve in $\mathbb{R}^2$ (or $\mathbb{R}^n$)?
- If not, is there a smooth, simple, immersed, dense curve in an open subset of $\mathbb{R}^2$ (or $\mathbb{R}^n$)?
- In general, which smooth manifolds $M$ admit a smooth injective immersion $\gamma:\mathbb{R}\to M$ such that $\gamma(\mathbb{R})$ is dense in $M$.
An answer to any of these would be extremely enlightening. Thanks in advance.