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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.

Christian Bueno
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  • By "simple," do you mean injectively immersed? – Jack Lee Oct 28 '15 at 00:22
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    If you demand injectivity this is not as obvious to me, though I still guess that it's true for any manifold. Here's a sketch of how to do it non-injectively for $\Bbb R^2$. Enumerate the rationals in $(0,1)$ as $q_n$. Now draw a curve that first traverses $q_1 \times [0,1]$, loops around, and comes back and tranverses $q_2 \times [0,1]$ at twice the speed, then $q_3 \times [0,1]$ at four times the speed, etc... eventually (in finite time) you have traversed a dense subset of $[0,1]^2$. Now put a bunch of these things together to traverse a dense subset of $\Bbb R^2$. –  Oct 28 '15 at 00:40
  • @JackLee Yes. My understanding is that "simple curve" and "injective curve" mean the same thing. Unless "simplicity" does not rule out self-tangency. – Christian Bueno Oct 28 '15 at 01:35

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