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Problem. If $\alpha \notin \mathbb{Q}$ and $\alpha > 0$. Considerthe map \begin{align} f: \mathbb{R} &\longrightarrow S^1 \times S^1 \\ t &\longmapsto (e^{i2\pi t}, e^{i2\pi \alpha t}) \end{align} Show that: $f$ is injective immersion and $f(\mathbb{R})$ is dense in $S^1 \times S^1$

For injective immersion: I think we can define $g: \mathbb{R} \rightarrow S^1 \times S^1$ with $t \mapsto e^{i2\pi t}$, it's clearly immersion; similarly, define $h: \mathbb{R} \rightarrow S^1 \times S^1$ with $t \mapsto e^{i2\pi \alpha t}$. Hence, $f$ is immersion, for injective, I want to use queotient map $f: \mathbb{R} \rightarrow\mathbb{R}/\sim$.

For dense: I try use $\mathbb{R}$ is connected and $f: \mathbb{R} \longrightarrow S^1 \times S^1$ continuous, so $f(\mathbb{R})$ is connected, but i think it's not correct, because if it's true, $f(\mathbb{R})$ will be a regular submanifold, that's contradiction with $\alpha \notin \mathbb{Q}$, How to prove it?

2403 Scott
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  • Your proof that $f$ is an immersion doesn't really make sense. The maps $g$ and $h$ as defined do not have the correct codomain. Try to compute $df_x: \mathbb{R}\to \mathbb{C}^2$ by considering $f$ as a map $\mathbb{R}\to \mathbb{C}^2$.

    As for the density portion, what does it mean for a subset of $S^1\times S^1$ to be dense?

     – J.V.Gaiter Feb 07 '24 at 19:55
  • @J.V.Gaiter, I see. Or I can define $g: \mathbb{R} \rightarrow \mathbb{R}^2$ with $t \mapsto (t, \alpha t)$ and $h: \mathbb{R}^2 \rightarrow \mathbb{T}^2$ with $(t, \alpha t) \mapsto (e^{i2\pi t}, e^{i2\pi \alpha t})$, $g$ is clearly embedding and $h$ is local diffeomorphism, $f=h \circ g$, so $f$ is immersion. That's right? – 2403 Scott Feb 08 '24 at 06:08
  • You're on the right track. A map $\mathbb{R}^2\to T^2$ should take inputs $(t,s)$ rather than $(t,\alpha t)$, right? – J.V.Gaiter Feb 08 '24 at 12:41
  • @J.V.Gaiter You're right. – 2403 Scott Feb 08 '24 at 12:47
  • How does connectedness tell you that $f(\Bbb R)$ would be a regular submanifold? For denseness, you're going to need to make a serious argument. What does the set ${e^{2\pi i\alpha n}: n\in\Bbb Z}$ look like in $S^1$? – Ted Shifrin Feb 08 '24 at 16:25

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