Prove that the following map is smooth and analyse its image as the real number $\alpha$ varies (how does it look like? Is it a submanifold?)
\begin{align*}
f_{\alpha}:\mathbb{R}&\to\mathbb{S}^1\times\mathbb{S}^1\\
t&\mapsto (e^{2\pi it}, e^{2\pi \alpha it})
\end{align*}
I've managed to prove that $f_{\alpha}$ is smooth and that $\text{Im}(f_{\alpha})$ is a closed curve on the torus $\Leftrightarrow \alpha\in\mathbb{Q}$. But I'm having trouble to formalize whether or not it is a submanifold of the torus. I actually don't know if it is true when the image is not a closed curve. How can I solve this?
Best Answer
Should it be $f_{\alpha}(t)=(e^{2\pi i t}, e^{2\pi \alpha i t})$ ?
If so, sure it is a manifold when $\alpha\in\mathbb Q$ (with the proper identification it is simply a curve on $\mathbb R^2$. In this case, it is a curve that spirals around the torus until it closes, much like in the figure:
When $\alpha\not\in\mathbb Q$ the curve is dense in the torus.
Since rational numbers and irrational numbers are dense, it is difficult to say much more about what happens when $\alpha$ varies.