Path on the torus

Question

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?

Answer 1

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.

Answer 2

Let $\alpha \in \mathbb{R}$ be irrational then we know the set $\{m+n\alpha|m,n \in \mathbb{Z} \}$ is dense in $ \mathbb{R}$. Let $x,y \in \mathbb{R}$ and for given $\delta >0 $ there exists $m,n\in \mathbb{Z}$ such that $|x \alpha-y+m+n\alpha|< \delta$. We have $||(e^{2 \pi ix},e^{2 \pi iy})-(e^{2 \pi i(x+n)},e^{2\pi i(x+n)\alpha} )||$=$|1-e^{2 \pi i(x \alpha-y +m+\alpha n)}| $. Since exponential function is continuous at zero for given any $\epsilon >0$ we can bound the quantity with that. Hence, it is dense.