What can you say about injection, immersion, embedding for the torus?

Question

Define $\varphi_a: \mathbb{R} \to T$ where $T = S^1 \times S^1$ is the torus via$$\varphi_a(x) = (e(x), e(ax)),\text{ }e(x):=e^{2\pi i x}$$and $a > 0$ is some parameter. Determine for which $a$ the map $\varphi_a$ is periodic. What can we say about $\varphi_a$ in terms of injective, immersion, embedding?

Answer 1

The map $\varphi_a$ is periodic if and only if $a$ is rational. Suppose $a = p/q$ is rational, where $\gcd(p,q) = 1$. Then $$\varphi_a (x+q) = (e^{2\pi i (x+q)} , e^{2\pi i a(x+q)}) = (e^{2\pi i x} e^{2\pi i q}, e^{2\pi i ax} e^{2\pi i p}) = (e(x), e(ax)) = \varphi_a (x)$$ so $\phi_a$ is periodic. Suppose $\phi_a$ is periodic with period $p$. Then $$\varphi_a (x+p) = \varphi_a (x) \implies e^{2\pi i p} = 1,\, e^{2\pi i ap} = 1,$$ $$\cos(2\pi p) = 1 \implies p \in \mathbb{Z},$$ $$\cos(2\pi a p) = 1 \implies a = \frac{k}{p},\, k\in\mathbb{Z},$$ so $a$ is rational.

With regards to talking about $\varphi_a$ in terms of injection, immersion, embedding, the crucial thing is really the density of the image of $\varphi_a$ for irrational $a$. Note, however, that the image is of course very far from being all of the $2$-torus. So the map is only an immersion. In fact, $\varphi_a$ in that case is a beautiful example of an injective immersion, which takes the line $\mathbb{R}$ and wraps it densely around the torus $T$. But this image is therefore not a $1$-dimensional smooth submanifold of $\mathbb{R}^3$, and $\varphi_a$ is therefore not an embedding for irrational $a$. For rational $a$ the map $\varphi_a$ also fails to be an embedding by lack of injectivity. But this is really a dumb reason, since the image is clearly a $1$-dimensional manifold we would still be able to view $\varphi_a$ as an embedding. To fix this, we simply mod out the period, by viewing $\varphi_a$ as a map from $S^1 \to T$. Then we again obtain an embedding, but from a one manifold into another. There is no mystery about what this is: a homeomorphism onto its image which is also an immersion.