Show the image of the map $\Phi_a(t): \mathbb{R}\rightarrow S^1\times S^1$ defined by $\Phi_a(t)=(e^{it}, e^{iat})$ is a regular submanifold of $S^1\times S^1$ if and only if $a$ is rational. I managed to show the implication from right to left but I'm not sure how to proceed to prove the other one.
I know that if $\Phi_a(\mathbb{R})$ is a regular submanifold of $S^1\times S^1$ then the inclusion map must be an embedding, i.e. it is an homeomorphism in its image and it is an imersion. I guess that being a homeomorphism doesn't tell me anything about what $a$ should be, so the answer must lie in the "imersion" part, but I don't know how to go on.
Any help would be appreciated! Thanks