I have proved that $g: \mathbb{R}^1 \rightarrow S^1, g(t) = (\cos 2 \pi t, \sin 2 \pi t)$, is a local diffeomorphism, as well as that
$G: \mathbb{R}^2 \rightarrow S^1 \times S^1, G = g \times g$ is a local diffeomorphism.
Now the question is
If $L$ is a line in $\mathbb{R}^2$, the restriction $G: L \rightarrow S^1 \times S^1$ is an immersion.
Since restricted to $L$ is a submanifold of $\mathbb{R}^1$, So I would like to use the conclusion Restriction of an immersion to any submanifold is still an immersion. Is that all? Regardless, the aforementioned problem is not well justified. It would be great if someone could help me check it out.
Thanks.