Consider the torus $T^2=S^1\times S^1$(where $S^1$ is the unit circle centered at $0$ in $\mathbb C$).
- Define a smooth structure on $S^1$ and $T^2$. ($\checkmark$)
- Let $f:T^2 \rightarrow S^1$ be defined as $f(z,w)=zw$. Compute the rank of $f$ at each point. ($\checkmark$)
- Let $N=f^{-1}(1)$. Is $N$ a submanifold of $T^2$? Describe $N$. (?)
- Consider $g:S^1 \rightarrow T^2$ defined as $g(z)=(z,z)$. Is this function an immersion? Is the image an immersed submanifold, embedding, regular submanifold? (?)
Despite "knowing" all the definitions of the question, I am not used to solve such differential geometry questions. I have very few thoughts on it but exactly need help. Any hints, "more than hint"s or solutions will surely be appreciated.
Edit: I more or less did the first part by writing out their charts etc. and the second part by finding out coordinate functions and this way formulating $f$ and then computing its rank. However, I still lack for some help to understand the latter two (submanifold issues) clearly.
A hint for the third part: By the Regular Value Theorem, all you need to show is that $1$ is a regular value of $f$.
A hint for the fourth part: Remember that $g$ is an immersion iff its derivative is 1-1 at every point, i.e. iff the rank of $g$ is always $1=\dim{S^1}$.