determining whether a function from the torus to R4 is an embedding

Question

I am not sure whether the following question requires me to find explicit charts or whether there is more theoretical machinery that may be used.

Starting from the description of $S_1$ as the unit circle in $\mathbb{R}^2$, we can identify the $2$-torus $T_2 = S_1 \times S_1$ as $T_2 = \{(x,y,z,w) \in \mathbb{R}^4 : x^2 + y^2 = 1, z^2 + w^2 = 1\}$, and we denote by $i$ a map from $T_2$ to $\mathbb{R}^4$ the corresponding embedding. Consider the function $f$ from $\mathbb{R}^4$ into $\mathbb{R}^3$ given by $f(x,y,z,w) = (x(2+z),y(2+z),w)$ and denote the composition of $f$ with $i$ by $F$.

a) Is $F$ smooth?

b) Is $F$ injective?

c) Is $F$ an immersion

d) Is $F$ an embedding?

I feel that any determination of the above should require some mention of charts because $T_2$ is a $2$-manifold, not a $3$-manifold.

Answer 1

$\textbf{Hint 1}$: This function is certainly injective due to the last coordinate. If you want to check for smoothness, note that $f$ is smooth, the embedding $i$ is smooth and so by chain rule $D_p(f \circ i) = D_{i(p)}f \circ D_p$. So that gives smoothness for $F = f \circ i$.

$\textbf{Hint 2}$: If you want to check about immersion properties then you see that $F$ is an immersion if and only if $f$ is an immersion since $i$ is trivally an immersion. So just check to see if $D_q f$ is injective. You have the definition of $f$. Similarly, $F$ is an embedding if and only if $f$ is an embedding.

I hope this helps.