Let $f:\mathbb{R^3} \to \mathbb{R}$ $$f(x, y, z)=\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^2-4 R^{2}\left(x^{2}+y^{2}\right) $$ and $R,r \in \mathbb{R} , 0<r<R$
now define :
$Z=\{ (x,y,z) \in \mathbb{R^3} | f(x,y,z)=0\}$
(i)- Prove $Z$ is a smooth manifold.
(ii)-draw set $Z$ in $\mathbb{R^3} $.
(iii)- find coordinate charts on $Z$.
(iv) prove $Z$ is diffeomorphic to $S^1 \times S^1$.
for (i) i think we can use following theorem :
theorem : if $M$ and $N$ are smooth manifolds, $f: M \to N$ is a smooth map and $n \in f(M)$ is a regular value for $f$, then $f^{-1} (\{n\})$ is a smooth submanifold of $M$ of dimension $\dim M - \dim N$.
for (ii) we must compute $Z$ . i think $Z=\{ (x,y,z) \in \mathbb{R^3} | x^2+y^2=R^2,z^2=r^2\}$ so $Z$ is two circle with radios $R$ and $z=-r,+r$
Some hints:
$i).\ $ Fix an arbitrary $p\in Z$ and show there are no critical points for $f$ on $Z.$
$ii).\ $ Consider that $\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^2-4 R^{2}\left(x^{2}+y^{2}\right)=0\Rightarrow $
$z^2=\pm2R\sqrt{x^2+y^2}-(\sqrt{x^2+y^2})^2-(R^{2}-r^{2})$ and compare with the equations and arguments of this post.
$iii).\ $ From $i).,\ $ note that one of $\partial_x(p),\ \partial_y(p),\ \partial_z(p)$ is non-zero, say $\partial_x(p)$. Now, consider $\varphi:(x,y,z)\mapsto (f,y,z)$ defined in some $U(\subseteq \mathbb R^3)$-neighborhood of $p$ and show that $\varphi|_{U\cap Z}$ is a slice chart about $p\in Z.$
$iv).\ $ Apply $ii).$