I am trying to solve the following questions.
Let $f:\mathbb R^3\rightarrow\mathbb R$ such that $f(x,y,z)=(x^2+y^2-4)^2+z^2-1$.
- a) Show that $0$ is a regular value of $f$ and that $f^{-1}(0)$ diffeomorphic to $T^2=S^1\times S^1$ $\qquad$
- b) Define an explicit embedding $g:T^2\rightarrow\mathbb R^3$ and find its parametrizations. $\quad$
My attempt:
a) $D_f(x,y,z)=(4x^3-12xy^2,4y^3-12x^2y,2z)$ and so for every point $p\neq 0$, we get that $D_f(x,y,z)$ is of full rank and in particular also for the set of all points such that $f(p)=0 $. Hence, by definition, $0$ is a regular value of $f$. Moreover, $f^{-1}(0)$ is a smooth manifold of dimension 3-1=2. Now I want to find a diffeomorphism between $f^{-1}(0)$ and $T^2$ using that $f^{-1}(0)$ is similiar to the torus cartesian parametrization $(R-\sqrt{x^2+y^2})^2+z^2=r^2$. Unfortunately I didn't find out how to connect between them. Any suggestions? $\quad$
b) Define $g:R^3\rightarrow R$ such that $g(x,y,z)=(R-\sqrt{x^2+y^2})^2+z^2=r^2$. It is obvious that $g^{-1}(0)=T^2$ is a smooth manifold of dimension 2. Is this proof sufficient to show that $g$ is an embedding? If so, can I use the torus paramatrization as the embedding parametrizations?