Let $f: (0, infty) times mathbb{R} to mathbb{R}$ be continuously differentiable, and $nabla f(x) ≠ 0$ on the set $M = f^{-1}(0)$. (Which means that $M$ is a 1-dimensional manifold of the $mathbb{R}^2$.) The rotation of $M$ (considered as a subset of the $x z$-plane) around the $z$-axis may now be given by
$$ R := {(x, y, z) in mathbb{R}^3: (x, y) ≠ (0, 0), fleft(sqrt{x^2 + y^2}, zright) = 0} $$
(1) I first want to show that $R$ is a hypersurface in $mathbb{R}^3$ (that is, a 2-dimensional manifold).
(2) Next, let $M$ also be the trace of a regular continuously differentiable curve $gamma = pmatrix{gamma_1 \ gamma_2}: I to mathbb{R}^2$ (with $I subseteq mathbb{R}$ being an open interval), that is, we have that $dot{gamma}(t) ≠ 0$ for all $t in I$. I want to find an immersion $Gamma: I times mathbb{R} to mathbb{R}^3$ which has $R$ as image.
(3) Finally, the rotation of the circle line
${(x, 0, z) in mathbb{R}^3: (x – a)^2 + z^2 = r^2}, 0 < r < a$
gives us a torus $T$ within $mathbb{R}^3$. I now want to show that $T$ is a hypersurface in $mathbb{R}^3$, and want to write $T$ as an immersion.
My attempt so far: Regarding (1): If we define $g: mathbb{R}^3 backslash {(0, 0, z) in mathbb{R}^3; z in mathbb{R} } to mathbb{R}, g(x, y, z) = f left(sqrt{x^2 + y^2}, z right) $, then I have a function that satisfies the property that for any open neighborhood $U_a$ of $a in R$, $R cap U_a = {x in U_a: f(x) = 0}$. On the other hand, $nabla f(x) ≠ 0$ for $x in f^{-1}(0)$, so $J_x f$ (the Jacobean matrix of $f$) has rank $1$. I believe this is already enough to show the statement given in (1)?
I don’t really know what to do about (2) and (3). I know that an immersion is a (continuously differentiable) curve from $mathbb{R}^k$ to $mathbb{R}^n$ that has a Jacobean matrix of rank $k$. Now since the curve $gamma$ given in (2) already has $M$ as a trace, I would need to cleverly construct a new curve that somehow utilizes $gamma$. I so far haven’t really found a way, though.