= Solution
An allowable parametrisation is a smooth homeomorphism from an open subset of $\mathbb R^2$ onto an open subset of $S$, with <derivative> of rank two. Away from the axis, the rotation orbit has nonzero tangent. A transverse curve supplied by the submanifold theorem, followed by the rotation action, gives
$$
\sigma(u,v)=(f(u)\cos v,f(u)\sin v,g(u)),
$$
with $|v|<\pi$ and $(f',g')\ne(0,0)$.
For a ruled parametrisation, regularity is exactly
$$
\psi_s\times\psi_t=(a'+tb')\times b\ne0.
$$
Rotate and translate along the axis so the specified ruling is
$$
L(t)=(d,st,ct),\qquad d>0.
$$
Here $s\ne0$ because the line is not parallel to the axis, and $c\ne0$: if $c=0$, rotating the horizontal tangent line makes the ruled parametrisation singular at its closest point. Rotating $L$ gives
$$
x^2+y^2=d^2+\frac{s^2}{c^2}z^2,
$$
a <one-sheet hyperboloid>. Rotation invariance puts this whole surface in $\Sigma$. Connectedness and the fact that a complete embedded hyperboloid cannot be a proper subset of another connected embedded surface force equality. Rescaling radial and axial coordinates gives a diffeomorphism with $x^2+y^2=1+z^2$.
Back to article page
