= Solution
Set
$$
\Gamma(t)=\sigma(t,0)=(3\cos t,3\sin t,t),
\qquad t\in\mathbb R.
$$
Along $v=0$,
$$
\sigma_u=(-3\sin u,3\cos u,1),
\qquad
\sigma_v=(0,0,1),
$$
whose span is the tangent plane of the cylinder $x^2+y^2=9$. Also
$$
\ddot\Gamma(t)=(-3\cos t,-3\sin t,0)
$$
is normal to that cylinder. The same tangency argument proves that $\Gamma$ is a geodesic of $\Sigma_1$. It is defined for every $t\in\mathbb R$, hence complete, and it is disjoint from $\gamma$ because their distances from the $z$-axis are respectively $3$ and $1$.
Back to article page
