 Solution

SetAlong $v=0$,whose span is the tangent plane of the cylinder $x^2+y^2=9$. Alsois 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 $1$.