Geodesic trapped in one half of the unit one-sheet hyperboloid
= Geodesic trapped in one half of the unit one-sheet hyperboloid
For a unit-speed geodesic on $x^2+y^2=z^2+1$, Clairaut's constant $c$ gives
$$
\dot z^2=\frac{z^2+1-c^2}{1+2z^2}.
$$
If $c>1$, the geodesic has a turning point at $z=\sqrt{c^2-1}$ and remains entirely in one of the regions $z>0$ or $z<0$.