Suppose that $a>0$ and that $S \subset \mathbb{R}^{3}$ is the half-cone defined by $z^{2}=a\left(x^{2}+y^{2}\right)$, $z>0$. By using an explicit smooth parametrization of $S$, calculate the curvature of $S$.

Describe the geodesics on $S$. Show that for $a=3$, no geodesic intersects itself, while for $a>3$ some geodesic does so.