SCHWEITZER 1979 Problem 6

Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space $\mathbb{E}^3$ not lying on the $z$ -axis by the metric tensor $$\left( \begin{array}{ccc} 1 & 0 & 0 0 & 1 & 0 0 & 0 & -\sqrt{x^2+y^2} \end{array} \right),$$ where $(x,y,z)$ is a Cartesian coordinate system $\mathbb{E}^3$ . Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the $(x,y)$ -plane are straight lines or conic sections with focus at the origin

P. Nagy