Realizing Keplerian orbits as geodesics on a surface of revolution

Abstract

It is well known that in Newtonian physics, the gravitational trajectories in a central gravitational field are conic sections with one focus at the central mass. This is known as Kepler's First Law. We consider those gravitational trajectories that lie in a fixed plane P (perhaps thought of as an ecliptic plane), and investigate this question: to what extent can P be embedded as a surface of revolution S inside an abstract three-dimensional Euclidean space E in such a way that gravitational trajectories are mapped to geodesics on S, subset of E?