It is not easy to find a nontrivial pole even on a surface of revolution, unless the latter has a nonpositive Gaussian curvature. We shall give a necessary and sufficient condition for a surface of revolution to have nontrivial poles. The proof is achieved by obtaining Jacobi fields along any geodesic (see [101]). The method is found in a classical work of von Mangoldt [59]. We will also determine the cut loci of a certain class of surfaces of revolution containing well-known examples: the two-sheeted hyperboloids of revolution and the paraboloids of revolution (see [102]). von Mangoldt proved in [59] that any point on a two-sheeted hyperboloid of revolution is a pole if the point is sufficiently close to the vertex. Furthermore, he proved in [59] that the two umbilic points of a two-sheeted hyperboloid are poles and that the poles of any elliptic paraboloid are the two umbilic points. These surfaces are typical examples of a Liouville surface. A definition of global Liouville surfaces was introduced by Kiyohara in [44]. See also [40] for poles on noncompact complete Liouville surfaces.

Properties of geodesics

A surface of revolution means a complete Riemannian manifold (M, g) homeomorphic to R2 that admits a point p such that the Gaussian curvature of M is constant on S(p, t) for each positive t. The point p is called the vertex of the surface of revolution.

Throughout this chapter (M, g) denotes a surface of revolution and p denotes the vertex of the surface.