On the cut loci of a von Mangoldt's surface of revolution
Minoru Tanaka · Journal of the Mathematical Society of Japan · 1992
It is a classical problem to investigate the behavior of a geodesic on a surface of revolution, or more generally on a surface of Liouville.A pioneer- ing and beautiful work in this field was done by von Mangoldt in 1881; He investigated the behavior of a geodesic on a hyperboloid of two sheets (or an elliptic paraboloid), which both are surfaces of Liouville.He proved in [7] that the two umbilic points on any hyperboloid of two sheets (or any elliptic paraboloid) are poles and that for any revolutionary hyperboloid of two sheets, the set of poles on the surface is a nontrivial closed ball centered at the unique umbilic point.Note that the set of poles on any surface of revolution with vertex $P$ is a closed ball centered at $p$ .This fact will be Proved in Lemma 1.1.Here a surface of revolution $M$ with vertex $p$ means that $M$ is a complete Riemannian manifold homeomorphic to $R^{2}$ such that the Gaussian curvature $G$ is constant on each metric circle $S_{p}(t):=\{q\in M;d(p, q)=t\},$ $t\in[0, \infty)$ , where $d$ denotes the Riemannian distance function on $M$ .Furthermore, by calculating the elliptic integrals defining geodesics on a revolutionary hyperboloid of two sheets, von Mangoldt explicitly determined the radius of the ball of poles.His results were extended by Elerath almost one hundred years after.By de- fining the class of flattening surfaces of revolution generalizing revolutionary hyperboloids of two sheets and revolutionary Paraboloids, Elerath ([3]) exPlicitly determined the cut locus of each point on a flattening surface of revolution.Recently another extension was made by Maeda; Let $M$ be a non-negatively curved, complete noncompact Riemannian manifold and let $D_{t}$ be the diameter of the metric sphere centered at a point $P$ with radius $t$ .In [6] Maeda proved that the number $\lim\sup_{carrow\infty}D_{t}^{2}/t(=:d_{0})$ does not depend on the choice of the point $P$ and that the diameter of the set of poles on