Simple closed geodesics on convexsur faces

Abstract

A geodesic is said to be simple if it does not have any self-intersection point. It will be shown that the shortest closed geodesic is simple on any smooth Riemannian 2-sphere of nonnegative curvature. We will also derive various estimates for lengths of simple closed geodesics, in terms of the diameter D, total area A, and curvature K of a given surface M2. In particular, if we let L be the length of the longest simple closed geodesic on a smooth Riemannian sphere of curvature 0 ≤ K ≤ 1, then 2D ≤ L ≤ A/2. Furthermore, equality L = A/2 holds if and only if M2 is isometric to the unit sphere. Finally, if M2 is a Riemannian sphere with nonnegative curvature, then we find that the isoperimetric inequality A ≤ 8D2/π is useful. © 1992, International Press of Boston, Inc. All Rights Reserved.