Closed Geodesics on Weingarten Surfaces with κ1/κ2 = c > 0

In 2006, Alexander proved a result that implies for a Weingarten surface kappa(1)/kappa(2) = c > 0, if n is the number of times a closed geodesic winds around the axis of rotation and m is the number of times the geodesic oscillates about the equator, then n/m is an element of [1, 1/root c) when 0 < c < 1 and n/m is an element of (1/root c, 1] when c > 1. In this paper, we present another proof of Alexander's result for the Weingarten surfaces kappa(1)/kappa(2) = c > 0 that is simpler and more direct. Our approach uses sharp estimates of certain improper integrals to obtain the intervals for permissible ratios n/m. We numerically compute a number of closed geodesics for various combinations of (c, n, m) to illustrate the variety of patterns that are possible.