Title: The WKB method for conjugate points in the volumorphism group

Authors: Stephen C. Preston

Issue: Volume 57 (2008), Issue 7, 3303-3328

Abstract: Using the fact that ideal fluid flow corresponds to geodesics of a Riemannian metric on the volumorphism group, we show that the location of conjugate points along such a geodesic in three dimensions can be found very simply from an ordinary differential equation along any particle trajectory. The basic method is to construct a sequence of sharply-peaked perturbations, using the WKB method. As a consequence, we show that the set of all conjugate points along a geodesic is generally a union of closed intervals; in particular, the conjugate points are not discrete along a geodesic, as they are for two-dimensional ideal fluid flow. Finally we explicitly give the locations of all conjugate points along those geodesics which consist of isometries.