Abstract

Consider the 2-sphere in Euclidean 3-space with the usual round metric, where we know the geodesics are arcs of great circles. By rotation we get a 1-parameter family of geodesics through any given geodesic, which turns out to imply that each geodesic is degenerate for the length functional since it then has a non-trivial Jacobi field. However if we change the metric on the 2-sphere to, say, that of a triaxial ellipsoid, all but 3 of these closed geodesics disappear. Perturbing the metric further via adding more “bumps” to the 2-sphere, all geodesics are in fact non-degenerate. In 1970 Ralph Abraham established that on a compact manifold, ‘almost all’ metrics have the property that any geodesic is non-degenerate. This result was then extended to the case of minimal submanifolds of any codimension in 1991 by Brian White to a result now known as the Bumpy Metrics Theorem. In this talk we shall discuss the Bumpy Metrics Theorem, and then some conjectures we have for extending it to the case of singular hypersurfaces.