On the Volumorphism Group, the First Conjugate Point is Always the Hardest

We find a simple local criterion for the existence of conjugate points on the group of volume-preserving diffeomorphisms of a 3-manifold with the Riemannian metric of ideal fluid mechanics, in terms of an ordinary differential equation along each Lagrangian path. Using this criterion, we prove that the first conjugate point along a geodesic in this group is always pathological: the differential of the exponential map always fails to be Fredholm.Much of this work was completed while the author was a Lecturer at the University of Pennsylvania. The author is grateful for their hospitality.