Since Arnold in 1966, the Euler equations for ideal fluids in a domain have had a well-known geometric interpretation as geodesics on the group of volume-preserving diffeomorphisms of this domain. This allows one to apply the methods of Riemannian geometry to the study of ideal fluid flows. In this presentation, we will give an introduction to this geometric framework and study the existence of conjugate points along the geodesics representing ideal fluid flow. The existence of conjugate points indicates whether a perturbation of the initial condition of the corresponding flow can lead to the same result as no perturbation at all. We will show how the problem of the existence of conjugate points along a particular family of steady solutions can be reduced to the existence of a negative eigenvalue of a certain operator, and solved using a simple numerical scheme.
