  Two fluid configurations along a flow are conjugate if there is a one
parameter family of geodesics (fluid flows) joining them to infinitesimal
order. Geometrically, they can be seen as a consequence of the (infinite
dimensional) group of volume preserving diffeomorphisms having sufficiently
strong positive curvatures which `pull' nearby flows together. Physically, they
indicate a form of (transient) stability in the configuration space of particle
positions: a family of flows starting with the same configuration deviate
initially and subsequently re-converge (resonate) with each other at some later
moment in time. Here, we first establish existence of conjugate points in an
infinite family of Kolmogorov flows - a class of stationary solutions of the
Euler equations - on the rectangular flat torus of any aspect ratio. The
analysis is facilitated by a general criterion for identifying conjugate points
in the group of volume preserving diffeomorphisms. Next, we show non-existence
of conjugate points along Arnold stable steady states on the annulus, disk and
channel. Finally, we discuss cut points, their relation to non-injectivity of
the exponential map (impossibility of determining a flow from a particle
configuration at a given instant) and show that the closest cut point to the
identity is either a conjugate point or the midpoint of a time periodic
Lagrangian fluid flow.
