  We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove
that the $L^2$ exponential map on the group of volume-preserving
diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with
sufficiently small swirl. Along the way, we define the notions of axisymmetric
and swirl-free diffeomorphisms of any manifold with suitable symmetries and
show that such diffeomorphisms form a totally geodesic submanifold of infinite
$L^2$ diameter inside the space of volume-preserving diffeomorphisms whose
diameter is known to be finite. As examples we derive the axisymmetric Euler
equations on $3$-manifolds equipped with each of Thurston's eight model
geometries.
