Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

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 f lows 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.