We present a geometric analysis of the incompressible averaged Euler equations for
an ideal inviscid fluid. We show that solutions of these equations are geodesics on the
volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We
prove that for precompact open subsets of ${\mathbb R}^n$, this system of PDEs with
Dirichlet boundary conditions are well-posed for initial data in the Hilbert space $H^s$,
$s>n/2+1$. We then use a nonlinear Trotter product formula to prove that solutions of
the averaged Euler equations are a regular limit of solutions to the averaged Navier-Stokes
equations in the limit of zero viscosity. This system of PDEs is also the model for
second-grade non-Newtonian fluids.