  The incompressible Euler equations on a compact Riemannian manifold $(M,g)$
take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p
\\ \mathrm{div}_g u &= 0, \end{align*} where $u: [0,T] \to \Gamma(T M)$ is the
velocity field and $p: [0,T] \to C^\infty(M)$ is the pressure field. In this
paper we show that if one is permitted to extend the base manifold $M$ by
taking an arbitrary warped product with a torus, then the space of solutions to
this equation becomes "non-rigid'"in the sense that a non-empty open set of
smooth incompressible flows $u: [0,T] \to \Gamma(T M)$ can be approximated in
the smooth topology by (the horizontal component of) a solution to these
equations. We view this as further evidence towards the "universal" nature of
Euler flows.
