  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*} We show that any quadratic ODE $\partial_t
y = B(y,y)$, where $B : {\bf R}^n \times {\bf R}^n \to {\bf R}^n$ is a
symmetric bilinear map, can be linearly embedded into the incompressible Euler
equations for some manifold $M$ if and only if $B$ obeys the cancellation
condition $\langle B(y,y), y \rangle = 0$ for some positive definite inner
product $\langle,\rangle$ on $ {\bf R}^n$. This allows one to construct
explicit solutions to the Euler equations with various dynamical features, such
as quasiperiodic solutions, or solutions that transition from one steady state
to another, and provides evidence for the "Turing universality" of such Euler
flows.
