On the universality of the incompressible Euler equation on compact manifolds

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form

$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$

We show that any quadratic ODE $\partial_t y =B(y,y)$, where $B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$ 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 $\mathbb{R}^n$. 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.