The Euler equations describe the dynamics of an inviscid and incompressible fluid on a Riemannian manifold. In this talk, I will introduce an approach to these equations based on taking the ambient metric as a new variable, and a connection with contact geometry proposed by Sullivan, Etnyre, and Ghrist. Using this relation, it is possible to obtain steady Euler flows with rich dynamical behavior. In particular, I will explain the recent construction of a steady Euler flow in the three-sphere capable of universal computation. This work is based on joint work with Eva Miranda, Daniel Peralta-Salas, and Francisco Presas.