  A new important relation between fluid mechanics and differential geometry is
established. We study smooth steady solutions to the Euler equations with the
additional property: the velocity vector is orthogonal to the gradient of the
pressure at any point. Such solutions are called Gavrilov flows. Local
structure of a Gavrilov flow is described in terms of geometry of isobaric
hypersurfaces. In the 3D case, we obtain a system of PDEs for an axisymmetric
Gavrilov flow and find consistency conditions for the system. Two numerical
examples of axisymmetric Gavrilov flows are presented: with pressure function
periodic in the axial direction, and with isobaric surfaces diffeomorphic to
the torus.
