I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines on the manifold"? If so, would it be fair to say that the Navier-Stokes equations are partial differential equations which "embed" this manifold in $R^3$?
This would be somewhat analogous to the way that the Poincare Disk Model transforms straight lines in the hyperbolic plane into arcs of circles that are orthogonal to the boundary circle. Except in this case, the "straight lines" are transformed into fluid particle trajectories in $R^3$.