Gromov used convex integration to prove that any closed
orientable three-manifold equipped with a volume form admits three
divergence-free vector fields which are linearly independent at every
point. We provide an alternative proof of this using geometric
properties of eigenspinors in three dimensions. In fact, our proof
shows that for any Riemannian metric, one can find three
divergence-free vector fields such that at every point they are
orthogonal and have the same non-zero length.