  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 (inspired by Seiberg-Witten theory) 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.
