Is there a nice geometric interpretation of what a vector field $V$ in $\mathbf{R}^3$, that satisfies $\epsilon_{ijk} V_i \partial_j V_k=0$ is? I.e. if I know a vector field at every point in some region in $\mathbf{R}^3$ is orthogonal to it's curl, can I say anything nice about it geometrically? I know this is somewhat of a vague question, but I can't make it any more concrete at the moment.
Context: This is a necessary and sufficient condition that the velocity vector field of a steady flow in fluid dynamics has to satisfy, in order for Bernoulli's theorem to apply not only on a streamline, but across the streamlines. In textbooks it is usually not treated however, as they usually just state that if the curl is zero, the theorem applies.