Let $ M$ be a smooth Riemannian manifold. The Riemannian metric enable us to equipe the tangent bundle $ TM$ with a symplectic structure $ \omega$ , which is the pull back of the standard symplectic $ 2$ form of the cotangent bundle. Let $ X$ be a vector field on $ M$ . Then $ X:M \to TM$ is an smooth map.

What is a dynamical interpretation for vanishing $ X^{*} (\omega)$ , the pull back of $ \omega$ via $ X$ . Is there a name for this property? For a given vector field, what kind of dynamical obstructions exist to have a Riemannian metric with vanishing the above $ 2$ form?

In $ 2$ and $ 3$ dimensional Euclidean space the above $ 2$ form is closely related to “gradien vector fields” and Curl vector field, repectively.