A generalization of Gradient vector fields and Curl of vector fields

Question

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

What is a dynamical interpretation for vanishing $X^{*} (\omega)$, the pullback 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 "gradient vector fields" and Curl vector field, respectively.

Answer 1

This is equivalent to the fact that the image of the 1-form $X^\flat$ in $T^*M$ is a Lagrangian submanifold; equivalently, the 1-form $X^\flat$ is closed. So, locally, $X^\flat = df$ for some function $f$, or, $X=\operatorname{grad}^g(f)$.