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.


1 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, $X=\operatorname{grad}^g(f)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.