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Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^* \omega =0$ where $S$ is the singular points of $d^* \omega$?(The operator $d^*$ is the adjoint of the exterior derivative $d$).

In dimension $2$ the answer is negative since it leads to a gradient vector field, but a gradient vector field has no any closed orbit.

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^* \omega =0$ where $S$ is the singular points of $d^* \omega$?(The operator $d^*$ is the adjoint of the exterior derivative $d$).

In dimension $2$ the answer is negative since it leads to a gradient vector field, but a gradient vector field has no any closed orbit.

As another question: Is there a name or terminology for the following compatibility of symplectic structure and Riemannian structure?

$$d^*\omega \wedge dd^* \omega=0$$

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^* \omega =0$ where $S$ is the singular points of $d^* \omega$?(The opertor $d^*$ is the adjoint of the exterior derivative $d$).

In dimension $2$ the answer is negative since we lead to a gradient vector field, but a gradient vector field has no any closed orbit.

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^* \omega =0$ where $S$ is the singular points of $d^* \omega$?(The operator $d^*$ is the adjoint of the exterior derivative $d$).

In dimension $2$ the answer is negative since it leads to a gradient vector field, but a gradient vector field has no any closed orbit.

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^* \omega =0$ where $S$ is the singular points of $d^* \omega$?(The opertor $d^*$ is the adjoint of the exterior derivative $d$).

In dimension $2$ the answer is negative since we lead to a gradient vector field which has no any closed orbit.

Is there a symplectic manifold $(M,\omega)$ equipped with a Riemannian metric such that $d^* \omega \wedge dd^*\omega=0$ and there is a compact leaf for the foliation of $M\setminus S$ defined by $d^* \omega =0$ where $S$ is the singular points of $d^* \omega$?(The opertor $d^*$ is the adjoint of the exterior derivative $d$).

In dimension $2$ the answer is negative since we lead to a gradient vector field, but a gradient vector field has no any closed orbit.