# Darboux theorem for the symplectisation of a contact manifold

I'm wondering if there is some "nice" version of Darboux's theorem that can be applied in the case of a symplectisation of a contact manifold $$(X,\alpha)$$ with the canonical symplectic form. i.e. $$(\Bbb R \times X, d(e^t\alpha))$$, where $$t$$ is the $$\Bbb R$$-parameter.

More specifically, I'd like to know if we can assume that the first 2 coordinates in a Darboux neighbourhood are given by the Reeb vector field and the canonical Liouville vector field $$\partial_t$$.

I tried looking around but wasn't really able to find anything

## 1 Answer

This is not possible. Let $$\omega = d(e^t\alpha)$$, and suppose $$(x^1,\dots,x^n,y^1,\dots,y^n)$$ are any Darboux coordinates for $$\omega$$. Then the vector fields $$\partial _{x^i}$$ and $$\partial_{y^i}$$ are all symplectic (or equivalently locally Hamiltonian), which means that the Lie derivative of $$\omega$$ with respect to each of them is zero. But $$\partial_t$$ is not symplectic, because $$\mathscr L_{\partial_t}\omega = d(i_{\partial_t}\omega) = d(e^t\alpha) = \omega.$$