## Tag Archives: Reeb

### Symplectic and contact manifolds

Let $(M,\omega)$ be a symplectic manifold. It said to be exact if $\omega=d\lambda$ for some one-form $\lambda$ on $M$.

(1) If $\omega=d\lambda$ is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. $X$ such that $\lambda=i_X\omega$. Then we have $\lambda(X)=\omega(X,X)=0$, and $L_X\lambda=i_X d\lambda+d i_X\lambda=i_X\omega +0=\lambda$, and $L_X\omega=d i_X\omega=d\lambda=\omega$.

(2) Suppose there exists a vector field $X$ on $M$ such that its Lie-derivative $L_X\omega=\omega$ (notice the difference with $L_X\omega=0$). Then Cartan’s formula says that $\omega=i_X d\omega+ di_X\omega=d\lambda$, where $\lambda=i_X\omega$. So $\omega=d\lambda$ is exact, and $L_X\lambda=i_Xd\lambda+di_X\lambda=i_X\omega+0=\lambda$.