Let $M$ be a closed and connected manifold, $H:T^*M\times \mathbb{ R} / \mathbb{ Z} \to \mathbb{ R}$ a Tonelli $1$-periodic Hamiltonian and $\mathcal{ L} \subset T^*M$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $\mathcal{ L}$ is invariant by the time-one map of $H$, then $\mathcal{ L}$ is a graph over $M$. An interesting consequence in the autonomous case is that in this case, $\mathcal{ L}$ is invariant by all the time $t$ maps of the Hamiltonian flow of $H$.
Categories: math.DS