Restriction of viscosity solutions of the Hamilton-Jacobi equation to a submanifold M of $\mathbb{R}$^{d}

We show in this article in what sense viscosity solutions of the Hamilton-Jacobi equation can be restricted to a submanifold M of $\mathbb{R}$^{d}. We treat in this article the case of M\times$\mathbb{R}$^{d} being invariant by the Hamiltonian vector field.