On isomorphic classical diffeomorphism groups. I

Abstract:

Let $({M_i},{\alpha _i})$, $i = 1,2$, be two smooth manifolds equipped with symplectic, contact or volume forms ${\alpha _i}$. We show that if a group isomorphism between the automorphism groups of ${\alpha _i}$ is induced by a bijective map between ${M_i}$, then this map must be a ${C^\infty }$ diffeomorphism which exchanges the structures ${\alpha _i}$. This generalizes a theorem of Takens.