Let $\omega_0$ and $\omega_1$ be symplectic structures on a compact manifold $M$ . If there is a path in the space of symplectic structures of a fixed DeRham cohomology class connecting $\om_0$ and $\om_1$ (in particular $\om_0$ and $\om_1$ must have the same class), then $(M,\omega_0)$ and $(M,\omega_1)$ are symplectomorphic, by a symplectomorphism isotopic to the identity.