questions 1 and 2 are negative. Namely, there are two homotopy classes of symplectic forms on ${{\mathbb R}}^{2n}$, one for each orientation, and in [@GROMOV1] Gromov constructed a symplectic form ${{\omega}}$ on ${{\mathbb R}}^{2n}$ (for $n\ge 2$) such that $({{\mathbb R}}^{2n},{{\omega}})$ is not symplectomorphic to any open subset of $({{\mathbb R}}^{2n},{{\omega}}_0)$ with the standard symplectic form (see also [@BP] and [@MS1 Example 13.8]). By contrast, if ${{\omega}}$ is a symplectic form on ${{\mathbb R}}^4$ that agrees with the standard symplectic form ${{\omega}}_0$ at infinity, then $({{\mathbb R}}^4,{{\omega}})$ is symplectomorphic to $({{\mathbb R}}^4,{{\omega}}_0)$ by another theorem of Gromov (see [@GROMOV1] and [@MS2 Theorem 9.4.2]).

\[ex:Riemann\]Let $M$ be a closed orientable two-manifold. Then the space ${{\mathcal S}}_a$ is nonempty and convex for every nonzero cohomology class $a\in H^2(M;{{\mathbb R}})$. Since $M$ admits an orientation reversing diffeomorphism, questions 1, 2, and 3 have positive answers.

In higher dimensions the uniqueness problem for symplectic forms on closed manifolds does not reduce to topological obstruction theory. There is often a dramatic difference between the space of nondegenerate two-forms and the space of symplectic forms, as the following examples show.

\[ex:McDuff\]This example is due to McDuff [@MCDUFF3] (see also [@MS2 Theorem 9.7.4]). Here the set ${{\mathcal 