Symplectomorphisms of $S^2$ It is well-known that $S^2$ is a Kähler manifold with complex structure $J$ (as the Riemann sphere), Kähler metric $g$ (induced from $S^2\subset\mathbb{R}^3$), and Kähler form $\omega$ (volume form of $g$). Now, the automorphism groups of $S^2$ either as a complex manifold $(S^2,J)$ or as a Riemannian manifold $(S^2,g)$ are known.

What do we know about the symplectomorphism group $\operatorname{Sympl}(S^2,\omega)$, i.e., the group of all diffeomorphisms $\Phi:S^2\to S^2$ such that $\Phi^*\omega=\omega$?

In particular:


*

*Is there an example of an element of $\operatorname{Sympl}(S^2,\omega)$ which preserves neither $J$ nor $g$?

*What is $\operatorname{Sympl}(S^2,\omega)$ as a topological group?

 A: There are certainly examples of symplectomorphisms which do not preserve $J$ or $g$. In fact, for any closed symplectic manifold the symplectomorphism group is infinite dimensional. The group of complex automorphisms and isometries is finite dimensional.
Here is a sketch of the proof that the symplectomorphism group is infinite dimensional. Consider a smooth function $H : M \rightarrow \mathbb{R}$. Then, since symplectic forms are non-degenerate, one may check that there is a unique vector field $X_{H}$, satisfying $\omega(X_{H},\cdot) = dH$, called the Hamiltonian vector field ascosiated to H. The flow of the vector field is by symplectomorphisms, by Cartan's magic formula, which I leave for you to verify. Since the space of smooth functions $C^{\infty}(M,\mathbb{R})$ is infinite dimensional, this argument along with a few additional technicalities shows that the symplectomorphism group is infinite dimensional. See "Introduction to symplectic topology" by McDuff and Salamon for a formal proof.
