Uniqueness of smooth/symplectic/etc structure

Question

It is well-known that every topological manifold $M$ of dimension $\le 3$ admits a unique smooth structure. That is to say for any choice of atlas on $M$ like $A$ and $B$, the smooth manifolds $(M, A)$ and $(M, B)$ are diffeomorphic.

My question is whether a similar result is known for symplectic or riemannian manifolds. For instance is it true that $(S^2, \omega_1)$ and $(S^2, \omega_2)$ are symplectomorphic for any two symplectic forms $\omega_1$ and $\omega_2$ on the $2$-sphere?

Answer 1

No, this is very far from true for symplectic or Riemannian structures; if this were true then symplectic and Riemannian geometry would be very boring!

Riemannian structures have local invariants coming from curvature and can be distinguished by those: if two Riemannian surfaces are isometric then in particular there must be a diffeomorphism between them which respects Gaussian curvatures, so for example the flat metric on $S^1 \times S^1$, which has zero Gaussian curvature everywhere, is not isometric to the metric coming from a typical embedding into $\mathbb{R}^3$, which has points with positive, negative, and zero Gaussian curvature.

Symplectic structures don't have local invariants, but they do have global invariants. A simple example that comes to mind is the cohomology class $[\omega] \in H^2(X, \mathbb{R})$ of the symplectic form, or more precisely the relationship between that cohomology class and the image of $H^2(X, \mathbb{Z})$. In particular, some symplectic forms have the property that their cohomology classes lie in the image of $H^2(X, \mathbb{Z})$ and some don't, and this distinguishes symplectic forms on $S^2$. (You can say more: you can compare $[\omega]$ to a generator of $H^2(X, \mathbb{Z})$, which is equivalent to looking at the symplectic volume of $S^2$. This can be any nonzero positive real number.)

(Riemannian metrics also have global invariants, for example the Riemannian volume if that is finite.)

Answer 2

In response to your last question, the answer is no, because a symplectic form on $S^2$ is a volume form, and its integral over $S^2$ is a global invariant under symplectomorphisms. However, it's interesting to note that the integral the only invariant: a 1965 paper by Moser [M] showed that if $M$ is a compact, connected, oriented manifold, then any two volume forms on $M$ with the same integral are related by a diffeomorphism.

[M] J. Moser, "On the volume elements on a manifold." Trans. Am. Math. Soc. 120, 286–294 (1965).