I've been skulking around MathOverflow for about a month, reading questions and answers and comments, and I guess it's about time I asked a question myself, so here is one has interested me for a long time.


Suppose $M$ is a compact even dimensional smooth manifold with two symplectic forms $\omega\_0$ and $\omega\_1$ When are they "isotopic", i.e., when does there exist a 1-parameter family of diffeos $\phi\_t$ of $M$, starting from the identity, such that $\phi\_1^\*(\omega\_0) = \omega\_1$? Of course a necessary condition is that $\omega\_0$ and $\omega\_1$ should define the same 2-dimensional cohomology classes. Is this also sufficient? One can ask the same question for volume forms. I asked Juergen Moser about this twenty-five years ago, and he came back with an elegant proof of sufficiency for the volume element case a few months later in a well-known paper in TAMS. He remarks in that paper as follows:


"The statement concerning 2-forms was also suggested by R. Palais. Unfortunately 
it seems very difficult to decide when two 2-forms which are closed, 
belong to the same cohomology class and are nondegenerate can be deformed 
homotopically into each other within the class of these differential forms."


So my question is, what if any progress has been made on this question. Poking around here and in Google hasn't turned up anything. Does anyone know if there are any progress?



When are two symplectic forms "isotopic"?
It is known, for a long time now, that there exist examples of symplectic forms in the same cohomology class which are non-isotopic. I do not remember if there exists such example in the dimension $4$, but in dimension $6$ there are different examples. Here is an example constructed by Dusa McDuff:


Let $X$ be a product $S^2\times S^2\times T^2$ ($T^2$ is a torus $(\mathbb R/2\pi\mathbb Z)^2$ with angle coordinates $(\psi,\gamma)$) and $\omega$ is a sum $\omega\_1\oplus\omega\_2\oplus\omega\_3$ of area forms on factors. We suppose that total areas of the first and of the second factor coincides. Consider the map $\varphi \colon X \to X$, where $\varphi (x,y,\psi,\gamma) = (x, T\_{x,\psi}(y),\psi,\gamma)$, where $T\_{x,\psi}$ is the rotation around $x$ on the angle $\psi$. Then forms $\omega$ and $\varphi^\*(\omega)$ define the same cohomology class and non-isotopic.


Moreover, forms $\omega$ and $\varphi^\*(\omega)$ could be joined by a path in a space of symplectic structures.


There is a survey containing the statement of this result and helpful references:
<http://www.math.sunysb.edu/~dusa/princerev98.pdf>


