LECTURE 2
Moser's Argument
In this lecture I will show you a powerful argument due to Moser [M] which
exhibits the "flabbiness" or lack of local structure in symplectic geometry. Here is
the basic argument.
Lemma 2.1. Suppose that Wt, 0 :::; t :::; 1, is a family of symplectic forms on a
closed manifold M whose time derivative is exact. Thus
Wt= dat,
where at is a smooth family of 1-forms. Then there is a smooth family of diffeo-
morphisms <Pt with ¢ 0 = id such that
¢;(wt)= wo.Proof. We construct <Pt as the flow of a time-dependent vector field Xt. We know
¢;(wt)= w <===? :t (¢;wt)= 0
<===? ¢;(wt+ .Cx,wt) = O
<===? Wt+ i(Xt)W.Vt + d(i(Xt)wt) = 0
<===? d(at + i(Xt)Wt) = 0.
This last equation will hold if <Tt + i(Xt)Wt = 0. Observe that for any choice of
1-forms <Tt the latter equation can always be solved for Xt because of the nonde-
generacy of the Wt. Therefore, reading this backwards, we see that we can always
find an X t and hence a family <Pt that will do what we want. D
Remark 2.2. (i) The condition Wt = dCJt is equivalent to requiring that the coho-
mology class [wt] be constant. For if this class is constant the derivative Wt is exact
for each t so that for each t there is a form <Tt with Wt = dat. Thus the problem is
to construct these <Tt so that they depend smoothly on t. This can be accomplished
in various ways (eg by using Hodge theory, or see Bott-Tu [BT].)
(ii) The previous lemma uses the fact that the forms Wt are closed and t he fact
that the equation <Tt+i(Xt)Wt = 0 can always be solved. This last is possible only for
nondegenerate 2-forms and for nonvanishing top dimensional forms. In particular
the argument does apply to volume forms. Note that this case is very different from
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