In my symplectic geometry class we gave the following result :

Let $M$ be a closed symplectic manifold, $Q\subset M$ a closed submanifold and $\omega_t$ a family of symplectic forms such that $[\frac{d}{dt}\omega_t]=0$ and $(\frac{d}{dt}\omega_t)|_Q=0$ for any $t\in [0,1]$. Then there exists an isotopy $\phi_t :M\rightarrow M$ such that $(\phi_t)_Q=id$ and $\phi_t^*(\omega_t)=\omega_0$ for any $t\in [0,1]$.

Now there's a step on the proof where we realized there is a mistake and so far no one was able to fix it. I have tried to look for this result in symplectic geometry books but I had no sucess so I am wondering if anyone knows if this result is actually true and how one can prove it , or if there is a counterexample ?

Any help is appreciated. Thanks in advance.


1 Answer 1


If the cohomology class $[\frac{d\omega_t}{dt}]$ is merely assumed to lie in the (absolute) cohomology $H^2_{dR}(M)$, then the result is false. If it is assumed to lie in the relative cohomology $H^2_{dR}(M,Q)$, then the result is true under the following formulation (to be explained below):

Let $M$ be a closed symplectic manifold, $i_Q : Q \subset M$ a closed submanifold and $\omega_t$ a family of symplectic forms such that $\left[ \frac{d \omega_t}{dt} \right] = 0$ in $H^2(M,Q)$ and $i_Q^*\frac{d\omega_t}{dt} = 0$ for any $t \in [0,1]$. Then there exists an isotopy $\phi_t : M \to M$ such that $\phi_t|_Q = id$ and $\phi_t^*\omega_t = \omega_0$ for every $t \in [0,1]$.

Absolute case: For $[\frac{d\omega_t}{dt}] = 0$ in $H^2_{dR}(M)$ , one can produce counter-examples to the statement by taking $Q$ to be a contractible embedded circle in a closed surface $M$. Indeed, pick an embedded smooth disc $D$ bounded by $Q$ and define $\omega_t$ by changing the symplectic form $\omega_0$ away from a neighborhood of the circle in such a way that the area of $D$ disc increases with $t$ while keeping the area of the whole surface fixed. Then any isotopy $\phi_t$ produced from the family $(\omega_t)_{t \in [0,1]}$ by Moser's trick is such that the $\omega_0$-area of $D_t := \phi_t^{-1}(D)$ increases with $t$, which implies $\phi_t|_Q \neq id$.

Relative case: (Preliminary remark: Let $i_Q : Q \subset M$ denote the inclusion map. Given a differential form $\alpha$ on $M$, we are going to consider two types of 'restrictions' to $Q$, (i) the pullback form $i_Q^*\alpha$ on $Q$ and (ii) the restriction $\alpha|_{Q}$ of $\alpha$ to $Q$, by which I mean the form induced on the pullback bundle $i_Q^*TM = T_{Q}M$. The pullback $i_Q^*\alpha$ only captures the value of $\alpha|_Q$ on the subbundle $(i_Q)_*TQ \subset T_Q M$.)

The desired property $\phi_t|_Q = id$ amounts to $X_t|_Q = 0$, where $X_t$ is the vector field produced by Moser's trick via the equation $\lambda_t = \iota_{X_t}\omega_t$ where $d\lambda_t = \frac{d \omega_t}{dt}$. Hence, for $\phi_t$ to have the desired property, we need to be able to find for every $t \in [0,1]$ a primitive 1-form $\lambda_t$ for $\frac{d \omega_t}{dt}$ whose restriction $\lambda_t|_Q$ to $Q$ vanishes. This is only possible if $\frac{d \omega_t}{dt}$ is 'well-behaved relative to $Q$', in a sense we now explore.

The counter-examples given in the absolute case suggest that for the statement to hold, one should further require that all members of the family $(\omega_t)_{t \in [0,1]}$ give the same area to every 2-cycle with boundary in $Q$. To satisfy this requirement, it suffices to ask that for every $t \in [0,1]$, the exact two-form $\frac{d\omega_t}{dt}$ admits a primitive 1-form $\lambda_{t}$ whose pullback $i_Q^*\lambda_{t}$ to $Q$ is exact. (Note: The condition $i_Q^*\frac{d\omega_t}{dt} = 0$ already implies that $i_Q^*\lambda_{t}$ is closed.)

This last condition can be expressed in relative cohomological terms as defined in Bott's & Tu's Differential forms in algebraic topology. Namely, we define the de Rham complex of $M$ relative to $Q$ as the cone of $i_Q$, i.e. as the complex $\Omega^*(M, Q) := \Omega^*(M) \oplus \Omega^{*-1}(Q)$ with differential given by $d(\alpha, \eta) = (d\alpha, i_Q^*\alpha - d\eta)$. Then the condition stated in the previous paragraph is precisely that for every $t \in [0,1]$, the relative 2-form $\left( \frac{d \omega_t}{dt}, 0 \right) \in \Omega^2(M, Q)$ should be exact; In other words, the relative cohomology class $\left[ \frac{d \omega_t}{dt} \right] := \left[ \left( \frac{d \omega_t}{dt}, 0 \right) \right] \in H^2(M, Q)$ should be required to vanish.

Under this assumption, we can find for every $t \in [0,1]$ a primitive 1-form $\lambda_t$ to $\frac{d\omega_t}{dt}$ whose pullback to $Q$ is exact, say $i_Q^*\lambda_t = df_t$ with $f_t : Q \to \mathbb{R}$. Now, by Whitney's extension theorem, there exists $F_t : M \to \mathbb{R}$ such that $i_Q^* F_t = f_t$ and $dF_t|_Q = \lambda_t|_Q$. Let's define $\lambda'_t := \lambda_t - dF_t$; This is another primitive 1-form for $\frac{d\omega_t}{dt}$ which satisfies $\lambda'_t|_Q = 0$. Since all the above choices can be made to depend smoothly on $t$, we can apply Moser's trick with this family $\lambda'_t$ to obtain the desired isotopy.


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