Spaces of symplectic embeddings: Bundle? Smoothness?

Question

Let $(M, \omega)$ and $(N, \sigma)$ be two symplectic manifolds, $M$ compact and without boundary. Consider the space $$ \mathcal{E} = \mathrm{Emb}((M, \omega), (N, \sigma)) $$ of all smooth embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$. We call such an embedding isosymplectic.

The group $\mathrm{Symp}(M, \omega)$ of symplectomorphisms of $M$ acts on $\mathcal{E}$ from the right by composition of mappings and this action is free and continuous in the compact-open $C^{\infty}$-topology. Therefore we get a projection $$ \mathrm{Symp}(M, \omega) \to \mathcal{E} \xrightarrow{p} \mathcal{E}\,/\, \mathrm{Symp}(M, \omega) =:\mathcal{B}.$$

Question 1: Is $p\colon \mathcal{E} \to \mathcal{B}$ a locally trivial fibre bundle?

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Question 2: Have there been any attempts to show that $\mathcal{E}$ is an infinite dimensional smooth manifold modeled on convenient locally convex spaces or Fréchet spaces?

I know that the space $\mathrm{Emb}(M, N)$ of all embeddings of $M$ into $N$ is a convenient infinite dimensional manifold (Kriegl, Michor [1]). So is the group $\mathrm{Symp}(M, \omega)$, but here to find local charts is not as easy as one would (maybe) expect, so it is probably even harder to find local charts on $\mathcal{E}$.

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[1] A. Kriegl, P. W. Michor: The convenient setting of global analysis.

Answer 1

It should not be hard to produce a Frechet manifold structure on the space of symplectic submanifolds. A symplectic submanifold of a symplectic manifold has a neighbourhood which is symplectomorphic to the total space of its normal bundle, with a natural (split) symplectic structure. This is a version of Darboux theorem, found, for example, in Dusa McDuff's Park City lectures. Now, the $C^\infty$ symplectic deformations of a zero section in a symplectic bundle are sections of this symplectic bundle. This is a Frechet vector space. We have constructed a Frechet atlas on the space of symplectic submanifolds.

Answer 2

This is not an answer, but I hope it helps. Let $P = {\rm Emb}((M,\omega), (N,\sigma))$. The quotient $Q = P / {\rm Symp}(M,\omega)$ appears to be the set of symplectic submanifolds of $N$ which are diffeomorphic to $M$. The quotient projection appears to me $p: f \in P \mapsto f(M) \in Q$.

For each $f \in P$ it appears $T_fP$ is the vector space of sections $\Gamma( T_{f(M)}N)$ for which given any section $X$ we find $X = \left. Y \right|_{f(M)}$ for some symplectic vector field $Y \in \mathfrak{X}(N)$. Perhaps knowing the tangent space can prove useful in finding a local chart.

Choose a $f_0(M) \in Q$ and a local neighborhood $U$ of $f_0(M) \in Q$. Assuming that for any $f(M) \in U$ we can assign a symplectomorphism $\varphi_{f(M)} : f(M) \to f_0(M)$, then $\Phi: f \in p^{-1}(U) \to (f(M) , \varphi_{f(M)} \circ f \circ f_0^{-1}) \in U \times \operatorname{Symp}(M,\omega)$ appears to be a local trivialization. I think my assumption on the existence of $\varphi_{f(M)}$ is equivalent to the assumption on the existence of a chart for $U \subset Q$.

Answer 3

This is my suggested solution with a few details left out (thanks also go to Jonny Evans).

Choose an auxiliary Riemannian metric on $N$. Let $f\colon M \to N$ be a fixed isosymplectic embedding, i.e. such that $f^{*}\sigma = \omega$, and let $\pi\colon N_{f}M \to N$ denote the normal bundle of $f$ with respect to the metric on $N$. By the tubular neighbourhood theorem there is a diffeomorphism $$\nu_{f}\colon \mathcal{O}_{f} \subseteq N \to \mathcal{O}(0_{M}) \subseteq N_{f}M$$ from an open neighbourhood $\mathcal{O}_{f}$ of the image $\mathrm{im}\,f \subseteq N$ to an open neighbourhood $\mathcal{O}(0_{M})$ of the zero section of $N_{f}M$.

Let $g\colon M \to N$ be another isosymplectic embedding such that $\mathrm{im}\,g \subseteq \mathcal{O}_{f}$ and suppose that $g$ is $C^{1}$-close enough to $f$ so that the composition $\pi\circ \nu_{f} \circ g\colon M \to M$ is a diffeomorphism of $M$ (recall that $\mathrm{Diff}(M)$ is $C^{1}$-open in $C^{\infty}(M, M)$). Then $$s_{g} := \nu_{f}\circ g\circ (\pi \circ \nu_{f} \circ g)^{-1}\colon M \to N_{f}M$$ is a section of the vector bundle $\pi\colon N_{f}M \to M$.

Furthermore, denote by $\Gamma(N_{f}M)$ the space of (smooth) sections of $\pi$ and let $V \subseteq \Gamma(N_{f}M)$ be a $C^{1}$-small neighbourhood of the zero section. Define $$ U_{f} = \{g \in \mathcal{E} \,|\, \mathrm{im}\,g \subseteq \mathcal{O}_{f}, \pi \circ \nu_{f} \circ g \in \mathrm{Diff}(M), s_{g} \in V\}$$ and $$V_{f} = \{ s \in \Gamma(N_{f}M) \,|\, \mathrm{im}\,s \subseteq \mathcal{O}(0_{M}), s \in V\}.$$

The set $V_{f}$ is open in $\Gamma(N_{f}M)$ (in the compact-open $C^{\infty}$-toplogy) and so we could consider the 'chart' $$ U_{f} \ni g \mapsto (s_{g}, \pi \circ \nu_{f} \circ g) \in V_{f} \times \mathrm{Diff}(M) $$ but we would like to adjust the second factor so that it lands in $\mathrm{Symp}(M, \omega)$ instead. We will use Moser stability to achieve this.

Let $\kappa_{t}\colon N_{f}M \to N_{f}(M), t\in [0,1]$, be the fibre-preerving smooth homotopy defined by 'sliding along the fibres to the zero section,' i.e. $\kappa_{0} = \mathrm{id}$ and $\kappa_{1} = \text{projection onto the zero section}$.

For $s\in V_{f}$ arbitrary, $t\mapsto \kappa_{t}\circ s$ is a smooth homotopy of sections from $s$ to the zero section and, moreover, $$t \mapsto \omega^{s}_{t} := (\nu_{f}^{-1} \circ \kappa_{t}\circ s)^{*} \sigma $$ is a smooth homotopy of (cohomologous) symplectic forms on $M$ (if the neighbourhood $V$ above was chosen $C^{1}$-small enough) from $\omega_{0}^{s}$ to $\omega_{1}^{s} = f^{*}\sigma = \omega$ (yes, this is the original $\omega$ on $M$). By the Moser stability theorem there exists a smooth isotopy $h_{t}^{s} \in \mathrm{Diff}(M), t\in [0,1],$ such that $(h_{1}^{s})^{*}\omega = \omega_{0}^{s}$. Then $$ g_{s} := \nu_{f}^{-1} \circ s \circ (h_{1}^{s})^{-1}\colon M \to N$$ is an embedding such that $g_{s}^{*}\sigma = \omega$, hence $g_{s} \in \mathcal{E}$. In fact, $g_{s} \in U_{f}$.

Now we can define $$\varphi_{f}\colon V_{f}\times \mathrm{Symp}(M, \omega) \to U_{f}, \quad \varphi_{f}(s, a) = g_{s} \circ a = \nu_{f}^{-1}\circ s \circ (h_{1}^{s})^{-1} \circ a.$$ This is a bijection with the inverse $$\psi_{f}\colon U_{f} \to V_{f} \times \mathrm{Symp}(M, \omega), \quad \psi_{f}(g) = (s_{g}, h_{1}^{{s}_{g}}\circ \pi \circ \nu_{f} \circ g).$$

Since $\mathrm{Symp}(M, \omega)$ is a smooth manifold (for example in the convenient calculus setting of Frölicher, Kriegl and Michor), we can proclaim $\psi_{f}$ a local chart on $\mathcal{E}$. It is not difficult to check that if another $k\in \mathcal{E}$ is given, the transition map $$\psi_{k} \circ \psi_{f}^{-1} = \psi_{k} \circ \varphi_{f}\colon V_{f}\times \mathrm{Symp}(M, \omega) \to V_{k}\times \mathrm{Symp}(M, \omega)$$ is smooth (in the convenient calculus setting) -- the Moser stability term $h_{1}^{s}$ can be constructed canonically using Hodge theory, this way it will be in particular smoothly dependent on the section $s$.

The collection $(U_{f}, \varphi_{f})_{f \in \mathcal{E}}$ then defines a smooth atlas on the set $\mathcal{E}$. The topology on $\mathcal{E}$ induced by charts from the compact-open $C^{\infty}$-topology on $\Gamma(N_{f}M)$ and $\mathrm{Symp}(M, \omega)$ turns out to be Hausdorff and so $\mathcal{E}$ is a convenient smooth manifold.

Finally, for any $g\in U_{f}$ and $a\in \mathrm{Symp}(M, \omega)$ we have $s_{g\circ a} = s_{g}$ and so $$\psi_{f}(g\circ a) = \psi_{f}(g)\cdot a.$$ In other words, the chart $\psi_{f}$ descends to a chart on the quotient $\mathcal{B}$ with values in $V_{f} \subseteq \Gamma(N_{f}M)$ thus defining a smooth structure on $\mathcal{B}$. Moreover, the above also shows that $p\colon \mathcal{E} \to \mathcal{B}$ is indeed a locally trivial smooth principal $\mathrm{Symp}(M, \omega)$-bundle.