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.



Spaces of symplectic embeddings: Bundle? Smoothness?
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. 


