Return to 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 embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$.

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?

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 $\mathcal{E}$.

A failed attempt for a solution: (I previously believed this to be a solution.)

Let $f\colon M \to N$ be a fixed symplectic embedding. The normal bundle of $f$ can be identified with its symplectic normal bundle $\pi\colon SN_{f} \to M$, which as a smooth manifold carries a natural split symplectic form $\upsilon = \pi^{*}\omega + \sigma_{\text{fibre}}$, where by $\sigma_{\text{fibre}}$ we mean the linear symplectic form on fibres of $SN_{f}$ inherited from $(TN, \sigma)$. (I haven't been careful enough to write down $\sigma_{\text{fibre}}$ explicitly here and now I can't see it is a closed two-form on $N_{f}$.) The zero section $0_{M}$ of $SN_{f}$ is then a symplectic submanifold.

By a Darboux neighbourhood theorem for symplectic submanifolds there exists a symplectomorphism $n\colon \mathcal{O}_{f} \to \mathcal{O}(0_{M})$ between an open neighbourhood of $f(M)$ in $N$ and an open neighbourhood of the zero section $0_{M}$.

Let $g\colon M \to N$ be another symplectic embedding such that $g(M) \subseteq \mathcal{O}_{f}$. If we choose the neighbourhood $\mathcal{O}_{f}$ small enough, then $n\circ g$ is still transverse to the fibres and the composition $\pi\circ n \circ g\colon M \to M$ is a symplectomorphism of $M$.

Now put $$U_{f} = \{g\colon M \to N \,|\, g^{*}\sigma = \omega, g(M) \subseteq \mathcal{O}_{f}\} \subseteq \mathcal{E}$$ and $$ V_{f} = \{s\colon M \to SN_{f}\,|\, \pi\circ s = \text{id}_{M}, s^{*}\upsilon = \omega, s(M) \subseteq \mathcal{O}(0_{M})\}.$$ Then the map $\psi_{f}\colon U_{f} \to V_{f}\times \mathrm{Symp}(M, \omega)$ defined by $$ \psi_{f}(g) = (n\circ g \circ (\pi \circ n \circ g)^{-1}, \pi \circ n \circ g)$$ is a bijection.

Since $\mathrm{Symp}(M, \omega)$ is a smooth manifold, $\psi_{f}$ can be used to endow $\mathcal{E}$ with a convenient smooth structure -- one has to check that the transition maps are smooth though.

Also because $\psi_{f}$ is $\mathrm{Symp}(M, \omega)$-equivariant, it is a bundle chart for $p\colon \mathcal{E} \to \mathcal{B}$.

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

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?

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 $\mathcal{E}$.

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

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 embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$.

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?

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 $\mathcal{E}$.

A solution: (based on Misha's answer below)

Let $f\colon M \to N$ be a fixed symplectic embedding. The normal bundle of $f$ can be identified with its symplectic normal bundle $\pi\colon SN_{f} \to M$, which as a smooth manifold carries a natural split symplectic form $\upsilon = \pi^{*}\omega + \sigma_{\text{fibre}}$, where by $\sigma_{\text{fibre}}$ we mean the linear symplectic form on fibres of $SN_{f}$ inherited from $(TN, \sigma)$. The zero section $0_{M}$ of $SN_{f}$ is then a symplectic submanifold.

By a Darboux neighbourhood theorem for symplectic submanifolds there exists a symplectomorphism $n\colon \mathcal{O}_{f} \to \mathcal{O}(0_{M})$ between an open neighbourhood of $f(M)$ in $N$ and an open neighbourhood of the zero section $0_{M}$.

Let $g\colon M \to N$ be another symplectic embedding such that $g(M) \subseteq \mathcal{O}_{f}$. If we choose the neighbourhood $\mathcal{O}_{f}$ small enough, then $n\circ g$ is still transverse to the fibres and the composition $\pi\circ n \circ g\colon M \to M$ is a symplectomorphism of $M$.

Now put $$U_{f} = \{g\colon M \to N \,|\, g^{*}\sigma = \omega, g(M) \subseteq \mathcal{O}_{f}\} \subseteq \mathcal{E}$$ and $$ V_{f} = \{s\colon M \to SN_{f}\,|\, \pi\circ s = \text{id}_{M}, s^{*}\upsilon = \omega, s(M) \subseteq \mathcal{O}(0_{M})\}.$$ Then the map $\psi_{f}\colon U_{f} \to V_{f}\times \mathrm{Symp}(M, \omega)$ defined by $$ \psi_{f}(g) = (n\circ g \circ (\pi \circ n \circ g)^{-1}, \pi \circ n \circ g)$$ is a bijection.

Since $\mathrm{Symp}(M, \omega)$ is a smooth manifold, $\psi_{f}$ can be used to endow $\mathcal{E}$ with a convenient smooth structure -- one has to check that the transition maps are smooth though.

Also because $\psi_{f}$ is $\mathrm{Symp}(M, \omega)$-equivariant, it is a bundle chart for $p\colon \mathcal{E} \to \mathcal{B}$.

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

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 embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$.

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?

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 $\mathcal{E}$.

A failed attempt for a solution: (I previously believed this to be a solution.)

Let $f\colon M \to N$ be a fixed symplectic embedding. The normal bundle of $f$ can be identified with its symplectic normal bundle $\pi\colon SN_{f} \to M$, which as a smooth manifold carries a natural split symplectic form $\upsilon = \pi^{*}\omega + \sigma_{\text{fibre}}$, where by $\sigma_{\text{fibre}}$ we mean the linear symplectic form on fibres of $SN_{f}$ inherited from $(TN, \sigma)$. (I haven't been careful enough to write down $\sigma_{\text{fibre}}$ explicitly here and now I can't see it is a closed two-form on $N_{f}$.) The zero section $0_{M}$ of $SN_{f}$ is then a symplectic submanifold.

By a Darboux neighbourhood theorem for symplectic submanifolds there exists a symplectomorphism $n\colon \mathcal{O}_{f} \to \mathcal{O}(0_{M})$ between an open neighbourhood of $f(M)$ in $N$ and an open neighbourhood of the zero section $0_{M}$.

Let $g\colon M \to N$ be another symplectic embedding such that $g(M) \subseteq \mathcal{O}_{f}$. If we choose the neighbourhood $\mathcal{O}_{f}$ small enough, then $n\circ g$ is still transverse to the fibres and the composition $\pi\circ n \circ g\colon M \to M$ is a symplectomorphism of $M$.

Now put $$U_{f} = \{g\colon M \to N \,|\, g^{*}\sigma = \omega, g(M) \subseteq \mathcal{O}_{f}\} \subseteq \mathcal{E}$$ and $$ V_{f} = \{s\colon M \to SN_{f}\,|\, \pi\circ s = \text{id}_{M}, s^{*}\upsilon = \omega, s(M) \subseteq \mathcal{O}(0_{M})\}.$$ Then the map $\psi_{f}\colon U_{f} \to V_{f}\times \mathrm{Symp}(M, \omega)$ defined by $$ \psi_{f}(g) = (n\circ g \circ (\pi \circ n \circ g)^{-1}, \pi \circ n \circ g)$$ is a bijection.

Since $\mathrm{Symp}(M, \omega)$ is a smooth manifold, $\psi_{f}$ can be used to endow $\mathcal{E}$ with a convenient smooth structure -- one has to check that the transition maps are smooth though.

Also because $\psi_{f}$ is $\mathrm{Symp}(M, \omega)$-equivariant, it is a bundle chart for $p\colon \mathcal{E} \to \mathcal{B}$.

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

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 embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$.

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?

Question 2: Has 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 $\mathcal{E}$.

A less important comment: For my purposes, it seems that I am happy enough with the structure of a Frölicher space on $\mathcal{E}$ induced from the manifold structure on $\mathrm{Emb}(M, N)$, but having actual charts would be nice.

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

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 embeddings $f\colon M \to N$ such that $f^{*}\sigma = \omega$.

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?

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 $\mathcal{E}$.

A solution: (based on Misha's answer below)

Let $f\colon M \to N$ be a fixed symplectic embedding. The normal bundle of $f$ can be identified with its symplectic normal bundle $\pi\colon SN_{f} \to M$, which as a smooth manifold carries a natural split symplectic form $\upsilon = \pi^{*}\omega + \sigma_{\text{fibre}}$, where by $\sigma_{\text{fibre}}$ we mean the linear symplectic form on fibres of $SN_{f}$ inherited from $(TN, \sigma)$. The zero section $0_{M}$ of $SN_{f}$ is then a symplectic submanifold.

By a Darboux neighbourhood theorem for symplectic submanifolds there exists a symplectomorphism $n\colon \mathcal{O}_{f} \to \mathcal{O}(0_{M})$ between an open neighbourhood of $f(M)$ in $N$ and an open neighbourhood of the zero section $0_{M}$.

Let $g\colon M \to N$ be another symplectic embedding such that $g(M) \subseteq \mathcal{O}_{f}$. If we choose the neighbourhood $\mathcal{O}_{f}$ small enough, then $n\circ g$ is still transverse to the fibres and the composition $\pi\circ n \circ g\colon M \to M$ is a symplectomorphism of $M$.

Now put $$U_{f} = \{g\colon M \to N \,|\, g^{*}\sigma = \omega, g(M) \subseteq \mathcal{O}_{f}\} \subseteq \mathcal{E}$$ and $$ V_{f} = \{s\colon M \to SN_{f}\,|\, \pi\circ s = \text{id}_{M}, s^{*}\upsilon = \omega, s(M) \subseteq \mathcal{O}(0_{M})\}.$$ Then the map $\psi_{f}\colon U_{f} \to V_{f}\times \mathrm{Symp}(M, \omega)$ defined by $$ \psi_{f}(g) = (n\circ g \circ (\pi \circ n \circ g)^{-1}, \pi \circ n \circ g)$$ is a bijection.

Since $\mathrm{Symp}(M, \omega)$ is a smooth manifold, $\psi_{f}$ can be used to endow $\mathcal{E}$ with a convenient smooth structure -- one has to check that the transition maps are smooth though.

Also because $\psi_{f}$ is $\mathrm{Symp}(M, \omega)$-equivariant, it is a bundle chart for $p\colon \mathcal{E} \to \mathcal{B}$.

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