\chapter{Preliminary: Symplectic geometry on the space of codimension-2 shapes}\label{ch:preliminary}

\autoref{ch:preliminary}, \autoref{ch:implicit_representations}, and \autoref{ch:prequantum_bundle} are based on the article: 

\bibentry{chernIshida2025prequantum}

\vspace{5pt}

In this chapter, we review basic results on the symplectic geometry of codimension-2 shape space, as the topics in this thesis revolve around this subject.  We first lay out the preliminary settings for the shape space of codimension-2 submanifolds, and then explain that this space is a symplectic manifold equipped with the so-called Marsden--Weinstein symplectic structure.

We formulate these known results using currents from geometric measure theory, which are the continuous dual of differential forms. Currents allow a unified description of both the classical results reviewed here and the new results for implicit representations that we introduce and study in \autoref{ch:implicit_representations} and \autoref{ch:prequantum_bundle}.

Alongside, we extend a previous result. The existence of a Liouville form $\eta$ for the MW form $\omega$ (\ie, 1-form $\eta$ such that $d\eta = \omega$) was proven only for space curves.
% the space of space curves \ie, the embeddings of $\SS^1$ into $\RR^3$ modulo reparametrization.
We extend this to a general closed and oriented codimension-2 manifold, embedded in an ambient manifold of arbitrary dimension, equipped with an exact volume form.

\paragraph{Dependencies of the other chapters on this chapter}
The main purpose of this chapter is to provide an overview of the codimension-2 shape space as an infinite-dimensional symplectic manifold, formulated using the language of currents. This builds a foundation for \autoref{ch:implicit_representations} and \autoref{ch:prequantum_bundle}, where implicit representations of shapes are introduced and studied.

On the other hand, \autoref{ch:hidden_degrees}, \autoref{ch:more_symplectic}, and \autoref{ch:spherical_polygon} are written in a self-contained manner and do not strictly rely on the results and the descriptions in this chapter. Hence the readers interested in these chapters can directly jump to any of them.


\paragraph{Organization of the chapter}
In \autoref{sec:currents}, we review currents from geometric measure theory and introduce the action of  diffeomorphism groups on currents, as well as the Lie derivative as its infinitesimal generator.
% This allows us to treat shapes in a parameterization-free manner in both explicit setting explained in this chapter and implicit settings introduced in \autoref{ch:implicit_representations}.

In \autoref{sec:explicit_symplectic}, we define the codimension-2 shape space as embeddings of a manifold into an ambient manifold modulo reparametrization. We do so in terms of diffeomorphism actions on currents, highlighting that the deformation of a shape can be seen as the transport of a current along a vector field.

In \autoref{sec:symplectic_structure_on_codim2}, we review the Marsden--Weinstein symplectic structure on the codimension-2 shape space. 
% 
Additionally, we extend the previous result on the existence of a Liouville form for space curves to a more general setting in arbitrary dimension.


\section{Diffeomorphism action on currents}\label{sec:currents}
We briefly review the notion of currents from geometric measure theory as a preliminary and define a diffeomorphism action on them. For readers interested in the general theory of currents, we refer to the literature \cite{federer2014geometric,deRham1984differentiable, morgan2008geometric}.
Currents are a natural language for shapes embedded in an ambient manifold as they allow us to treat differential forms, submanifolds, and their generalizations in a unified manner. 
Using currents, we will describe relations between quantities defined on an ambient manifold and a submanifold (possibly with a boundary). This includes, for example, the flux of a vector field through a surface, and a superposition of the fluxes through infinitely many surfaces. 

% \Sadashige{I dropped the paragraph "Currents are a natural generalization of distributions...". I think the flow is smooth enough if we already define currents here.}
% Currents are a natural generalization of distributions. Distributions are duals of smooth functions. As higher order analogues, currents are duals of differential forms. One can also see currents as differential forms with distributions as coefficients which extends smooth differential forms that take smooth functions as coefficients.


Let $M$ be an oriented $m$-dimensional manifold and $\Omega^k_c(M)$ be compactly supported differential $k$-forms. We say $k$-currents $\cD_k(M)$, also denoted by $\Omega^k_c(M)^*$, are linear functionals on $\Omega_c^k(M)$ that are continuous in the sense of distributions.
% with respect to the $C^\infty$-topology.
% For a sequence $\{f_n\}_n \subset \Omega^{0}_c(M)$, we say $f_n\to f$ if there is a compact set $K\subset M$ such that $\on{support}(f_n)\subset K, \forall n$ and $\partial_a f_n \to \partial_a f$ as $n\to \infty$ for any multi-index $a$. This notion of continuity extends to $\Omega_c^k(M)$ by taking the product $[\Omega_c^0(M)]^{\on{Ind}(k,m)}$. A linear functional $h$ on $\Omega^k_c(M)$ is said to be continuous if $\alpha_n \to \alpha$ in $\Omega_k^c(M)$ implies
% \begin{align*}
%     \langle h, \alpha_n \rangle \to  \langle h, \alpha \rangle.
% \end{align*}
% We also equip a topology to $\cD_k(M)$ by 
% \begin{align}\label{eq:current_topology}
%     h_n \to h \textup{ if } \langle h_n, \alpha \rangle \to  \langle h, \alpha \rangle, \quad \forall \alpha\in \Omega_c^k(M).
% \end{align}
% 
For details, see \cite{deRham2012currents,Alberti2006520GMT} for example.
% \begin{align}
%     \|h\|\coloneqq \sup_{\|\alpha\|_{\Omega^k(M)}=1} \langle h, \alpha \rangle <\infty
% \end{align}
% with respect to the norm 
% \begin{align}
% \| \alpha\|_{\Omega^k(M)}^2\coloneqq\left|\int_M \alpha\wedge \star \alpha\right|.
% \end{align}


We note that $\Om^{m-k}(M)\subset \cD_k(M)$ via 
\begin{align}\label{eq:current_as_integral}
   \langle h, \alpha\rangle \coloneqq \int_M h\wedge \alpha, \quad h\in \Om^{m-k}(M), \alpha\in\Omega_c^k(M). 
\end{align}
The space of differential forms $\Omega^{m-k}(M)$ is strictly smaller than the space of current $\cD_k(M)$, but $\cD_k(M)$ is attained as the closure of $\Omega^{m-k}(M)$ with respect to the locally convex topology. Keeping this in mind, we will formally write $\int_M h\wedge \alpha$  also for $h$ in $\cD_k(M)$ but not in $\Om^{m-k}(M)$ as if $h$ was a $m-k$ form. 

For $h\in \cD_k(M)$, we define the boundary operator $\partial_k\colon \cD_k(M) \to \cD_{k-1}(M)$ by
\begin{align}
    \langle \partial_k h, \alpha \rangle\coloneqq  \langle h, d_{k-1}\alpha \rangle, \quad \forall \alpha \in \Omega_c^k(M)
\end{align}
using the exterior derivative $d_{k-1}\colon \Omega^{k-1}(M)\to \Omega^k(M)$.
Unless necessary, we will simply write $\partial$ for $\partial_k$ and $d$ for $d_k$.
Notice that $\partial$ works for $h\in \Om^{m-k}(M)\subset \cD_k(M)$ as the exterior derivative $d$ up-to sign change with $\operatorname{sgn}=(-1)^{m-k+1}$,
% \begin{equation*}
% \begin{tikzcd} % [row sep=small]
% \operatorname{sgn} d:\Om^{m-k}(M)\arrow[r] & \Om^{m-k+1}(M)\\
% \partial:\cD_k(M)\arrow[r]\arrow[u, phantom, sloped,yshift=-2ex, "\supset"] & \cD_{k-1}(M)\arrow[u, phantom, sloped, "\supset"].
% \end{tikzcd}
% \end{equation*}
\[
\begin{tikzcd}
\operatorname{sgn} d: &[-1.2cm] \Om^{m-k}(M)\arrow[r] & \Om^{m-k+1}(M)\\
\quad\ \partial:&\cD_k(M)\arrow[r]\arrow[u, phantom, sloped,yshift=-2ex, "\supset"] & \cD_{k-1}(M)\arrow[u, phantom, sloped, "\supset"].
\end{tikzcd}
\]
We say that the current homology, which is the dual of de-Rham cohomology, is defined by
\begin{align}
    H^{CR}_k(M)\coloneqq Z^{CR}_k(M)/B^{CR}_k(M)
\end{align}
where 
$Z^{CR}_k(M)=\ker\partial_{k}$ and $B^{CR}_k(M)=\im\partial_{k+1}$.
% In this article, we will assume $H_{m-2}^{CR}(M)=0$. 
% \SemiDI{In the smooth regime, is this same as $H^{m-2}_{dR}(M)=0$ and $H_{m-2}(M;\RR)=0$ on a closed manifold $M$? In practice, what we need is only $H_{m-2}(M;\RR)=0$ to ensure that every closed curve $\gamma$ (or more generally $m-2$-dim embedding) admits a chain $\Sigma$ s.t. $\partial \Sigma=\gamma$.}
\subsection{de Rham-Dirac currents}
We call a special class of currents de Rham-Dirac currents (or simply de Rham currents). These currents extend the Dirac measures for points (codimension-$m$ geometry) to other codimensions. 
For a $k$-dimensional submanifold $\Sigma$ of $M$ possibly with a boundary, we define a $k$-de Rham current $\delta_\Sigma\in \cD_k(M)$ by
\begin{align}
    \langle \delta_\Sigma, \alpha\rangle=\int_\Sigma \alpha, \quad \alpha \in \Omega^k(M).
\end{align}
We denote the collection of de Rham-$k$ current by $\cD^{dR}_k(M)$.
%  In the language of measure theory, it is written as,
%  \begin{align}
%     \langle \delta_\Sigma, \alpha\rangle
%     =\int_{|\Sigma|} \alpha(\tau) w\  d\cH^{k-1}
% \end{align}
%  where $|\Sigma|\subset M$ is interpreted as a $k$-rectifiable set and $\tau$ is the \emph{orientation} \ie, a unit tangent $k$-vector that spans $T_x \Sigma\subset T_x M$ for almost $\cH^{k-1}$ every $x$ on $\Sigma$, and $w$ is a \emph{weight} function s.t. $\int_{\Sigma} |w| d\cH^{k-1}<\infty$.
 % \SemiDI{With this, can we maybe handle self-intersecting curves? Coarea formula has a version for a non-manifold version (rectifiable sets version).}
 
The boundary operators $\partial$ for the de Rham currents and for submanifolds commute with the mapping of submanifolds to currents  \ie, $ \partial \delta_\Sigma=\delta_{\partial \Sigma}$ as
\begin{align}
    \langle \partial \delta_\Sigma, \alpha\rangle=\int_\Sigma d\alpha=\int_{\partial \Sigma} \alpha=\langle \delta_{\partial \Sigma},\alpha \rangle, \quad \forall \alpha \in \Omega^k(M).
\end{align}
We will also formally write this as
\begin{align}
    \int_M \delta_{\partial\Sigma}\wedge \alpha =\int_M \delta_\Sigma\wedge d\alpha
\end{align}
with the mind of \eqref{eq:current_as_integral}. 

\begin{remark}
    de Rham currents can be defined from rectifiable sets on $M$, which form a much broader class containining submanifolds and singular chains of $M$ \cite{morgan2008geometric}. We focus on the ones induced from submanifolds as it suffices for our purpose. 
\end{remark}

% Hence we can view $k$-currents $\cD_k(M)$ as natural extensions of $k-$chains $\cC_k(M)$ by
% \begin{align}
%     \cC_k(M) \xrightarrow[]{*} \Omega^k(M) \xrightarrow[]{*}  \cD_k(M),
% \end{align}
% where each $ \xrightarrow[]{*}$ means taking the dual \ie, the collection of bounded linear functionals. The space of chains is identified with the space of de Rham currents \ie, $\cC_k(M)\approx \cD_k^{\operatorname{dR}}(M) \subsetneq\cC^{**}_k(M)=\cD_k(M)$.

\subsection{Orbits under diffeomorphism groups}\label{sec:orbits_of_currents}
We represent unparametrized shapes by currents and describe the deformation of shapes in terms of diffeomorphism actions. 
To do so, we define the action of  diffeomorphism groups on currents and the Lie derivative as its infinitesimal generator. 

Let us denote by $\Diff(M)$ the space of diffeomorphisms over $M$. For each $f\in \Diff(M)$, we define the action $f_*\colon D_k(M)\to D_k(M)$ by pushforward as the adjoint of pullback $f^*\colon \Omega^k(M)\to \Omega^k(M)$,
\begin{align}
    \langle f_* h, \alpha\rangle\coloneqq \langle h, f^{*}\alpha\rangle, \quad h\in \cD_k(M), \alpha\in \Om^k(M).
\end{align}
For $h\in \Om^{m-k}(M)\subset \cD_{k}(M)$, the pushforward action $f_*$ on $h$ as a current agrees with the pullback action ${f^{-1}}^*$ on $h$ as a differential form. For a de-Rham current $\delta_\Sigma\in \cD_k^{dR}(M)$, we have $f_*\delta_\Sigma=\delta_{f\circ \Sigma}$. 
Note also that $f_*$  and $\partial$ commute similarly to the commutativity between $d$ and $f^*$ for differential forms.

On an orbit $\cO$ under the $\Diff(M)$-action by pushforward, the tangent space is
\begin{align}\label{eq:tangent_current_orbit}
    T_h \cO(M)=\{-\cL_v h \mid v\in \diff(M)\}
\end{align}
 where $\diff(M)$ is the Lie algebra of $\Diff(M)$, which is the space of smooth vector fields on $M$, and we say that Lie derivative $\cL_v\colon \cD(M)\to \cD(M)$ for currents is the skew-adjoint of the Lie derivative for differential forms, 
\begin{align}
    \langle -\cL_v h,\alpha\rangle\coloneqq\langle h, \cL_v \alpha\rangle,\quad \alpha \in \Om^k(M).
\end{align}
The minus sign of $-\cL_v h$ in \eqref{eq:tangent_current_orbit} expresses the feeling of \emph{advection} of $h$ along the velocity field $v$ by the formal transport equation $\partial_t h + \cL_v h=0$. Similarly to differential forms, we have $\cL_{[u,v]}h = \cL_u \cL_v h - \cL_v\cL_u h$ for currents, which can be verified by direct computation.

Lastly, we have a current analogy of the relation $\cL_u \alpha=\frac{d}{dt}\big|_{t=0} {\on{Fl}_u^t}^* \alpha$ for differential forms  where $\on{Fl}_u^t\in \Diff(M)$ is the time-$t$ flow map of $u\in \diff(M)$ defined as the solution to the ODE,
\begin{align}
    &\partial_t (\on{Fl}_u^t(x))=u(\on{Fl}_u^t(x)), \quad x\in M\\
    &\on{Fl}_u^0=\id_M.
\end{align}
For a current $h$, we have $-\cL_u h = \frac{d}{dt}\big|_{t=0} {\on{Fl}_u^t}_* h$ as 
\begin{align}
    \frac{d}{dt}\Big|_{t=0}\langle{\on{Fl}_{u}^t}_* h, \alpha  \rangle
    =
    \frac{d}{dt}\Big|_{t=0}\langle h, {\on{Fl}_{u}^t}^*\alpha  \rangle
    = \langle h,\cL_u \alpha \rangle
    =\langle-\cL_u h,\alpha \rangle, \quad \forall \alpha \in \Omega^k(M).
\end{align}

\subsection{Currents as functionals on the space of vector fields}
In later sections, we will consider fluxes of vector fields through possibly infinitely many hypersurfaces. Currents provide a natural description for this.

Let $\diff(M)$ and $\sdiff(M)$ denote the spaces of smooth vector fields and divergence free-vector fields on $M$ equipped with a volume form $\mu$. They are Lie algebras of $\Diff(M)$ and $\SDiff(M)$, the groups of diffeomorphisms and volume-preserving diffeomorphisms on $M$. Let us also define $\exdiff(M)$, a subspace of $\sdiff(M)$, consisting of \emph{exact divergence-free} vector fields, that is, $\iota_u\mu$ is exact.
\begin{definition}[Flux for $m-1$ currents and exact $m-2$ currents]\label{def:flux}
For $h\in \cD_{m-1}(M)$, we say  that the flux of a vector field $u\in \diff(M)$ through $h$ is 
    \begin{align}
    \fluxMone(h,u)\coloneqq \langle h, \iota_u \mu\rangle.
    \end{align}

    For $\beta\in B^{CR}_{m-2}(M)$, we say that the flux of $v\in\exdiff(M)$ through $\beta$ is 
    \begin{align}
    \fluxMtwo(\beta,v)\coloneqq  \fluxMone(h,v)
    \end{align}
    with any $h \in \partial^{-1}\beta\subset \cD_{m-1}(M)$.
\end{definition}
By design $\fluxMtwo$ is independent of the choice of $h$ in $\partial^{-1}\beta$.
% 
% It is well-known that $\SDiff(M)$ has a coadjoint representation on $\sdiff^*(M)=\Om^1(M)/B_{dR}^1(M)$, the smooth dual of $\sdiff(M)$. We can extend this to $B_{k-1}^{CR}(M)$. First note that we have an isomorphism between $B_{k-1}^{CR}(M) =\on{image}\partial_{k}$ and $\cD_k(M)/B_k^{CR}(M)=\cD_{k}(M)/\partial_{k+1}\cD_{k+1}(M)$ for any $k$. 
% When $H^{CR}_{m-2}(M)=0$, we have $Z_{m-2}^{CR}(M)\approx \cD_{m-1}(M)/B_{m-2}^{CR}(M)$.
Via the notion of flux, we can now regard $m-1$ and exact $m-2$ currents as linear functions on $\diff(M)$ and $\exdiff(M)$.




For a de Rham current $\delta_\Sigma\in \cD^{dR}_{m-1}(M)$, $\fluxMone(\delta_\Sigma,v)$ is indeed the flux of $v$ through $\Sigma$. But this notion of flux does not require $h$ to be a de Rham current. For example, $h$ may be a formal infinite sum of de Rham currents representing infinitely many hypersurfaces. This observation plays an important role when we consider a Liouvile form for a symplectic structure on the space of implicit representations in \autoref{sec:Liouville_form_prequantum}.


% Via flux  we can consider coadjoint orbits in $B_{m-2}^{CR}(M)$. Recall that the adjoint representation $\Ad\colon \SDiff(M) \to \on{Aut}( \sdiff(M))$ is given by the pushforward $\Ad_f v= df[v]$. We define the coadjoint representation $\Ad^*\colon \SDiff(M)  \to \on{Aut}( B_{m-2}^{CR}(M)) $ using flux as the dual product;
% \begin{align}
%     \langle \Ad^*_f \beta,v\rangle \coloneqq  \fluxMtwo( \beta,  \Ad_{f^{-1}} v) = \fluxMtwo( \beta,  df^{-1} [v]).
% \end{align}
% For $\beta\in B_{dR}^2(M)\subset B^{CR}_{m-2}(M)$, $\Ad_f^*\beta$ is nothing but the pullback $f^*\beta$. In this way, the coadjoint representation (and pullback operation) extends from $\sdiff^*(M)=\Om^1/d\Om^0$ to $B^{CR}_{m-2}(M)$. 

% Moreover, if $H_{m-2}(M)=0$ namely $\cD_{\UEmb(S,M)}\subset B^{CR}_2(M)$, we have for a de Rham current $\delta_\gamma\in \cD_{\UEmb(S,M)}$, 
% \begin{align}
%     \fluxMtwo( \on{Ad}_f^* \delta_\gamma, v )=\fluxMtwo (\delta_\gamma, df^{-1}[v] )=\fluxMtwo( \delta_{f\circ \gamma}, v ).
% \end{align}
% In this case, the coadjoint representation $\Ad_f^*$ is just the action $f_*$ of $\SDiff(M)$ on $\cD_{\UEmb(S,M)}\subset B^{CR}_{m-2}(M)$ defined in Section \ref{sec:orbits_of_currents}.

\section{The space of codimension-2 shapes}\label{sec:explicit_symplectic}
This section provides a preliminary review of the space of \emph{explicit representations} for codimension-2 shapes, namely, embeddings of an \(n\)-dimensional manifold \(S\) into an \((n+2)\)-dimensional ambient manifold \(M\). We revisit earlier work on the canonical symplectic structure on this space, the Marsden-–Weinstein (MW) structure, as studied in \cite{MarsdenWeinstein1983coadjoint, HallerVizman2003NonlinearGA, Tabachnikov2017bicycle, Padilla2019bubbleRings}. Our presentation is framed in terms of currents, which will play a central role in the next section when we introduce implicit representations (\autoref{ch:implicit_representations}) and relate them to the explicit framework.

In addition, we obtain new results (\autoref{th:liouville_classical}, \autoref{th:extended_classical_Liouville}): the MW structure on the codimension-2 shape space is exact if the volume form $\mu$ of the ambient manifold is exact. This extends earlier results for closed curves in \(\RR^3\) \cite{Tabachnikov2017bicycle,Padilla2019bubbleRings} to a broader class of submanifolds in arbitrary dimensions.

We conclude this section by considering the shape space of all codimension-2 submanifolds.  That is, we allow the submanifold \(S\) to range over all oriented and closed \(n\)-manifolds, rather than fixing one \(S\) in advance. This broader viewpoint naturally leads to the introduction of implicit representations in the next chapter.

\subsection{Spaces of parametrized and unparametrized codimension-2 submanifolds}\label{sec:explicit_shapes}
We begin with the case where the dimension of the ambient manifold is greater than 2, postponing remarks on the $2$-dimensional case to the end of this section. Let \( m > 2 \), and let \( M \) be an \( m \)-dimensional manifold equipped with a volume form \( \mu \). Let \( S \) be a closed (i.e., compact and without boundary), oriented manifold of dimension \( n = m - 2 \).

Consider the space of smooth embeddings
\begin{align}
    \on{Emb}(S, M) \coloneqq \{ \paramgam \in C^\infty(S, M) \mid \on{rank}(d\paramgam) = n,\ \paramgam(s) = \paramgam(s') \Rightarrow s = s' \},
\end{align}
which is an infinite dimensional manifold with the Fr\'echet topology \cite{bauerBruverisMichor2014overview, Michor2019ManifoldsOM}.
Its tangent space at each \( \paramgam \in \on{Emb}(S, M) \) is given by sections of the pullback bundle:
\begin{align}
    T_\paramgam \on{Emb}(S, M) = \Gamma(\paramgam^* TM).
\end{align}
That is, a tangent vector \( \dot\paramgam \in T_\paramgam \on{Emb}(S, M) \) assigns to each point \( s \in S \) a vector \( \dot\paramgam(s) \in T_{\paramgam(s)}M \). When $M=\RR^m$,  the tangent space \( T_\paramgam \on{Emb}(S, \RR^m) \) is identified with \( C^\infty(S, \RR^m) \).

On \(\Emb(S,M)\) we define a right action of the orientation-preserving diffeomorphism group \(\Diffp(S)\) representing reparametrizations. Taking the quotient by this action, we obtain the \emph{shape space of unparametrized shapes}:
\begin{align}
    \UEmb(S,M) \coloneqq \on{Emb}(S,M)/\Diffp(S).
\end{align}
The space \(\UEmb(S,M)\) is also referred to as the \emph{nonlinear Grassmannian of type \(S\)} \cite{HallerVizman2003NonlinearGA}.

Note that \(\Diffp(S)\) may have multiple connected components. For example, if \(S = \bigsqcup_{i=1}^{k_1} \SS^2 \sqcup \bigsqcup_{i=1}^{k_2} \TT^2\), then \(\Diffp(S)\) is isomorphic to the product of \(\Diff^+_0(S)\), the component containing \(\id_S\), and the symmetric groups \(\on{Sym}(k_1)\) and \(\on{Sym}(k_2)\). The operation of permuting identical copies of \(\SS^2\) or \(\TT^2\) is included in \(\Diffp(S)\).

The shape space \(\UEmb(S,M)\) is an infinite-dimensional manifold \cite{bauerBruverisMichor2014overview}. In what follows, we denote an element of \(\UEmb(S,M)\) by \(\gamma\) and any representative of \(\gamma\) by \(\tilde\gamma\in \Emb(S,M)\).

With the fibration $\pi\colon  
    \Emb(S,M) \to \UEmb(S,M)$
the tangent space is written as
\begin{align}
    T_{\pi(\paramgam)}\UEmb(S,M)=d\pi|_\paramgam (T_\paramgam \Emb(S,M)).
\end{align}
Note that $\ker d\pi|_\paramgam= d\paramgam (\diff(S))$ where $\diff(S)$ is the space of smooth vector fields on $S$, which is the Lie algebra of $\Diffp(S)$. Hence $\ker d\pi$ consists of the components of tangent vectors which do not change \emph{the shape} of $\gamma$. 
\begin{example}[Tangent space at a unparametrized space curve]
    When $S=\SS^1$ and $M=\RR^3$ equipped with the standard Euclidean metric, we have $\ker d\pi|_\paramgam=\{a\partial_s \paramgam \mid a\in C^\infty(S)\}$. Hence
$ T_{\gamma}\UEmb(S,M)= \{ \dot\paramgam+d\paramgam \diff(S)\mid \dot \paramgam\in T_\paramgam \Emb(S,M)\}$ is identified with $\{\dot \paramgam \colon S\to \RR^3 \mid \dot \paramgam(s)\perp \partial_s\paramgam(s), \forall s\in \SS^1\}$, vector fields on $\paramgam(S)$ that are everywhere perpendicular to the tangent vector $\partial_s \paramgam$.
\end{example}

We now define a left action of $\Diff(M)$ on $\UEmb(S,M)$
by
\begin{align}
    f \rhd \gamma = \pi(f \circ \tilde\gamma), \quad f\in\Diff_0(M), \gamma\in \UEmb(S,M)
\end{align}
 where \(\pi\) is the projection $\pi\colon \Emb(S,M)\to \UEmb(S,M)$ and \(\paramgam \in \pi^{-1}\gamma\) is any parametrization of \(\gamma\).

Each connected component of $\UEmb(S,M)$ is an orbit of the action of $\Diff_0(M)$,  the connected component of $\Diff(M)$ containing $\id_M$. This is a classical result due to Thom (see, e.g., \cite{hirsch2012differential}). Consequently, any tangent vector $\dot\gamma \in T_\gamma \UEmb(S,M)$ can be written as
$\dot \gamma = v \circ \gamma \coloneqq d\pi|_{\paramgam}(v \circ \paramgam)$
for some $v \in \diff(M)$. With this in mind, we will restrict our attention to a single $\Diff_0(M)$-orbit in $\UEmb(S,M)$, denoted by $\cO$.

In fact, the action by the subgroup $\SDiff_0(M)\subset \Diff_0(M)$ of volume-preserving diffeomorphisms is transitive on each \(\Diff_0(M)\)-orbit in \(\UEmb(S,M)\) \cite[Proposition 2]{HallerVizman2003NonlinearGA}.  Much of the theory in this section remains the same after restricting the diffeomorphism group \(\Diff_0(M)\) to \(\SDiff_0(M)\) and the space \(\diff(M)\) of vector fields to the space \(\sdiff(M)\) of divergence-free vector fields.

% When the codimension is $2$, i.e.\@ $n = m - 2$, the action of $\SDiff_0(M)$—the subgroup of $\Diff_0(M)$ consisting of volume-preserving diffeomorphisms—is already transitive on each connected component \cite[Proposition 2]{HallerVizman2003NonlinearGA}. Nonetheless, we work with the $\Diff_0(M)$-action throughout this article, as it aligns more naturally with the implicit representations we will introduce, for which the $\SDiff_0(M)$-action fails to be transitive on each $\Diff_0(M)$-orbit.\Albert{[The last part of this paragraph is not going to be clear to the readers in this context.  Maybe this paragraph can just be a note laying the fact.  ``In fact, the action by the subgroup $\SDiff_0(M)<\Diff_0(M)$ of volume-preserving diffeomorphisms is also transitive on each \(\Diff_0(M)\)-orbit in \(\UEmb(S,M)\) \cite[Proposition 2]{HallerVizman2003NonlinearGA}.  Much of the theory in this section remains the same after restricting the diffeomorphism group \(\Diff_0(M)\) to \(\SDiff_0(M)\) and the space \(\diff(M)\) of vector fields to the space \(\sdiff(M)\) of divergence-free vector fields.'']}

% \begin{example}[Knots and links]
%     When $S= \bigsqcup_{i=1}^N \SS^1$ with some positive integer $k$, each unparametrized curve $\gamma\in \UEmb(S,M)$ is called an oriented link (when $N=1$ in particular, an oriented knot). Two links are said to be ambient isotopic if they are in a same $\SDiff^+_0(M)$-orbit. This naturally extends to a higher dimension for a generic $S$. In this article, we occasionally call each $\SDiff^+_0(M)$ orbit \emph{a link equivalence class}. 
% \end{example}

\begin{remark}[Labeled and unlabeled shapes]\label{rmk:labeled_and_unlabeled_shapes}
There are two standard definitions of unparametrized shapes $\UEmb(S,M)$. One defines $\UEmb(S,M)$ as the quotient of $\Emb(S,M)$ by the orientation-preserving diffeomorphism group $\Diffp(S)$, and the other uses $\Diff_0(S)$, the connected component in $\Diff(S)$ containing the identity map  \cite{bauerBruverisMichor2014overview}.

The quotient by $\Diff_0(S)$ retains labels on identical components of the same shape. For example, if $S= \SS^1 \sqcup \SS^1$, then two embeddings $\gamma_a = (\gamma_1, \gamma_2)$ and $\gamma_b = (\gamma_2, \gamma_1)$, with non-intersecting embeddings $\gamma_1$ and $\gamma_2$ of $\SS^1$, represent distinct elements in $\Emb(S,M)/\Diff_0(S)$ but the same element in $\Emb(S,M)/\Diffp(S)$.

In this thesis, we consider \emph{unlabeled shapes}, defined as $\Emb(S,M)/\Diffp(S)$.  This convention better aligns with the implicit representations of shapes introduced in \autoref{sec:implicit_representations} (See also \autoref{rmk:psi_for_unlabeled_shapes}). 
However, we note that the theorems and propositions in this section hold under both definitions.
\end{remark}


\subsection{Shapes as de Rham currents}
% The tangent vectors of these mappings can be described extrinsically using vector fields on the ambient manifold $M$. Each tangent vector $\dot \gamma\in T_\gamma \Emb(S,M)$ can be written extrinsically with some $v\in \diffp(M)$ such that $v\circ\gamma=\dot \gamma$. Actually $v$ can be taken within the divergence-free vector fields $\sdiffp(M)$ due to the codimensionality $n=m-2$.  So we have
% \begin{align}
%     &T_\gamma \Emb(S,M)=\{v\circ \gamma \mid v\in \sdiffp(M)\},\\
%     &T_{[\gamma]}\UEmb(S,M)=\{v\circ\gamma +  \sdiffp_{[\gamma]}(S) \mid v\in \sdiffp(M)\}
% \end{align}
% where $\sdiffp_{[\gamma]}(S) = d\gamma \sdiffp(S)$ is the Lie algebra for the stabilizer group $\SDiffp_{[\gamma]}(M)$.

%  The identification $\cI$ defines a topology of $\cD_{\UEmb(S,M)}$. A set $A\subset \cD_{\UEmb(S,M)}$ is open if $\cI^{-1}A\subset \UEmb(S,M)$ is open. The tangent space at $\delta_\gamma\in \cD_{\UEmb(S,M)}$ is given by $T_{\delta_{\gamma}} \cD_{\UEmb(S,M)}=d\cI|_{\gamma} T_{\gamma}\UEmb(S,M)$. That is,
% \begin{align}
%     d\cI|_{\gamma} \colon {\dot \gamma}= \dot \gamma + d\gamma \sdiffp(S) \mapsto \cL_v \delta_\gamma \textup{ with some } v\in \sdiffp(M) \textup{ s.t. } {v\circ \gamma} = {\dot \gamma}.
% \end{align}

% \begin{remark}[Casimir elements and link invariants]
%     A function on $B_{m-2}^{CR}(M)$ which is piecewise constant on each coajoint orbit $\cO_h=\{\Ad^*_f h\mid f\in \SDiff_0(M)\}$ is called a Casimir element. In particular, it is called a link (knot) invariant on $\cD_{\UEmb(S,M)}$ when $S=\prod_{i=1}^N\SS^1$.
    
%     In infinite-dimensional settings, functions may or may not be Casimir elements depending on the dual space we consider. An example is \emph{helicity}, a well-studied quantity in fluid dynamics and electromagnetism, which measures the linkage of vorticity and magnetic fields.
% %     \begin{align}
% %         \on{Hel}(\beta)=\int_M \beta \wedge h, \quad 
% %     \end{align}
% % \textcolor{red}{
% % \begin{align}
% %     \int_\gamma \lambda_\psi = \int \delta_\gamma \wedge \lambda_\psi
% % \end{align}
% % \begin{align}
% %     \lambda_\psi= \psi^*\tau^* \sigma
% % \end{align}
% % where $\tau:z\mapsto z/|z|$ and $\sigma$ is a standard volume form of $\SS^1$.
% % }
%     % for $\beta\in B_{m-2}^{CR}(M)$ where $h$ is any element of $ \partial^{-1}\beta$. 
%     % Note that $\on{Hel}(\beta)$ is independent of the choice of $h$ provided that $ H_{m-1}^{CR}(M)=0$. 
%     % 
%     Helicity is a Casimir elemnt on the smooth dual $B_{dR}^2(\RR^3)\eqsim \Om^1(\RR^3)/d\Om^0(\RR^3)$ of $\sdiff(\RR^3)$ but is not on $B_2^{CR}(\RR^3)$ in particular on $\cD_{\UEmb(\SS^1,\RR^3)}$ as it carries only the information of writh on $\cD_{\UEmb(\SS^1,\RR^3)}$ while helicity is the sum of writh and twist \cite{MoffattRicca1992helicity}. Hence it is not a link invariant.
% \end{remark}

% \begin{align}
%     &dv^\flat=\sum_i \Gamma_i \delta_{x_i}, \quad \Gamma_i \in \RR\setminus 0\\
%     & (\partial_t + \cL_v)\delta_{x_i}=0, \forall i
% \end{align}

% The coadjoint representation on the Lie algebra level also extends from $B_{dR}^2(M)$ to $B^{CR}_{m-2}(M)$. Recall that $\ad\colon \sdiff(M)\to \on{Der}(\sdiff(M))$ is  $\ad_u v=[u,v]$. Then the coadjoint representation $\ad^*\colon \sdiff(M)\to \on{Der}(B^{CR}_{m-2}(M))$ is defined by $\langle \ad^*_u \beta, v\rangle=\langle  \beta, -\ad_{u} v\rangle$. It is easy to see that $\cL_u \beta= \ad^*_u \beta$, hence each tangent vector on a coadjoint orbit is of form $-\ad^*_u \beta $.

% Rigorously, limit of $(\partial^{-1}h)_n\subset \Omega^{1}(M)=\sdiff^*(M)$ and $\lim_n \langle \partial (\partial^{-1}h)_n, v \rangle= \lim_n \langle \partial (\partial^{-1}h)_n, \alpha \rangle$ with $(\partial^{-1}h)_n$ approching to a representative of $\partial^{-1}h$.


% \begin{remark}[Immersions and embeddings]
% We can also consider the space of immersions
% \begin{align}
%     \Imm(S,M)\coloneqq \{\gamma\in C^\infty(S,M) \mid \on{rank} d\gamma=n \},
% \end{align}
% which is a larger manifold than $\Emb(S,M)$.
% There are a few different properties between $\Imm(S,M)$ and $\Emb(S,M)$ and between $\Gri(S,M)$ and $\UEmb(S,M)$. First note that both $\Imm(S,M)$ and $\Emb(S,M)$ are not closed manifolds; constant maps are not contained in both and self-intersecting immersions are not in $\Emb(S,M)$. We may still guarantee that a time-series of immersions evolving under a large class of dynamics stay in $\Imm(S,M)$. For example, Hamiltonian dynamics for which the $(d-2)$ dimensional volume $\int_\gamma ds$ \ie, length, area, etc...  is a constant in motion fall into this category. Such a trajectory-wise closedness may be not easily guaranteed at the level of $\Emb(S,M)$ except for Hamiltonian functions with strong repulsive properties.

% At the level of Grassmanians, embeddings have certain advantages. Note that $\UEmb(S,M)$ equipped with the quotient topology is an infinite dimensional manifold and is a principal $\Diffp(S)$-bundle. However, $\Gri(S,M)$ is not a manifold but an orbifold since $\Diffp(S)$ action is not free on the fiber of every basepoint. For example, consider a double curve $\gamma\in \Imm(\SS^1,\RR^3)$ given by $\gamma(s)=(\cos(2s), \sin(2s), 0)$. Then the stabilizer group $\Diffp_\gamma(\SS^1)$ consists of the identity map and the shift $s\mapsto s+\pi$, hence is isomorphic to $\ZZ_2$. In this way, $\Gri(S,M)$ is an infinite dimensional orbifold that have singularities with finite groups as their stabilizers. In this way, $\UEmb(S,M)$ have higher homogeneity. In addition, embeddings can be straightforwardly treated in our upcoming implicit settings. For these reasons, we mainly work on embeddings rather than immersions in this article.
% \end{remark}

Since the shape space $\UEmb(S,M)$ is a subset of the space of $(m-2)$-dimensional submanifolds in $M$, there is a natural injection into the space of de Rham currents:
\begin{align}
    \cI\colon \UEmb(S,M) &\to \cD^{dR}_{m-2}(M)\\
    \gamma &\mapsto \delta_\gamma.
\end{align}
We denote the image of $\cI$ by $\cD_{\UEmb(S,M)}$, and for a $\Diff_0(M)$-orbit $\cO \subset \UEmb(S,M)$, we write $\cD_\cO \coloneqq \cI(\cO)$.

The tangent space at $\delta_\gamma \in \cD_\cO$ is given by the pushforward of the tangent space at $\gamma$:
\begin{align}
    T_{\delta_{\gamma}} \cO = d\cI|_{\gamma} \left(T_{\gamma} \cO\right).
\end{align}
More explicitly,
\begin{align}
    d\cI|_{\gamma} \colon {\dot \gamma}= v \circ \gamma \mapsto -\cL_v \delta_\gamma, \quad \text{for some } v \in \diff(M),
\end{align}
where $\cL_v$ denotes the Lie derivative in the sense of currents, as defined in \autoref{sec:orbits_of_currents}. Therefore, the tangent space at $\delta_\gamma$ is
\begin{align*}
    T_{\delta_\gamma} \cD_{\cO} = \{-\cL_v \delta_\gamma \mid v \in \diff(M)\}.
\end{align*}
Note also that the injection $\cI$ commutes with the $\Diff_0(M)$-action in the sense that $f_* \cI = \cI \circ f$ for all $f \in \Diff_0(M)$, as discussed in \autoref{sec:orbits_of_currents}.
In the remainder of the chapter, we will occasionally identify $\cO$ with $\cD_\cO$, and $\UEmb(S,M)$ with $\cD_{\UEmb(S,M)}$, and their tangent bundles, without explicitly stating so.


\section{Symplectic structure on the codimenion-2 shape space}\label{sec:symplectic_structure_on_codim2}
The space $\cO \cong \cD_{\cO}$ is a \emph{weak symplectic manifold}.  That is, it is equipped with a closed and \emph{weakly non-degenerate} 2-form. The Marsden--Weinstein (MW) form on \(\cO\) is given by
\begin{align}\label{def:MW_on_currents}
    \omega_{\delta_\gamma}(\dot \delta_\gamma, \ring \delta_\gamma) 
    = \langle \delta_\gamma, \iota_v \iota_u \mu \rangle 
    = \int_\gamma \iota_v \iota_u \mu,
\end{align}
where $\dot \delta_\gamma = -\cL_u \delta_\gamma$ and $\ring \delta_\gamma = -\cL_v \delta_\gamma$ for vector fields $u, v \in \Gamma(TM)$. When convenient, we also write $\omega_\gamma(\dot\gamma, \ring\gamma) \coloneqq \omega_{\delta_\gamma}(\dot \delta_\gamma, \ring \delta_\gamma)$ for $\dot \gamma = u \circ \gamma$ and $\ring \gamma = v \circ \gamma$.

The MW form $\omega$ is closed and weakly-nondegnerate in the sense that the associated flat operator
\begin{align}
    \flat^{\omega} \colon T\cO &\to T^*\cO \\
    \dot \gamma &\mapsto \iota_{\dot \gamma} \omega
\end{align}
is injective. In this chapter and the next few chapters, we refer to such weak symplectic forms simply as \emph{symplectic}. For background on weak symplectic geometry, we refer the reader to \cite[Chapter VI]{Michor1984convenient} and \cite[Appendix A]{bauerMichorIshida2024symplectic}.

On the parametrized shape space $\tilde \cO \coloneqq \pi^{-1}(\cO)$, where $\pi \colon \Emb(S,M) \to \UEmb(S,M)$ is the natural projection, the pullback 2-form $\tilde\omega \coloneqq \pi^* \omega \in \Omega^2(\tilde \cO)$ is written as
\begin{align}
    \tilde\omega_{\tilde\gamma}(\dot{\tilde\gamma}, \ring{\tilde\gamma}) 
    = \int_S \mu(\dot{\tilde\gamma}, \ring{\tilde\gamma}, \partial_{s_1} \tilde\gamma, \ldots, \partial_{s_n} \tilde\gamma) \, ds_1 \ldots ds_n
\end{align}
in local coordinates $s = (s_1, \ldots, s_n)$ on $S$. From this expression, we see that $\tilde\omega$ has a nontrivial kernel consisting of infinitesimal reparametrizations, that is, $ \ker \tilde\omega|_{\tilde\gamma} = \ker d\pi|_{\tilde\gamma}=d\paramgam (\diff(S))$ at each $\paramgam\in \tilde \cO$.
 Thus, $\tilde\omega$ is not symplectic, but merely presymplectic \ie, a closed 2-form.
 


% The space $\Emb(S,M)$ is a presymplectic manifold. Let us define a 2-form $\omega$ on $\Emb(S,M)$ by
% \begin{align}\label{def:MW_presymplectic}
%     \omega_\gamma(\dot \gamma,\ring \gamma)\coloneqq \int_\gamma \iota_v \iota_{u} \mu
% \end{align}
%  where $u, v$ are any vector fields on $M$ such that $\dot \gamma=u\circ \gamma$ and $\ring \gamma=v\circ \gamma$.
% With local charts, we can write 
%  \begin{align}
%      % \omega_\gamma(u,v) \coloneqq \int_\gamma \iota_u \iota_{v} \mu=\int_S \mu( u,v, D_{s_1}\gamma, \ldots, D_{s_{d-2}}\gamma)ds_1\ldots ds_{d-2}
%     \omega_\gamma(\dot \gamma,\ring \gamma) =\int_S \mu( \dot \gamma,\ring \gamma, \partial_{s_1}\gamma, \ldots, \partial_{s_n}\gamma)ds_1\ldots ds_{n}.
%  \end{align}

 % where $D_{s_i}=\partial_{x_i}/|\partial_{x_i}\gamma|$ and $ds_i=|\partial_{x_i}\gamma|dx_i$ for each $i$ are the derivative and the measure with the unit-length parameter.

%  Note that the form $\omega$ is presymplectic but not symplectic \ie, it is closed but degenerate, with kernel $\ker d\pi=d\gamma \diffp(S)$. Since $\omega$ is invariant under the right action of $\Diffp(S)$ and horizontal in the sense that it vanishes on the vertical tangent space $d\gamma \diffp(S)$, the 2-form $\omega$ on $\Emb(S,M)$ factors to a 2-form $\bar \omega$ on the shape space $\UEmb(S,M)$ by $\omega=\pi^*\bar\omega $ where $\pi^*$ denotes the pullback by $\pi$. Explicitly, this is 
%  \begin{align}\label{def:MW_on_UEmb}
%     \bar \omega_{[\gamma]}([\dot \gamma], [\ring\gamma])=\int_\gamma \iota_v \iota_u\mu,
%  \end{align}
% which is independent of the choice of representatives, $\gamma, \dot\gamma, \ring\gamma$, namely $u,v$. 


\begin{example}[MW structure on the space of space curves]
On the space of space curves $\UEmb(\SS^1,\RR^3)$ where $\RR^3$ is equipped with the standard Euclidean volume form $\mu = dx_1 \wedge \cdots \wedge dx_3$, the MW form takes the expression
\begin{align}
    \omega_\gamma(\dot \gamma, \ring \gamma) = \int_{\SS^1} \det(\partial_s \paramgam, \dot \paramgam, \ring \paramgam) \, ds,
\end{align}
where $\paramgam\in \Emb(S,M)$ is any parametrization of $\gamma\in \UEmb(S,M)$ and the integrand denotes the determinant of the $3 \times 3$ matrix whose columns are the vectors $\partial_s \paramgam(s)$, $\dot \paramgam(s)$, and $\ring \paramgam(s)$ in $\RR^3$.
Note that $\ker \tilde \omega|_\paramgam$ consists of infinitesimal reparametrizations of the form $a \partial_s \tilde \gamma$ with $a \in C^\infty(\SS^1)$, corresponding to tangent vector fields along the curve $\gamma$ viewed as a submanifold of $\RR^3$. 
\end{example}

% KKS form
% \begin{remark}[KKS form]
% When $\gamma\in \UEmb(S,M)$ is the boundary of a $m-1$ dimensional manifold, the MW form defined on the orbit $\cO_\gamma$ is a KKS form.

% The MW form is defined more generally on any $\SDiff(M)$-orbit in $B_2^{CR}(M)$, defined in the same way;
% \begin{align}
%     \omega_{\beta}(-\cL_u \beta, -\cL_v \beta) \coloneqq \langle \beta, \iota_v \iota_u \mu \rangle.
% \end{align}
% This is formally a Kirillov--Kostant--Souriau (KKS) form, as follows.

% In general, the dual $\mathfrak{g}^*$ of a Lie algebra $\mathfrak{g}$ is foliated by coadjoint orbits, and on each orbit, a natural symplectic form called the KKS form is defined by
% \begin{align}
%     \zeta_\beta^{\mathrm{KKS}}(\ad_u^* \beta, \ad_v^* \beta) \coloneqq \langle \beta, \ad_u v \rangle = \langle \beta, [u,v] \rangle \qquad \beta\in \frak{g}^*, u,v \in \mathfrak{g},
% \end{align}
% where $\ad_u$ and $\ad_u^*$ denote the adjoint and coadjoint representations of $\mathfrak{g}$ and $\mathfrak{g}^*$, defined by $\ad_u^v=[u,v]$ and $\langle\ad_u^*\beta, v \rangle=\langle \beta,\ad_u v\rangle$ (for details, see \cite{khesin2008geometry}, for example).

% For $\mathfrak{g} = \exdiff(M)$, we can formally regard $B_2^{CR}(M)$ as $\frak{g}^*$ via the flux (\autoref{def:flux})  while $B_2^{CR}(M)$ is larger than the topological dual $\exdiff(M)^*=B_{dR}^1(M)$.

% we have $\ad_u = \cL_u$ and $\ad^*_u=-\cL_u$. Then it is easy to see that $\omega_\beta=\zeta_\beta^{KKS}$ by a direct computation. 

% \end{remark}


\subsection{Hamiltonian vector fields}\label{sec:explicit_Hamiltonian_field}
For a function $\cH\colon \cO\to \RR$ with $d\cH\in \im \flat^\omega$,  the Hamiltonian vector field $X_\cH\in \Gamma(T\cO)$ with respect to the MW form $\omega$ is the unique vector field satisfying 
\begin{align}
    d\cH=\io_{X_\cH} \omega
\end{align} 
and the Hamiltonian system is the flow along $X_\cH$. That is,
\begin{align}
    \partial_t \gamma = X_\cH(\gamma).
\end{align}

\begin{example}
    On $\UEmb(\SS^1,\RR^3)$ where $\RR^3$ is equipped with the standard Euclidean metric, the length function as the Hamiltonian yields the binormal equation,
\begin{align}\label{eq:binormal_equation}
    \partial_t \gamma = D_s \gamma \times D_s^2 \gamma.
\end{align}
Here the right-hand side is understood as $d\pi|_\paramgam (D_s \paramgam \times D_s^2 \paramgam)$ with fibration $\pi\tilde \cO\to\cO$ and any parametrization $\paramgam \in \pi^{-1}(\gamma)$, where $D_s = \partial_s / |\partial_s \paramgam|$ is the derivative with respect to the arc-length parameter. 

More generally, on $\UEmb(S,\RR^m)$ with an oriented closed $n$-manifold $S$, choosing the $n$-dimensional volume of $\gamma$ as the Hamiltonian $\cH$ yields the Hamiltonian vector field $X_\cH$ as the mean curvature normal on the submanifold $\gamma(S)$, rotated by 90 degrees in the normal bundle $N\gamma(S)$ \cite{HallerVizman2003NonlinearGA,Khesin2012vortexMembrane}. While the binormal equation \eqref{eq:binormal_equation} is known to be an infinite-dimensional integrable system forming a KdV-type hierarchy \cite{chern_knoeppel_pedit_pinkall_2020}, the integrability of its higher-dimensional analogues remains less understood.
\end{example}


% For a function $\cH \colon \Emb(S,M)\to \RR$ such that $d\cH\in \Im \flat^\omega \subset T^*\Emb(S,M)$, there is a vector field $X_\cH\in \Gamma(T\Emb(S,M))$ satisfying
% \begin{align}
%     d\cH=\io_{X_\cH} \omega.
% \end{align}
% As $\omega$ is merely presymplectic, there are infinitely many such $X_\cH$ up to $\ker \omega_\gamma$. 

% When $\cH$ is invariant under reparametrization by $\Diffp(S)$, $\cH$ factors to a function $\bar \cH$ on $\UEmb(S,M)$ by $\pi^* \bar \cH=\cH$. Since $(\UEmb(S,M),\bar \omega)$ is a symplectic manifold, the Hamiltonian vector field $X_{\bar \cH}$ is uniquely defined by $\io_{X_{\bar \cH}}\bar \omega=d\bar \cH$ and satisfies
%  \begin{align}
%    X_{\bar \cH} \circ \pi= d\pi|_\gamma X_\cH
% \end{align}
% with a Hamiltonian vector field $X_\cH$ on $\Emb(S,M)$.


% \begin{remark}[Infinite-dimensional integrable system]
%     There is a hierachy of Hamiltonian vector fields  $\{Y_i\}_{i=0}^\infty$ on $X=\Imm(\bigsqcup_{i=0}^k\SS^1,\RR^3)$ such that $\omega(Y_i, Y_j)=0$ namely their Hamiltonian functions are Poisson commutative [Cite Chern, Kn\"oppel, Peddit, Pinkall]. This defines an infinite-dimensional integrable system in the Loouville sense. 
% \end{remark}


\subsection{Liouville form}\label{sec:Liouville_classical}
On a symplectic manifold, a \emph{Liouville form} is a 1-form whose exterior derivative equals to the symplectic form. 
Previous work \cite{Padilla2019bubbleRings,Tabachnikov2017bicycle} showed that the MW structure $\omega$ on the space of space curves $\UEmb(\SS^1,\RR^3)$ is exact by providing an explicit Liouville form. The 1-form $\eta$ defined by
\begin{align}\label{eq:liouville_classical_space_curves}
    \eta_\gamma(\dot \gamma)
    = \frac{1}{3} \int_{\SS^1} \det(\paramgam, \dot \paramgam, \partial_s \paramgam) \, ds
    % =\frac{1}{3}\int_{\SS^1} \det (\gamma,\partial_s \gamma, \dot \gamma) ds 
\end{align}
where $\paramgam$ is any parametrization of $\gamma$, satisfies $\omega = d\eta$.

A natural question is to consider its higher-dimensional generalization. However, the proof in \cite{Tabachnikov2017bicycle} relies on integration by parts using an explicit parametrization of $\SS^1$, which does not directly extend to higher dimensions.

Here, we present a new result for a generic closed and oriented $n$-dimensional manifold $S$ and $M = \RR^m$ equipped with the Euclidean volume form $\mu = dx_1 \wedge \ldots \wedge dx_m$ (\autoref{th:liouville_classical}), and then extend it to a slightly more general setting (\autoref{th:extended_classical_Liouville}).

For $\dot \delgam = - \cL_u \delta_\gamma \in T_{\delgam} \cD_{\cO}$, define
\begin{align}\label{eq:liouville_classical}
    \eta_{\delgam}(\dot \delgam) 
    = \frac{1}{m} \langle \delta_\gamma, \iota_u \iota_x \mu \rangle
    = \frac{1}{m} \int_\gamma \iota_u \iota_x \mu.
\end{align}

This 1-form \eqref{eq:liouville_classical} admits an explicit representation analogous to  \eqref{eq:liouville_classical_space_curves}:
\begin{align}
    \eta_\gamma(\dot \gamma)
    = \frac{1}{m} \int_S \det(\paramgam, \dot \paramgam, \partial_{s_1} \paramgam, \ldots, \partial_{s_{m-2}} \paramgam) \, ds_1 \ldots ds_{m-2}
\end{align}
where $\dot \gamma = u \circ \gamma$ and $s=(s_1,\ldots,s_n)$ is local coordinates.

\begin{theorem}[Liouville form on $\RR^m$]\label{th:liouville_classical}
    Suppose that $\RR^m$ is equipped with the standard volume form $\mu = dx_1 \wedge \cdots \wedge dx_m$. Then we have $d \eta = \omega$.
\end{theorem}

A special case of \autoref{th:liouville_classical} for $\Emb(\SS^1,\RR^3)$ is given in \cite[Proposition 2.1]{Tabachnikov2017bicycle}, where the proof is based on integration by parts using an explicit parametrization of a space curve.

However, this approach does not extend to the general setting $\Emb(S,M)$ unless $S$ admits an explicit global parametrization, such as $S = \TT^n$. In contrast, we prove the general case of \autoref{th:liouville_classical} in a parametrization-free fashion.

We formulate the proof using currents, as this computational routine will be reused throughout the present and the next chapters.
To that end, we introduce an auxiliary result. First, note that each vector field $u \in \Gamma(TM)$ induces a vector field $X^u$ on the $\Diff_0(M)$-orbit $\cD_\cO$ in $\cD_{m-2}(M)$, defined by
\begin{align}
    X^u(\delgam) = -\cL_u \delgam.
\end{align}
In other words, $X^u$ is the fundamental vector field associated with the $\Diff_0(M)$-action on $\cD_\cO$, since $-\cL_u \delgam = \frac{d}{dt} \big|_{t=0} {\on{Fl}_{u}^t}_* \delgam$, where $\on{Fl}_{u}^t \in \Diff_0(M)$ is the time-$t$ flow of $u$ on $M$ (see \autoref{sec:orbits_of_currents}).
This defines the following homomorphism; 
\begin{lemma}\label{lem:Lie_algebra_homomorphism}
    Let $\hat\cD$ be a $\Diff_0(M)$-orbit in the space of $k$-currents $\cD_k(M)$. Then the map
    \begin{eqnarray}\label{eq:fundamental_vectorfield_mapping}
        \Gamma(TM) &\to& \Gamma(T\hat\cD) \\
        u &\mapsto& X^u \nonumber
    \end{eqnarray}
    is a Lie algebra homomorphism. That is, for  $u,v \in \Gamma(TM)$, we have
    \begin{align}
        [X^u, X^v] = X^{[u,v]}.
    \end{align}
\end{lemma}

\begin{proof}
    To compute $[X^u, X^v] = \cL_{X^u} X^v$, we use the formula for the Lie derivative:
    \begin{align}\label{eq:LieDerivative_vector_fields_on_currents}
        \cL_{X^u} X^v(h)
        = \frac{d}{dt} \Big|_{t=0} D\Phi^{-t}_{X^u}\left[ X^v(\Phi^{t}_{X^u}(h)) \right], \quad h \in \hat\cD.
    \end{align}
    Here $\Phi^t_X \colon \hat\cD \to \hat\cD$ denotes the flow map on $\hat \cD$ along a given vector field $X \in \Gamma(T\hat\cD)$, defined by the ODE
    \begin{align}
        &\frac{d}{dt}(\Phi^t_X(h)) = X(\Phi^t_X(h)), \\
        &\Phi^0_X = \id_{\hat\cD}
    \end{align}
     and  $D\Phi^t_X|_h \colon T_h\hat\cD \to T_{\Phi^t_X(h)}\hat\cD$ is the differential of $\Phi^t_X$ at $h$.

    For the fundamental vector field $X^u$, the flow is given by $\Phi^t_{X^u}(h) = {\on{Fl}^t_u}_* h$ by definition. Using this, we have
    \begin{align}
        X^v(\Phi^t_{X^u}(h))
        &= \frac{d}{ds} \Big|_{s=0} {\on{Fl}_v^s}_* (\Phi^t_{X^u}(h)) 
        = \frac{d}{ds} \Big|_{s=0} {\on{Fl}_v^s}_* {\on{Fl}_u^t}_* h,
    \end{align}
    and obtain
    \begin{align}
        \frac{d}{dt} \Big|_{t=0} D\Phi^{-t}_{X^u} \left[ X^v(\Phi^{t}_{X^u}(h)) \right]
        &= \frac{d}{dt} \Big|_{t=0} \frac{d}{ds} \Big|_{s=0} \Phi^{-t}_{X^u} \left( {\on{Fl}_v^s}_* {\on{Fl}_u^t}_* h \right) \\
        &= \frac{d}{dt} \Big|_{t=0} \frac{d}{ds} \Big|_{s=0} {\on{Fl}_u^{-t}}_* {\on{Fl}_v^s}_* {\on{Fl}_u^t}_* h \\
        &= -\cL_u \cL_v h + \cL_v \cL_u h \\
        &= -\cL_{[u,v]} h = X^{[u,v]}(h),
    \end{align}
    which completes the proof.
\end{proof}

\begin{remark}\label{rmk:Lie_algebra_homomorphism}
  At first glance, \autoref{lem:Lie_algebra_homomorphism} may seem to contradict the standard result that the fundamental vector field mapping $\mathfrak{g} \to \Gamma(TN)$ on a manifold $N$ is an \emph{anti} Lie algebra homomorphism when the $G$-action on $N$ is a left action \cite[Theorem 20.18]{lee2012introduction}. The sign difference in our setting occurs because of our identification of vector fields $\diff(M)$ with derivations $\operatorname{Der}(C^\infty(M))=\Gamma(TM)$, where the mapping
 $\on{der}\colon \diff(M) \to  \operatorname{Der}(C^\infty(M))$ by $u\mapsto \cL_u$ is an anti Lie algebra homomorphism. 
 % In fact,  $\on{der}([u,v]_{\diff(M)})=\on{der}(\frac{d}{dt}\frac{d}{ds}|_{t=s=0} \on{Fl}_u^t\circ  \on{Fl}_v^s\circ  \on{Fl}_u^{-t})$ while $[\on{der}(u),\on{der}(v)]_{\on{Der}(C^\infty(M))}|_x=\frac{d}{dt}\frac{d}{ds}|_{t=s=0}  \on{Fl}_u^{-t}\circ  \on{Fl}_v^s\circ  \on{Fl}_u^{t}(x)$.
\end{remark}

\begin{proof}[Proof of \autoref{th:liouville_classical}]
   For $\dot \delgam=-\cL_u \delgam, \ring \delgam=-\cL_v \delgam$ with some $u,v\in \diff(M)$, we compute $d\eta_\delgam(\dot \delgam, \ring\delgam) = d\eta_\delgam(X^u, X^v)$ explicitly, using the fundamental vector fields $X^u, X^v \in \Gamma(T\cD_\cO)$ given by $X^u(\delgam) = -\cL_u \delta_\gamma$, $X^v(\delgam) = -\cL_v \delgam$. We use the formula
    \begin{align}
      d\eta_{\delta_\gamma}(X^u, X^v)
    &= \cL_{X^u} \iota_{X^v} \eta_{\delgam} - \cL_{X^v} \iota_{X^u} \eta_{\delgam} 
         - \iota_{[X^u, X^v]} \eta_{\delgam}.
  \end{align}
  
  First, by \autoref{lem:Lie_algebra_homomorphism}, we have
  \begin{align}
      \eta_{\delgam}( [X^u, X^v]) 
      &= \eta_{\delgam}( X^{[u,v]})
      = \eta_{\delgam}( -\cL_{[u,v]} \delgam)
      = \frac{1}{m} \langle \delgam, \iota_{[u,v]} \iota_x \mu \rangle.
  \end{align}
Next, we compute
\begin{align}
\cL_{X^u} \iota_{X^v} \eta_{\delgam} 
&= \frac{d}{dt} \Big|_{t=0} \eta(X^v(\Phi_{X^u}^t(\delgam))) 
= \frac{1}{m} \frac{d}{dt} \Big|_{t=0} \langle {\on{Fl}_u^t}_* \delgam, \iota_v \iota_x \mu \rangle \\
&= \frac{1}{m} \frac{d}{dt} \Big|_{t=0} \langle  \delgam, {\on{Fl}_u^t}^* \iota_v \iota_x \mu \rangle 
= \frac{1}{m} \langle \delgam, \cL_u \iota_v \iota_x \mu \rangle,
\end{align}
where $\Phi^t_{X^u}$ is the time-$t$ flow map of $X^u$, given by $\Phi_{X^u}^t(\delgam) = (\on{Fl}_u^t)_* \delgam$, as explained in the proof of \autoref{lem:Lie_algebra_homomorphism}. Similarly, we have
\(\cL_{X^v} \iota_{X^u} \eta_{\delgam} 
= \frac{1}{m} \langle \delgam, \cL_v \iota_u \iota_x \mu \rangle.\)

Combining these results, we obtain
\begin{align}
d\eta_{\delgam}(X^u, X^v)
&= \frac{1}{m} \left( \langle \delgam, \cL_u \iota_v \iota_x \mu \rangle 
- \langle \delgam, \cL_v \iota_u \iota_x \mu \rangle 
- \langle \delgam, \iota_{[u,v]} \iota_x \mu \rangle \right)\\
&=\frac{1}{m} \left( \langle \delgam, \cL_u \iota_v \iota_x \mu 
- \cL_v \iota_u \iota_x \mu -\iota_{[u,v]} \iota_x \mu \rangle \right).
\end{align}

By direct computation, we simplify the expression inside the dual pairing;
\begin{align}
\cL_u \iota_v \iota_x \mu - \cL_v \iota_u \iota_x \mu - \iota_{[u,v]} \iota_x \mu 
&= \cL_u \iota_v \iota_x \mu - \cL_v \iota_u \iota_x \mu - \cL_u \iota_v \iota_x \mu + \iota_v \cL_u \iota_x \mu \\
&= \iota_v \cL_u \iota_x \mu - \cL_v \iota_u \iota_x \mu \\
&= \iota_v d \iota_u \iota_x \mu + \iota_v \iota_u d \iota_x \mu - d \iota_v \iota_u \iota_x \mu - \iota_v d \iota_u \iota_x \mu \\
&= \iota_v \iota_u \cL_x \mu - d \iota_v \iota_u \iota_x \mu.
\end{align}

Note that $\cL_x \mu = m \mu$ for the volume form $\mu = dx_1 \wedge \cdots \wedge dx_m$, and that $\partial \delgam = \delta_{\partial \gamma} = 0$. Therefore, we obtain
\begin{align}
d\eta_{\delta_\gamma}(X^u, X^v)
&= \frac{1}{m} \langle \delta_\gamma, \iota_v \iota_u \cL_x \mu - d \iota_v \iota_u \iota_x \mu \rangle \\
&= \langle \delta_\gamma, \iota_v \iota_u \mu \rangle - \frac{1}{m} \langle \partial \delta_\gamma, \iota_v \iota_u \iota_x \mu \rangle \\
&= \omega_{\delta_\gamma}(X^u, X^v)-0.
\end{align}
Thus we have shown $d\eta=\omega$.
\end{proof}


% \begin{remark}
%     In the proof of Theorem \ref{th:liouville_classical}, we used $u,v$ and $x$ as fixed vector fields in $\RR^m$. We can alternatively consider the perturbation of $\gamma$ namely $\delta_\gamma$ by setting $\tilde \gamma\colon S \times (-\epsilon,\epsilon)^2\to M$ by $\tilde \gamma=\gamma+t_1 \dot \gamma + t_2 \ring \gamma$ and tangent vectors $\dot \gamma\coloneqq \partial_{t_1} \gamma(s,t_1,t_2) \eqqcolon u\circ \gamma$ and $\ring \gamma\coloneqq\partial_{t_2} \gamma(s,t_1,t_2) \eqqcolon v\circ \gamma$. In that case, the tangent vectors $\dot\delta_\gamma$ and $\ring \delta_\gamma$ commute but $\nabla_{\dot \delta_\gamma}\eta(\ring \delta_\gamma)$ and $\nabla_{\ring\delta_\gamma}\eta(\dot \delta_\gamma)$ have additional terms, namely we have 
%     \begin{align}
%        \nabla_{\dot \delta_\gamma}\eta(\ring \delta_\gamma)&= \langle \delta_\gamma, \cL_u\iota_v \iota_x\mu \rangle+\langle \delta_\gamma, \iota_{\dot v} \iota_x\mu \rangle,\\
%        \nabla_{\ring \delta_\gamma}\eta(\dot \delta_\gamma)&= \langle \delta_\gamma, \cL_v\iota_u \iota_x\mu \rangle+\langle \delta_\gamma, \iota_{\ring u} \iota_x\mu \rangle
%     \end{align} 
%     where $\dot v=\partial_{t_1,t_2}\tilde \gamma$ and $\ring u=\partial_{t_2,t_1}\tilde \gamma$, so they are equal. Hence we have the same computational result in the end.
% \end{remark}

\autoref{th:liouville_classical} can be extended from \(\RR^m\) to a general manifold \(M\) equipped with an exact volume form;
\begin{theorem}[Liouville form on \(M\) with an exact volume form]\label{th:extended_classical_Liouville}
Let \(\mu\) be a volume form on \(M\) such that \(\mu = d\nu\) for some \(\nu \in \Omega^{m-1}(M)\). Define a 1-form \(\eta\) on \(\cD_\cO\) by
\begin{align}
    \eta_\delgam(-\cL_u \delta_\gamma) = \langle \delta_\gamma, \iota_u \nu \rangle.
\end{align}
Then \(d\eta = \omega\).
\end{theorem}

\begin{example}
The Liouville form \(\eta\) in \autoref{th:liouville_classical} is a special case of \autoref{th:extended_classical_Liouville} with \((M,\mu) = (\RR^m, dx_1 \wedge \cdots \wedge dx_m)\) and
\begin{align}
    \nu = \frac{1}{m} \sum_i x_i\, dx_{i+1} \wedge \cdots \wedge dx_{i+m-1},
\end{align}
where the indices are taken modulo \(m\).
\end{example}

\begin{proof}[Proof of \autoref{th:extended_classical_Liouville}]
 Let $\dot \delgam=-\cL_u \delgam, \ring \delgam=-\cL_v \delgam$ with some $u,v\in \diff(M)$. Using the same computational routine as in the proof of \autoref{th:liouville_classical}, with the fundamental vector fields \(X^u, X^v\in \Gamma(T\cD_\cO)\),
we compute:
\begin{align}
    d\eta_{\delta_\gamma}(\dot \delta_\gamma, \ring \delta_\gamma)
    &= \cL_{X^u} \iota_{X^v} \eta - \cL_{X^v} \iota_{X^u} \eta - \iota_{[X^u, X^v]} \eta \\
    &= \langle \delta_\gamma, \cL_u \iota_v \nu \rangle 
     - \langle \delta_\gamma, \cL_v \iota_u \nu \rangle 
     - \langle \delta_\gamma, \cL_{[u,v]} \nu \rangle \\
    &= \langle \delta_\gamma, \iota_v \iota_u d\nu - d \iota_v \iota_u \nu \rangle \\
    &= \langle \delta_\gamma, \iota_v \iota_u \mu \rangle - \langle \partial \delta_\gamma, \iota_v \iota_u \nu \rangle \\
    &= \omega_{\delta_\gamma}(\dot \delta_\gamma, \ring \delta_\gamma)-0.
\end{align}
\end{proof}

\begin{remark}[Proof via tilde calculus]
\autoref{th:extended_classical_Liouville} can alternatively be proved using the tilde calculus introduced by Haller and Vizman \cite{HallerVizman2003NonlinearGA, Vizman2011hatcalculus}. We explain this approach in the appendix of this chapter (\autoref{sec:Liouville_via_hat_calculus}).
\end{remark}


\begin{remark}[Liouville form on a closed manifold]
We are not aware of the existence or non-existence of a Liouville form on a general closed manifold \((M,\mu)\). 
There are a few special cases where exactness has been proved. For example, the space of two distinct points on the sphere, equipped with the MW form (see \autoref{sec:base_dimension_2}), is symplectomorphic to the cotangent bundle of a certain space and therefore admits a Liouville form \cite{Oakley2013twoPoints}.  

However, we speculate that exactness of the MW form is unlikely in the general case when the volume form \(\mu\) is not exact, or at least that a Liouville form may not admit an explicit expression.

In contrast, we will explicitly define a Liouville form in the prequantum sense (\autoref{def:formal_prequantization}) on the space of implicit representations of submanifolds in a generic closed ambient manifold. Our result does not, however, directly extend to unbounded manifolds, including simple cases such as~\(\RR^m\).
\end{remark}

\begin{remark}[The Liouville form \(\eta\) is not a connection form]
The fibration \(\pi\colon \Emb(S,M) \to \UEmb(S,M)\) is a principal \(G\)-bundle, where the structure group \(G\) is the group of reparametrizations \(\Diffp(S)\). Using the Liouville form \(\eta\) defined in \autoref{th:extended_classical_Liouville}, we define a 1-form \(\tilde{\eta} = \pi^*\eta\) on \(\Emb(S,M)\). By construction, \(d\tilde{\eta} = \pi^* \omega\), and \(\tilde{\eta}\) is equivariant under the \(\Diffp(S)\)-action. However, it is not a connection form, as it lacks vertical reproducibility: since \(\ker d\pi \subset \ker \tilde{\eta}\), we have \(\tilde{\eta}_\paramgam(\hat{\xi}) = 0\) for any \(\xi \in \diff(S)\), where \(\hat{\xi}(\paramgam) = d\paramgam(\xi)\) is the fundamental vector field corresponding to \(\xi\) at \(\paramgam\).

On the other hand, the Liouville form on the space of implicit representations that we will define does yield a connection form (\autoref{prop:connection_form}).
\end{remark}

\subsection{Momentum maps on $\UEmb(S,\RR^m)$}\label{sec:explicit_momentum_maps}
The Liouville form \(\eta\) of the MW form \(\omega\) can be used to describe conserved quantities of Hamiltonian systems on \(\cO\). As in finite-dimensional symplectic geometry, when a Lie group \(G\) acts on \(\cO\), a map \(J \colon \cO \to \frak{g}^*\) is called a momentum map if 
\begin{align}
    d \langle J(\cdot), \xi \rangle \big|_{\gamma} = \iota_{\hat{\xi}} \omega \big|_{\gamma}
\end{align}
for any \(\gamma \in \cO\), \(\xi \in \frak{g}\), and its fundamental vector field \(\hat{\xi} \in \Gamma(T\cO)\). If a given Hamiltonian function \(\cH \colon \cO \to \RR\) is invariant under the \(G\)-action, then \(J(\gamma_t) \in \frak{g}^*\) is conserved along the Hamiltonian flow \(\gamma_t\).

When the Liouville form \(\eta\) is also invariant under the \(G\)-action, the momentum map can be expressed as
\begin{align}
    d \langle J(\gamma), \xi \rangle = \iota_{\hat{\xi}} \omega = \cL_{\hat{\xi}} \eta - d \iota_{\hat{\xi}} \eta = - d (\eta(\hat{\xi})),
\end{align}
so that \(\langle J(\gamma), \xi \rangle = - \eta(\hat{\xi})\), up to addition of a constant.


\begin{example}\label{eg:momentum_map_SO3}
Consider the rotation action of \( \on{SO}(3) \) on \( \UEmb(\SS^1,\RR^3) \cong \cD_{\UEmb(\SS^1,\RR^3)} \). The fundamental vector field \( \hat\xi \in \Gamma(T\cD_{\UEmb(\SS^1,\RR^3)}) \) associated with \( \xi \in \frak{so}(3) \) is given at each \( \delgam \) by
\(\hat\xi(\delgam) = -\cL_{R_\xi}\delgam\)
where \( R_\xi \in \sdiff(M) \) is the rotational vector field defined by \( R_\xi(x) = \xi \times x \) on \( \RR^3 \).

Since the Liouville form \( \eta \) \eqref{eq:liouville_classical} is invariant under rotation, we have \( \langle J(\cdot), \xi\rangle = -\eta(\hat\xi) \), up to an additive constant. Therefore, the \emph{angular momentum} \( J(\delgam) \in \frak{so}^*(3) \) at \( \delgam \) can be computed from $R_\xi$ and and the expression \eqref{eq:liouville_classical_space_curves}. Since \( R_\xi(x) = \xi \times x = \on{curl}\left(-\frac{1}{3} x \times (\xi \times x) \right) \), we have
\begin{align}
    \langle J(\gamma), \xi\rangle
    &= -\frac{1}{3} \int_{\SS^1} \det( \partial_s \paramgam, \paramgam, \hat\xi \circ \paramgam ) \, ds \\
    &= \int_\gamma \left( -\frac{1}{3} x \times (\xi \times x) \right)^\flat \\
    &= \fluxMtwo(\delgam, R_\xi), \quad \xi \in \frak{so}(3),
\end{align}
where $\paramgam$ is any parametrization of $\gamma$ and  the flat operator \( \flat \colon \diff(\RR^3) \to \Omega^1(\RR^3) \) is defined with respect to the Euclidean metric.

The quantity \( \langle J(\gamma), \xi\rangle \) can be interpreted as the signed volume of the surface of revolution obtained by rotating \( \gamma \) about the axis \( \xi/|\xi| \in \frak{so}(3)\cong \RR^3\), multiplied by $|\xi|$.
\end{example}


On \( \RR^m \), the Liouville form \( \eta \) is not invariant under \( \SDiff_0(\RR^m) \); for instance, its value depends on the choice of origin and is therefore not translation-invariant. It is preserved only by volume-preserving linear transforms \( x \mapsto fx \) with \( f \in \on{SL}(m)\), which form a finite-dimensional subgroup of \( \SDiffp(\RR^m) \).

In contrast, the Liouville form on the space of implicit representations we will introduce in \autoref{sec:Liouville_form_prequantum} is invariant under \( \SDiffp(M) \) for any closed manifold \( M \) (see \autoref{sec:implicit_Hamiltonian_vector_field}).


\subsection{Riemannian and formal K\"ahler structure}\label{sec:kaehler_explicit}

So far, we introduced the MW structure using only a volume form on the ambient manifold~$M$. When a Riemannian metric is given on~$M$, we can additionally equip~$\cO$ with a Riemannian structure.

Assume now that $M$ is equipped with a Riemannian metric~$g$ inducing the volume form~$\mu$. We first define an almost complex structure, that is, a vector bundle homomorphism $\cJ \colon T\cO \to T\cO$ satisfying $\cJ^2 = -\id$.

Let \(\tilde\cO \coloneqq \pi^{-1}\cO\) be the parametrization space of $\cO$ via embeddings. We define an operator $\tilde\cJ \colon T\tilde\cO \to T\tilde\cO$ by projecting each $\dot\paramgam \in T_{\paramgam} \tilde\cO$ onto the normal bundle $N_{\paramgam} \tilde\cO$, and then applying a 90-degree rotation in $N_{\paramgam} \tilde\cO$ with respect to the metric~$g$, oriented so that
\begin{align}\label{eq:orientation_of_J}
    \mu(\partial_{s_1}\paramgam(s), \ldots, \partial_{s_n}\paramgam(s), \tilde\cJ \dot\paramgam(s), \dot\paramgam(s)) \geq 0,
\end{align}
where $s = (s_1, \ldots, s_n)$ is any local chart consistent with the orientation of~$\paramgam$.

Since $\ker \tilde\cJ = \ker d\pi$ and $\tilde\cJ$ is invariant under reparametrization, it descends to an almost complex structure $\cJ$ on~$\cO$, defined by
\begin{align}
    \cJ \dot\gamma = d\pi|_{\paramgam} (\tilde\cJ \dot\paramgam)
\end{align}
with any representatives $\paramgam$ of $\gamma$ and $\dot\paramgam$ of $\dot\gamma$, respectively.

With $\cJ$ and the MW form $\omega$, we define a Riemannian metric on~$\cO$ by
\begin{align}
    \cG_\gamma(\dot\gamma, \ring\gamma) = \omega_\gamma(\dot\gamma, \cJ(\ring\gamma)).
\end{align}
In terms of de~Rham currents, this becomes
\begin{align}
    \cG_{\delgam}(-\cL_u \delgam, -\cL_v \delgam)
    = \langle \delgam, \iota_{\hat v} \iota_u \mu \rangle,
\end{align}
where $\hat{v}$ is any vector field on~$M$ satisfying $\hat{v} \circ \gamma = \cJ(v \circ \gamma)$.

We remark that the triple \((\cG, \omega, \cJ)\) defines a formal K\"ahler structure in the sense that
\begin{align}
    \cG(\cJ \cdot, \cdot) = \omega(\cdot, \cdot).
\end{align}
Strictly speaking, this is not a K\"ahler structure in the classical sense, which additionally requires a complex structure \ie, the existence of holomorphic coordinates~\cite{Lempert1993LoopSA, Millson1996Kaehler}.

In some infinite-dimensional settings, including our case, the Nijenhuis tensor
\begin{align}
    \cN(\dot \gamma, \ring \gamma) \coloneqq 
    [\cJ \dot \gamma, \cJ \ring \gamma]
    - \cJ [\cJ \dot \gamma, \ring \gamma]
    - \cJ [\dot \gamma, \cJ \ring \gamma]
    - [\dot \gamma, \ring \gamma], \quad \dot\gamma, \ring\gamma \in T_\gamma \cO
\end{align}
vanishes \cite[Theorem 3.4.3]{brylinski2009loop}\cite[Theorem 2.5]{henrich2009loop}, but this does not imply the existence of a complex structure \cite[Theorem 10.5]{Lempert1993LoopSA}. This contrasts with the finite-dimensional case, where vanishing of the Nijenhuis tensor is equivalent to integrability. For details, see~\cite[Chapter 2]{henrich2009loop} for example. 

% \begin{remark}[Horizontal lift via a Riemmanian structure]
%     With the the degenerate Riemannian metric $\cG$ on $\Emb(S,M)$, the tangent space at each $\gamma$ is uniquely decomposed into the vertical component $V_\gamma \Emb(S,M)=\ker d\pi_\gamma$ and the horizontal component $H_\gamma \Emb(S,M)=(\ker d\pi)^{\perp_\cG}$.

%     With this setting, we may define the unique \emph{horizontal Hamiltonian vector field} of a Hamiltonian $H\colon \Emb(S,M)\to \RR$ among the vector fields $X_H$ satisfying $dH=\iota_{X_H}\omega$ and $X_H \perp_\cG \ker \omega$.
% % 
%     This is equivalent to the pointwise condition
% \begin{align}
%     X_H(s)\perp_{g} T_{\gamma(s)}\gamma(S), \quad \forall s\in S.
% \end{align}
% Intuitively, the horizontal Hamiltonian field is the only field that causes no infinitesimal reparametrization. 

% We can however uniquely define the \emph{horizontal} Hamiltonian vector by choosing the unique tangent vector satisfying 
% \begin{align}
%     X_H \perp_{\cG} \ker \omega
% \end{align}
%  with respect to a Riemannian metric $\mathfrak{G}$ on $\Emb(S,M)$ defined by 
%  \begin{align}
%      \mathfrak{G}_\gamma(\dot \gamma, \ring \gamma)=\int_S g_M(\dot \gamma, \ring \gamma)\mu_{\gamma^*g_M}
%  \end{align}
% using the volume-form $\mu_{\gamma^*g_M}$ on $S$ defined by the induced metric $\gamma^*g_M$ (for details of $\mathfrak{G}$, see \cite{BauerHarmsMichor2011SobolevOnShapeSpace, BauerHarmsMichor2012nonLocal}, for example).
% \end{remark}


\subsection{Base Dimension \texorpdfstring{$2$}{2}}\label{sec:base_dimension_2}

We briefly discuss the case where the dimension \( m \) of the ambient manifold \( M \) is \(2\), which is a finite-dimensional setting in contrast to \( m > 2 \). Although many aspects are simplified in this case, the symplectic structure requires additional care: orientations of points must be supplied explicitly, as they are not encoded in the embeddings themselves.

We regard the domain \( S \) as a collection of \( N \) distinct points. The space of embeddings is  
\begin{align}
    \Emb(S, M) \coloneqq 
    M^N \setminus \Delta 
    = \{ (x_1, \dots, x_N) \in M^N \mid x_j \neq x_k \text{ for } j \neq k \},
\end{align}
which is a smooth \( 2N \)-dimensional manifold.

In contrast to higher-dimensional settings, this space does not carry orientation data, that is, whether each point is embedded positively or negatively. Hence we need to manually attach the orientations \( I = (I_1, \dots, I_N) \in \{ \pm 1 \}^N \) to the symplectic structure. Namely we define 
\begin{align}\label{eq:pointwise_MW_2d}
    \tilde{\omega}_{\paramgam}(\dot\paramgam, \ring\paramgam)
    \coloneqq \sum_{j=1}^N I_j \pi_j^* \mu(\dot\paramgam, \ring\paramgam)
    = \sum_{j=1}^N I_j \mu(\dot\paramgam_j, \ring\paramgam_j),
\end{align}
where \( \dot\paramgam, \ring\paramgam \in T_{\paramgam} \Emb(S, M) \cong T_{x_1} M \times \cdots \times T_{x_N} M \), and \( \pi_j\) denotes projection onto the \( j \)-th factor. Since the volume form \( \mu \) is symplectic on a 2-dimensional manifold $M$, \( \tilde{\omega}_{\paramgam} \) is non-degenerate and thus already symplectic.

We may still consider the unparametrized shape space
\begin{align}
    \UEmb(S, M) \coloneqq \Emb(S, M) / \Diffp(S),
\end{align}
by quotienting out the reparametrization group \( \Diffp(S) \coloneqq \mathrm{Sym}(k_1) \times \mathrm{Sym}(k_2) \), which permutes the \( k_1 \) positively oriented and \( k_2 \) negatively oriented points, with \( k_1 + k_2 = N \).

Using currents (which are simply distributions in base dimension 2), we can encode orientations directly to the shape $\delgam\in \cD_{\UEmb(S,M)}\subset \cD_0(M)$ by
\begin{align}
    \delta_{\gamma} \coloneqq \sum_{j=1}^N I_j \delta_{x_j} 
\end{align}
with the Dirac delta distribution \( \delta_{x_j} \) at each \( x_j \).

Then, for \( \dot{\delta}_\gamma = -\cL_u \delta_\gamma \) and \( \ring{\delta}_\gamma = -\cL_v \delta_\gamma \) with vector fields \( u, v \in \Gamma(TM) \), the MW structure is defined
\begin{align}
    \omega_{\delta_\gamma}(-\cL_u \delta_\gamma, -\cL_v \delta_\gamma)
    = \langle \delta_\gamma, \iota_v \iota_u \mu \rangle
\end{align}
in the same way as for higher dimensional cases.
This evaluates to
\begin{align}
    \langle \delta_\gamma, \iota_v \iota_u \mu \rangle
    = \sum_{j=1}^N I_j \langle \delta_{x_j}, \mu(u, v) \rangle
    = \sum_{j=1}^N I_j \mu(\dot{\gamma}_j, \ring{\gamma}_j),
\end{align}
which agrees with the pointwise expression of the MW form \eqref{eq:pointwise_MW_2d}.


\subsection{Toward implicit representations: the space of all the codimension-2 shapes}
We considered the explicit shape space consisting of embeddings modulo reparametrization for each fixed manifold \(S\). We conclude this section by defining their disjoint union:
\begin{align}\label{eq:explicit_represetations}
    \cX \coloneqq \bigsqcup_S \UEmb(S,M)
\end{align}
where \(S\) runs through all oriented, closed \(n\)-dimensional manifolds. Thus, the explicit shape space \(\cX\) is the set of all closed \(n\)-dimensional submanifolds of \(M\), also known as the space of \(n\)-dimensional nonlinear Grassmannians \cite{HallerVizman2003NonlinearGA}.
Each \(\gamma \in \cX\) lies in a unique \(\UEmb(S,M)\), and we have \(T_\gamma \cX = T_\gamma \UEmb(S,M)\). Hence, the symplectic structure \(\omega\) defined on each \(\UEmb(S,M)\) naturally extends to \(\cX\). For simplicity, we will identify \(\cX\) with its current counterpart \(\cD_{\cX}\) via the injection \(\cI \colon \cX \to \cD_n(M)\) without further mention.

\begin{example}
    If \(m=2\), the space \(\cX = \bigsqcup_{k \in \NN} \left\{ \sum_{j=1}^k I_j \delta_{x_j} \mid I_j \in \{-1,1\},\ x \in M^k \setminus \Delta \right\}\) is all the possible configurations of finitely many unordered oriented points.
    If \(m=3\), \(\cX = \bigsqcup_{k \in \NN} \UEmb\left(\bigsqcup^k \SS^1, M\right)\) consists of all smooth links with finitely many knot components.
    If \(m > 3\), then \(S\) may be a disjoint union of different types of manifolds.
\end{example}


%%%% Apendix 
\section{Appendix: Liouville form via the tilde calculus}\label{sec:Liouville_via_hat_calculus}
An alternative approach to prove \autoref{th:extended_classical_Liouville} is using the so-called tilde calculus presented in \cite{HallerVizman2003NonlinearGA, Vizman2011hatcalculus}. 
The tilde calculus is driven by the tilde operator which takes as input a differential form in a finite-dimensional manifold and outputs another differential form in an infinite-dimensional manifold. 

Let $M$ be an $m$-dimensional oriented manifold and $S$ be $n(<m)$ dimensional closed (i.e., compact \& boundary-less) oriented manifold. 
% In this setting each tangent vector $h\in T_\gamma \Emb(N,M)$ is written as $h= v \circ f$ with some $v\in \Gamma(TN)$. 
Let us define the tilde operator 
\begin{align}
    \widetilde{}\  \colon \Om^k(M) &\to  \Om^{k-n}(\UEmb(S,M))\\
    \alpha &\mapsto \widetilde \alpha
\end{align}
for $k> n$ by 
\begin{align}
    \widetilde\alpha_{\gamma}(u_1 \circ \gamma,\ldots, u_{k-n} \circ \gamma)=\int_S \gamma^* (i_{u_{k-n }}\ldots i_{u_1} \alpha)
\end{align}
where $\gamma\in \UEmb(S,M)$ and $u_1 \circ \gamma,\ldots, u_{k-n} \circ \gamma\in T_\gamma \UEmb(S,M)$ with some $u_1,\ldots,u_{k-n}\in\Gamma(TM)$.


The framework of the tilde calculus allows a simple expression of the Marsden--Weinstein structure:
\begin{example}[The Marsden--Weinstein form]\label{eg:MW-form}
    Let $\mu$ be a volume form on the ambient manifold $M$ and let the dimension of $S$ be $n=m-2$. Then the Marsden--Weinstein form \eqref{def:MW_on_currents} is written as $\tilde \mu\in \Om^2(\UEmb(S,M))$. Eplicitly, it is
    \begin{align}
        \widetilde \mu_{\gamma}(u\circ\gamma, v\circ \gamma)=\int_M \gamma^*i_v i_u \mu.
    \end{align}
\end{example}
The tilde calculus has useful properties. In particular, we can use the commutativity between $d$ and ~$\widetilde{}$ ~\cite[Lemma 1]{HallerVizman2003NonlinearGA} to show the exactness of the Marsden--Weinstein form.
\begin{proposition}\label{prop:d_and_hat_commute}
    We have $d\widetilde\alpha=\widetilde{d\alpha}$.
\end{proposition}
With this, the exactness of the Marsden--Weinstein form $\widetilde \mu$ is a direct consequence of the exactness of $\mu$.
\begin{proof}[An alternative proof of \autoref{th:extended_classical_Liouville}]
    The 1-form $\eta$ given by $\eta_{\delgam}(-\cL_u \delta_\gamma)=\langle\delta_\gamma, \iota_u\mu\rangle$ in \autoref{th:extended_classical_Liouville} agrees with $\widetilde\mu$. Applying to \autoref{prop:d_and_hat_commute} to $\mu=d\nu$, we get 
    \begin{align}
        \widetilde\mu=\widetilde{d \nu}= d \widetilde \nu.
    \end{align}
\end{proof}

