Symmetry, Integrability and Geometry: Methods and Applications Geometric Structures on Spaces of Weighted Submanifolds

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on"convenient"vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold $(M,\omega)$, we construct a weak symplectic structure on each leaf ${\textbf I}_{w}$ of a foliation of the space of compact oriented isotropic submanifolds in $M$ equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangian submanifolds is equivalent to a heuristic weak symplectic structure of Weinstein [Adv. Math. 82 (1990), 133-159]. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson [Asian J. Math. 3 (1999), 1-15] on the space of embeddings of a fixed compact oriented manifold into $M$. When $M$ is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf ${\textbf I}_{w}$ consisting of positive weighted isotropic submanifolds onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of $M$ equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space ${\textbf I}_{w}$ can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.


Introduction
In the same way that finite dimensional manifolds are locally modeled on R n , many collections of geometric objects can be viewed as infinite dimensional manifolds locally modeled on interesting geometric spaces. For example, if N and M are smooth manifolds then the following local models are known:

Collection M
Modeling Space at x ∈ M diffeomorphisms of N vector fields on N Riemannian metrics on N symmetric 2-tensors on N smooth maps from N to M sections of x * T M Lagrangian submanifolds closed 1-forms on x These local models represent certain choices, as many geometric structures coincident in finite dimensions diverge in infinite dimensions. For example, there are typically more derivations than equivalence classes of paths; there are many ways to define the dual of a tangent space; there may fail to exist holomorphic charts even when the Nijenhuis tensor vanishes, etc. Accordingly, there are many frameworks available to study differential geometric structures in infinite dimensions.
Once a framework has been chosen, and a local model identified, the geometry of a collection M can be explored using the following correspondence: structures inherent to objects in M induce global structures on M. For example, if N and M are as above then • the set of Riemannian metrics on N inherits weak Riemannian structures (Ebin 1970 [5], Smolentzev 1994 [14]); • if M is symplectic and L → M is a prequantization line bundle, then the space of sections Γ (L) inherits a weak symplectic structure (Donaldson 2001 [4]); • the set of embeddings of N into M is the total space of a principal fiber bundle with structure group Diff (N ), the diffeomorphisms of N , and base the set of submanifolds of M diffeomorphic to N (Binz, Fischer 1981 [1]).
In this paper we study a particularly interesting example of this phenomenon involving Lagrangian submanifolds equipped with certain measures. From the very beginning, we study these objects in the "Convenient Setup" of Frölicher, Kriegl, and Michor (see [9]). The starting point for this framework is the definition of smooth curves in locally convex spaces called convenient vector spaces. Once the smooth curves have been specified, smooth maps between convenient vector spaces can be defined as maps which send smooth curves to smooth curves. Smooth manifolds then are defined as sets that can be modeled on convenient vector spaces via charts, whose transition functions are smooth. Once the appropriate notions of smoothness are specified, objects in differential geometry are defined by choosing how to generalize finite dimensional constructions to infinite dimensions (e.g. Lie groups, principal G bundles, vector fields, differential forms, etc.) An important feature of this approach is that the modeling space E U for each chart (ϕ, U ) can be different for different chart neighbourhoods U . This differs from the usual description of finite dimensional manifolds which are always modeled on the same vector space R n . This flexibility is useful in describing the local structure of many infinite dimensional manifolds, including the collection of Lagrangian submanifolds in a symplectic manifold.
In 1990 Alan Weinstein [23] introduced a foliation F of the space of Lagrangian submanifolds in a fixed symplectic manifold (M, ω). A leaf of F consists of Lagrangian submanifolds that can be joined by flowing along Hamiltonian vector fields. F lifts to a foliation F w of the space of pairs (L, ρ), where L is a Lagrangian submanifold in M equipped with a smooth density ρ of total measure 1. Weinstein called such pairs weighted Lagrangian submanifolds and leaves of F and F w isodrasts. He showed that each leaf I w of F w can be given a weakly nondegenerate symplectic structure Ω W . He also showed that the leaves consisting of Lagrangian submanifolds equipped with positive densities can be identified with coadjoint orbits of the group of Hamiltonian symplectomorphisms. All of these constructions were done on a heuristic level.
Instead of starting with the Lagrangian submanifolds directly, we instead begin by showing that the set of Lagrangian embeddings of a fixed compact oriented manifold L 0 into M is the total space of a principal fiber bundle with structure group Diff + (L 0 ), the orientation preserving diffeomorphisms of L 0 . The base Lag (M ) is naturally identified with the space of oriented Lagrangian submanifolds in M diffeomorphic to L 0 . We define a foliation E of the total space which descends to the isodrastic foliation F of the space of Lagrangian submanifolds. Similarly, the product of the space of Lagrangian embeddings with the space of top degree forms on L 0 that integrate to 1 is the total space of a principal Diff + (L 0 ) bundle. The base of this bundle can be identified with the set of pairs (L, ρ), where L is an oriented Lagrangian submanifold in M diffeomorphic to L 0 equipped with a top degree form ρ (not necessarily non-vanishing) satisfying L ρ = 1. The foliation E gives a foliation E w of the total space that descends to the isodrastic foliation F w of the base. We define a basic 2-form Ω on the leaves of E w which descends to a weakly nondegenerate symplectic structure on the leaves of F w . We then show that the tangent spaces to the space of pairs (L, ρ) in the "Convenient Setup" can be identified with the tangent spaces in Weinstein's heuristic construction, and that Ω corresponds to Ω W . In this way we make rigourous Weinstein's original construction.
The set of pairs (L, ρ) consisting of Lagrangian submanifolds equipped with volume forms of total measure 1 is an open subset of the set of all weighted Lagrangian submanifolds. The leaves of F w in this open subset of positive weighted Lagrangian submanifolds inherit the symplectic structure Ω and provide a link between Weinstein's symplectic structure and a symplectic structure defined by Simon Donaldson on the space of smooth mappings between manifolds described briefly as follows.
In 1999 Donaldson [3] heuristically wrote down a symplectic structure Ω D on the space of smooth mappings C ∞ (S 0 , M ) of a compact oriented manifold S 0 , equipped with a fixed volume form η 0 , into a symplectic manifold (M, ω). Under some topological restrictions on ω and S 0 , Donaldson described a moment map µ for the Diff (S 0 , η 0 )-action of volume preserving diffeomorphisms on C ∞ (S 0 , M ). This Diff (S 0 , η 0 )-action restricts to a Hamiltonian action on the space of embeddings Emb (S 0 , M ) ⊂ C ∞ (S 0 , M ), with respect to the restrictions of Ω D and µ. By a lemma of Moser, symplectic quotients of Emb (S 0 , M ) by Diff (S 0 , η 0 ) can be identified with spaces of submanifolds in M equipped with volume forms of fixed total measure. In fact when S 0 is half the dimension of M the level surface µ −1 {0} consists of Lagrangian embeddings. This suggests that when η 0 has total measure 1 the symplectic quotients of Emb (S 0 , M ) , Ω D should be related to the leaves of F w consisting of positive weighted Lagrangian submanifolds.
The main result of this paper is that reductions of Emb (S 0 , M ) , Ω D can be defined, in the "Convenient Setup", without any topological restrictions on ω or S 0 and that these reductions are symplectomorphic to leaves of F w consisting of positive weighted Lagrangian submanifolds when the dimension of S 0 is half the dimension of M . In this way we obtain not only a rigorous formulation of Donaldson's heuristic constructions, but also a precise relationship between Weinstein's symplectic structure and Donaldson's symplectic structure. Namely, symplectic quotients of Donaldson's symplectic space can be identified with Weinstein's symplectic spaces in the particular case of leaves consisting of positive weighted Lagrangian submanifolds.
For S 0 of dimension less than or equal to half the dimension of M , symplectic reductions of Emb (S 0 , M ) , Ω D are still well defined in the "Convenient Setup" and yield symplectic spaces consisting of positive weighted isotropic submanifolds in M . This suggests that the symplectic structure Ω on weighted Lagrangian submanifolds should have a generalization to weighted isotropic submanifolds. We show that indeed such a generalization exists, and that the corresponding symplectic spaces in the particular case of leaves consisting of positive weighted isotropic submanifolds are symplectomorphic to reductions of Emb (S 0 , M ) , Ω D . In this way we obtain a generalization of our observed relationship between Weinstein's symplectic structure and Donaldson's symplectic structure to the case of weighted isotropic submanifolds.
Our next result takes its cue from this generalization to weighted isotropic submanifolds. Namely, we show that the symplectic spaces of positive weighted isotropic submanifolds are symplectomorphic to coadjoint orbits of the group Ham (M ) of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. This symplectomorphism is given by a generalization of the moment map written down by Weinstein in his identification of positive weighted Lagrangian submanifolds with coadjoint orbits of Ham (M ). The heuristic idea is that any submanifold I equipped with a volume form ρ can be viewed as an element of the dual of the Lie algebra of Hamiltonian vector fields via the mapping (I, ρ) → f → I f ρ . This mapping is equivariant, injective, and hence induces a coadjoint orbit symplectic structure on spaces of positive weighted submanifolds that can be joined by Hamiltonian deformations.
These positive weighted isotropic submanifolds have yet another interpretation akin to leaves of Poisson manifolds in finite dimensions. Given a finite dimensional Poisson manifold (P, {·, ·}), for each smooth function f ∈ C ∞ (P, R) on P there exists a unique vector field X f on P satisfying dg (X f ) = {f, g} for all g ∈ C ∞ (P, R). The leaves swept out by integral curves to such vector fields X f are symplectic manifolds. This picture can be adapted to infinite dimensions in the following sense. Given an infinite dimensional manifold P , for a subalgebra A ⊂ C ∞ (P, R) we define a Poisson bracket {·, ·} on A and a Poisson algebra (A, {·, ·}) in the usual way. If for every f ∈ A there exists a unique vector field X f on P satisfying dg (X f ) = {f, g} for all g ∈ A, then the directions swept out by such vector fields on each point in P define a distribution on P . We call maximal integral manifolds of this distribution leaves. By defining a Poisson algebra on Emb (S 0 , M ), which restricts to a Poisson algebra on the space of isotropic embeddings, which descends to a Poisson algebra on the space of positive weighted isotropic submanifolds, we show that the reductions of Emb (S 0 , M ) , Ω D are symplectic leaves of a Poisson structure.
As a result we arrive at three different interpretations of the symplectic spaces consisting of positive weighted isotropic submanifolds. Namely, they can be identified with reductions of the space of embeddings Emb (S 0 , M ) , Ω D , with coadjoint orbits of the group Ham (M ) of Hamiltonian symplectomorphisms, and with symplectic leaves of Poisson structures.
We then take a kinematic approach to the leaves of the foliation F of the space of Lagrangian submanifolds to obtain a phase space symplectic structure. That is, by viewing the leaves of F as possible configurations for a submanifold moving in M , on each Lagrangian we can associate "conjugate momenta" with top degree forms that integrate to 0. We call such pairs (L, χ) with L in a leaf of F satisfying L χ = 0 momentum weighted Lagrangian submanifolds. By writing down what should be the canonical 1-form on this set of momentum weighted Lagrangian submanifolds and calculating its exterior derivative, we obtain a weakly symplectic structure.
Finally, we apply this kinematic approach to the set of pseudo Riemannian metrics of a fixed signature on a finite dimensional manifold N . This collection can be viewed as a set of submanifolds by identifying each metric with its graph as a section. Weightings then can be assigned to each metric by pulling up a structure assigned to N . By equipping each metric in this way with a compactly supported symmetric 2-tensor on N , we show that the set of all such weighted metrics has a natural exact symplectic structure.

Conventions
Unless stated otherwise, all finite dimensional manifolds are smooth, connected, and paracompact. For manifolds M and N and vector bundle E → M , we will use the following notation: interior derivative with respect to X; L X Lie derivative with respect to X. In the absence of summation signs repeated indices are summed over.

Basic definitions
We begin by describing the "Convenient Setup" of Frölicher, Kriegl, and Michor in order to establish what we will mean by smoothness, tangent vectors, etc. on some infinite dimensional manifolds. Many definitions will be taken verbatim from [9]. All references like [9, X.X] in this section refer to sections in [9].

Locally convex spaces
A real topological vector space E is a vector space equipped with a topology under which addition 3) absolutely convex if C is circled and convex.
A locally convex space is a Hausdorff topological vector space E, for which every neighbourhood of 0 contains an absolutely convex neighbourhood of 0.

Smooth curves
Let E be a locally convex space. A curve c : at t exists for all t. A curve c : R → E is called smooth if all iterated derivatives exist. The set of all smooth curves in E will be denoted by C ∞ (R, E) [9, 1.2].
One would hope that reasonable definitions of smoothness would imply that "diffeomorphisms" are homeomorphisms. For this purpose we will make use of another topology on locally convex spaces.

The c ∞ -topology
The c ∞ -topology on a locally convex space E is the finest topology for which all smooth curves c : R → E are continuous [9, 2.12]. The c ∞ -topology is finer than the locally convex topology on E [9, 4.7]. If E is a Fréchet space, (i.e. a complete and metrizable locally convex space), then the two topologies coincide [9, 4.1, 4.11].

Convenient vector spaces
A convenient vector space is a locally convex space E with the following property: For any Any c ∞ -closed subspace of a convenient vector space is convenient [9, 2.12, 2.13, 2.14].

Space of curves
The set of smooth curves C ∞ (R, E) in a convenient vector space E has a natural convenient structure. Moreover, a locally convex space E is convenient if and only if C ∞ (R, E) is convenient [9, 3.7].
We would like to study sets that can be locally modeled on convenient vector spaces. To define "smooth transition functions" we need to define smooth mappings between convenient vector spaces.

Convention
For the rest of this section E and F will denote convenient vector spaces.

Mappings between convenient vector spaces
given by pullback along smooth curves (i.e. c * (f ) = f • c), are continuous. Then C ∞ (U, F ) is a convenient vector space [9, 3.11]. [9, 2.11] and inherits a convenient structure [9, 3.17]. The set of invertible maps in L(E, F ) with bounded inverse will be denoted by GL(E, F ).

The differentiation operator and chain rule
exists, is linear and bounded (smooth). Note that the above limit is taken in the locally convex topology of F . Also the chain rule

Manifolds
A chart (U, ϕ) on a set M is a bijection ϕ : U → E U from a subset U ⊂ M onto a c ∞ -open set in a convenient vector space E U . A family of charts (U α , ϕ α ) α∈A is called an atlas for M , if the U α cover M and all transition functions Two atlases are equivalent if their union is again an atlas on M . A smooth manifold M is a set together with an equivalence class of atlases on it [9, 27.1].

Smooth mappings between manifolds
So a mapping f : M → N is smooth if and only if it maps smooth curves to smooth curves. A smooth mapping f : M → N is a diffeomorphism if it is a bijection and if its inverse is smooth [9, 27.2]. The set of smooth maps from M to N will be denoted by C ∞ (M, N ). [9, 27.11]. A curve in a submanifold N of M is smooth if and only if it is smooth as a curve in M .

Tangent spaces of a convenient vector space
Let a ∈ E. A tangent vector with base point a is a pair (a, X) with X ∈ E. For each neighbourhood U of a in E, a tangent vector (a, X) defines a derivation C ∞ (U, R) → R by X a f := df (a) (X) [9, 28.1].
Remark 1. In [9] these tangent vectors are called kinematic tangent vectors since they can be realized as derivatives c ′ (0) at 0 of smooth curves c : R → E. This is to distinguish them from more general derivations which are called operational tangent vectors.

The tangent bundle
Let M be a smooth manifold with an atlas (U α , ϕ α ) indexed by α ∈ A. On the disjoint union α∈A U α × E α × {α} define the following equivalence relation: A tangent vector at x ∈ M is an equivalence class [(x, v, α)]. The quotient α∈A U α ×E α ×{α} / ∼ will be called the tangent bundle of M and will be denoted by T M .
Let π : T M → M denote the projection [(x, v, α)] → x. T M inherits a smooth manifold structure from M . For x ∈ M the set T x M := π −1 (x) is called the tangent space at x. Since each transition function ϕ αβ is smooth, each differential dϕ αβ (x) is bounded linear, which means each tangent space T x M has a well defined bornology independent of the choice of chart (cf. [9, 1.1, 2.11]).
Alternatively, we can describe tangent vectors to a smooth manifold by means of equivalence classes of smooth curves. We will say that two smooth curves c 1 and c 2 in M are equivalent at x ∈ M , (and write c 1 ∼ x c 2 ), if c 1 (0) = x = c 2 (0) and d dt t=0 ϕ α • c 1 (t) = d dt t=0 ϕ α • c 2 (t) for a chart ϕ α in an atlas (U α , ϕ α ) α∈A on M . The tangent space at x then is equal to C ∞ (R, M ) / ∼ x (compare with [9, 28.12]).

Tangent mappings
Let f : M → N be a smooth mapping between manifolds. Then f induces a linear map df (x) : This defines a fiberwise linear map df : T M → T N called the differential of f (compare with [9, 28.15]). Let D be a distribution on a manifold M . The set of locally defined vector fields X on M satisfying X (x) ∈ D x will be denoted by X D (M ).

Remark 2.
In finite dimensions such distributions defined without any assumptions regarding continuity or smoothness are sometimes called "generalized distributions". If a generalized distribution D is "smooth" in the sense that every v ∈ D x ⊂ T x M can be realized as X (x) for a locally defined vector field X ∈ X D (M ), then there exist results on integrability of such distributions (see e.g. [18,19,16,17]).

Foliations
Let M be a smooth manifold. A foliation of M is a distribution F = {F x } on M , for which there exists an atlas (U α , ϕ α ) of charts ϕ α : U α → E α on M and a family of c ∞ -closed subspaces {F α ⊂ E α }, such that the inverse image under ϕ α of translations of F α are integral manifolds of F, and such that if N ⊂ U α is an integral manifold of F then ϕ α (N ) is contained in a translation of F α . The charts ϕ α will be called distinguished charts.
Let ϕ α : U α → E α be a distinguished chart of a foliation F of M and y + F α a translation of F α ⊂ E α . Then ψ α,y := ϕ α | ϕ −1 (y+Fα) − y defines a chart into F α , and the set of all such charts ψ α,y defines an alternative smooth structure on the set M modeled on the spaces F α . The set M equipped with this alternative manifold structure will be denoted by

Fiber bundles
A fiber bundle (Q, p, M ) consists of manifolds Q (the total space), M (the base), and a smooth mapping p : Q → M (the projection) such that for every x ∈ M there exists an open neighbourhood U of x, a smooth manifold S U , and a diffeomorphism ψ such that the following diagram commutes: Such a pair (U, ψ) as above is called a fiber bundle chart. A fiber bundle atlas (U α , ψ α ) α∈A is a set of fiber bundle charts such that {U α } α∈A is an open cover of M . If we fix a fiber bundle atlas, The mappings ψ αβ are called the transition functions of the bundle. When S Uα = S for all charts (U α , ψ α ) for some smooth manifold S, then S is called the standard fiber (compare with [9, 37.1]).

Vector bundles
Let (Q, p, M ) be a fiber bundle. A fiber bundle chart (U, ψ) is called a vector bundle chart if S U is a convenient vector space. Two vector bundle charts (U α , ψ α ) and (U β , ψ β ) are compatible if the transition function ψ αβ is bounded and linear in the fibers, i.e. ψ αβ (x, s) = (x, φ αβ (x) s) for some mapping φ αβ : U αβ → GL (S β , S α ) ⊂ L(S β , S α ). A vector bundle atlas is a fiber bundle atlas (U α , ψ α ) α∈A consisting of pairwise compatible vector bundle charts. Two vector bundle atlases are equivalent if their union is again a vector bundle atlas. A vector bundle (Q, p, M ) is a fiber bundle together with an equivalence class of vector bundle atlases (compare with [9, 29.1]).

Remark 5.
Here again our definition differs from that in [9,29.1] in that we allow for different S U in different neighbourhoods U . However, this more general version of vector bundles is subsequently used implicitly throughout the text (see e.g. [9, 29.9] and [9, 29.10] where the tangent bundle T M of any smooth manifold M is taken to be a vector bundle).

Constructions with vector bundles
If Q → M and R → M are vector bundles then we have vector bundles Q * , L (Q, R), and L k alt (Q, R) whose fibers over x ∈ M are (Q x ) * (the space of bounded linear functionals on Q x ), L (Q x , R x ) and L k alt (Q x , R x ) respectively [9, 29.5].
Remark 6. We will always use E * to denote the space of bounded linear functionals on a locally convex space E. In [9] E * is reserved for the space of continuous (in the locally convex topology) linear functionals while E ′ is used to denote the space of bounded linear functionals.

Cotangent bundles
Since T M is a vector bundle for any manifold M , the bundle (T M ) * with fiber over x ∈ M equal to (T x M ) * is also a vector bundle. This vector bundle is called the cotangent bundle of M and will be denoted by T * M [9, 33.1].  We will be interested primarily in sets that can be locally modeled on spaces of sections of vector bundles. To understand notions of smoothness on such sets, it is enough to identify the smooth curves.

Curves in spaces of sections
) .

Vector fields
Let M be a smooth manifold. A vector field X on M is a smooth section of the tangent bundle T M [9, 32.1]. The set of all vector fields on M will be denoted by X (M ). Each vector field X specifies a map

The Lie bracket
Let X and Y be smooth vector fields on a manifold M . Each such vector field is a smooth mapping M → T M between manifolds, and so it makes sense to compute the differentials dX and dY . The Lie bracket [X, Y ] of X and Y is the vector field on M given by the expression

Differential forms
A differential k-form on a manifold M is a section ω ∈ Γ L k alt (T M, M × R) . The set of all differential k-forms will be denoted by Ω k (M ) [9, 33.22].

The pullback of a differential form
Let f : N → M be a smooth mapping and ω ∈ Ω k (M ) be a differential k-form on M . The pullback f * ω ∈ Ω k (N ) of ω is defined by see [9, 33.9].

The insertion operator
For a vector field X ∈ X (M ) on a manifold M , the insertion operator ı (X) is defined by see [9, 33.10].

The exterior derivative
Let U ⊂ E be a c ∞ -open subset and let ω ∈ C ∞ U, L k alt (E, R) be a differential k-form on U . The exterior derivative dω ∈ C ∞ U, L k+1 alt (E, R) of ω is the skew symmetrization of the differential dω: (Note that the differential dω with plain text d is used to define the exterior derivative dω with italicized d.) If ω is a differential k-form on a manifold M , then this local formula defines a differential k + 1-form dω on M . The above local expression for the exterior derivative induces the global formula where X 0 , . . . , X k ∈ X(M ) [9, 33.12].

Lie groups
A Lie group G is a smooth manifold and a group such that multiplication µ : G × G → G and inversion ν : G → G are smooth. The Lie algebra of a Lie group G is the tangent space to G at the identity e, which inherits a Lie bracket from the identification with left invariant vector fields. The Lie algebra will be denoted either by g or Lie(G) [9, 36.1, 36.3].

Basic differential forms
Let l : G × M → M be a smooth action of a Lie group G on a smooth manifold M . Let

Principal G bundles
Let G be a Lie group. A principal G bundle is a fiber bundle (P, p, M, G) with standard fiber G whose transition functions act on G via left translation: There is a family of smooth mappings The pull back through the projection p * : Ω k (M ) → Ω k hor (P ) G is an isomorphism [9, 37.30].

Diffeomorphism groups
The following diffeomorphism groups are examples of infinite dimensional Lie groups:

The adjoint representation
Let GL (E) denote the set of bounded invertible linear transformations of E. Let G be a Lie group with Lie algebra g. Every element g ∈ G defines an automorphism ψ g : G → G by conjugation: ψ g (a) := gag −1 . The adjoint representation of G denoted by Ad : G → GL (g) ⊂ L (g, g) is given by Ad (g) := d e ψ g : g → g for g ∈ G. The adjoint representation of g denoted by ad : g → gl (g) := L (g, g) is given by ad := d e Ad [9, 36.10].

Weak symplectic manifolds
A 2-form σ ∈ Ω 2 (M ) on a manifold M is called a weak symplectic structure on M if it is closed (dσ = 0) and if its associated vector bundle homomorphism σ ♭ : T M → T * M is injective. This last condition is equivalent to weak nondegeneracy: for every x ∈ M and v ∈ T x M there exists a w ∈ T x M such that σ x (v, w) = 0. If σ ♭ : T M → T * M is invertible with a smooth inverse then σ is called a strong symplectic structure on M [9, 48.2]. A vector field X ∈ X (M ) will be called Hamiltonian if ı (X) σ = dH for some H ∈ C ∞ (M, R), and the function H will called a Hamiltonian of X.

Isodrastic foliations
In this section we will describe our approach towards describing Lagrangian submanifolds as Lagrangian embeddings modulo reparametrizations. We will show that the space of Lagrangian embeddings into a fixed symplectic manifold (M, ω) is a smooth manifold which has a natural foliation E. Moreover, the space of Lagrangian embeddings of the form L 0 ֒→ M is the total space of a principal Diff + (L 0 ) bundle over the space of Lagrangian submanifolds in M . The leaves of E will turn out to be orbits of the group of Hamiltonian symplectomorphisms under the natural left composition action. Meanwhile the foliation E descends to a foliation F of the space of Lagrangian submanifolds in M . In all of these constructions, the key will be to use Weinstein's Lagrangian Neighbourhood Theorem which says that any symplectic manifold near a Lagrangian L looks like a neighbourhood of the zero section in the cotangent bundle T * L.
Let (M, ω) be a finite dimensional symplectic manifold. Let L 0 be an oriented, compact manifold of half the dimension of M .

Notation
By Lag (L 0 , M ) we will denote the set of Lagrangian embeddings of L 0 into (M, ω). That is, Let Z k (N ) and B k (N ) denote the set of closed and exact k-forms respectively on a manifold N . That is, We will show that Lag (L 0 , M ) is a smooth manifold by defining an atlas of charts using the following Lagrangian neighbourhood theorem of Weinstein: Theorem 1 (see Theorem 6.1 and Corollary 6.2 in [21]). Let L be a Lagrangian submanifold of a symplectic manifold (M, ω). Then there exists an open neighbourhood U of L and a symplectic embedding ψ : U → T * L such that ψ| L = 1 L and ψ * ω T * L = ω.
Proof . The idea of the proof is as follows. By Theorem 1, Lagrangian submanifolds near a given Lagrangian submanifold can be identified with the graphs of closed 1-forms in T * L 0 . It follows that Lagrangian embeddings near a given one can be identified with closed 1-forms viewed as maps from L 0 to T * L 0 precomposed with diffeomorphisms of L 0 .
Given i ∈ Lag(L 0 , M ), by Theorem 1 the embedding i can be extended on a neighbourhood W i of the zero section in T * L 0 to a symplectic embedding λ i : W i → M . Let V e be a chart neighbourhood of the identity map e ∈ Diff (L 0 ) and denote by ψ e : V e → X (L 0 ) the corresponding chart as part of an atlas on Diff (L 0 ). Define The space X (L 0 ) = Γ (T L 0 ) is convenient by Section 2.10. The space The collection (U i , ϕ i ) i∈Lag(L 0 ,M ) defines a smooth atlas on Lag (L 0 , M ), since the chart changings ϕ ik are smooth by smoothness of the exponential map, by smoothness of each symplectic embedding λ i , and by Section 2.24.
To explicitly describe the tangent space to Lag (L 0 , M ) at a point i, we will make use of the following notation.

Notation
If S 0 is a manifold (not necessarily of half the dimension of M ), then for every i ∈ Emb (S 0 , M ) we can view the tangent bundle T S 0 as a subbundle of the pullback bundle i * T M . The symplectic form ω defines a vector bundle isomorphism ω ♭ : T M → T * M , which induces a vector space There is a natural surjection from the pullback bundle i * T * M onto the cotangent bundle T * S 0 . This induces a linear map ν : To check that f 2 is onto, let α ∈ Z 1 (L 0 ). By Theorem 1, it is enough to prove the assertion when M = T * L 0 and i is the zero section inclusion. Let π : Then α Zα•i = α, which means f 2 is surjective and so the sequence is exact.
Remark 9. Each symplectic embedding λ i : T * L 0 ⊃ W i → M defines a splitting map s i : (1) given by Conversely, suppose that X ∈ Γ closed (i * T M ) and denote by λ i : The set Ham (M ) of Hamiltonian symplectomorphisms is a subgroup of Symp (M ) (see e.g. [12]). So left composition defines an action of Ham (M ) on Lag (L 0 , M ) via We will next show that the Ham (M ) orbits in Lag then there exists a Hamiltonian vector field X H defined on a neighbourhood of i(L 0 ) satisfying X = X H • i. By multiplying H by a cutoff function which is equal to 1 near i(L 0 ) we may assume that X H is defined on all of M . It follows that E i ⊂ T i (Ham(M ) · i). So Ham (M ) orbits are integral manifolds. To show they are maximal, we first consider the case when M = T * L 0 . Let i : L 0 ֒→ T * L 0 denote the zero section inclusion and (U i , ϕ i ) the corresponding chart on Lag (L 0 , T * L 0 ). Let j t be a smooth curve in an integral manifold N contained in U i . For every t, d dt j t ∈ Γ exact (j * t T (T * L 0 )) which means α d dt jt = dh t for a family of functions h t ∈ C ∞ (L 0 , R). This family h t can be chosen to be a smooth curve in Since c t is a smooth curve in Diff (Graph (β 0 )) passing through the identity map, the cotangent lift C t is a smooth curve in Ham (T * Graph (β 0 )).
For the general case when M is any symplectic manifold, the previous discussion implies that the intersection of any integral manifold with a chart neighbourhood U i on Lag (L 0 , M ) lies in a Ham (M ) orbit. Thus any integral manifold containing a point i ∈ Lag (L 0 , M ) is contained in Ham (M ) · i, which means that such orbits are maximal integral manifolds.
Finally, we will show that the atlas The zero section in T * L 0 can be deformed to the graph of any 1-form α ∈ Ω 1 (L 0 ) on L 0 by taking the time 1 flow of the transformation (x, p) → (x, p + tα x ) of the cotangent bundle. When α is closed this transformation is symplectic; when α is exact it is a Hamiltonian symplectomorphism. So the graph of any exact form can be obtained by deforming the zero section in T * L 0 along a Hamiltonian vector field. Conversely, suppose that φ ∈ Ham (T * L 0 ) is a Hamiltonian symplectomorphism and {ψ t } is a collection of symplectomorphisms satisfying ψ 0 = Id, ψ 1 = φ, andψ t = X Ht • ψ t for some family of Hamiltonian vector fields X Ht on T * L 0 . If O denotes the zero section, then j t : Proof . We will first describe the manifold structure on Lag (M ). For each Lagrangian L ∈ Lag (M ), by Theorem (1) Thus the collection (U L , ϕ L ) L∈Lag(M ) defines a smooth atlas on Lag (M ) as the transition functions ϕ LN are smooth by smoothness of the symplectic embeddings λ L . As for the tangent spaces, suppose that K t is a smooth curve in Lag (M ) such that K 0 = L.
We will now describe the identification of tangent spaces of Lag (M ) with vector spaces Γ closed (i * T M ) /X (L 0 ). Let i ∈ Lag (L 0 , M ) be a representative in the class L ∈ Lag (M ). Let λ i : T * L 0 ⊃ W i → M be the symplectic embedding chosen in the definition of the chart (U i , ϕ i ) on Lag (L 0 , M ), and s i : Z 1 (L 0 ) → Γ closed (i * T M ) the corresponding splitting map (see Remark 9). Then the linear map is a vector space isomorphism.

Weighted Lagrangian submanifolds
In this section we introduce the notion of weightings and weighted submanifolds. The set Lag w (L 0 , M ) of pairs (i, η) consisting of Lagrangian embeddings i : L 0 ֒→ M and top degree forms η that satisfy L 0 η = 1 has the smooth structure of the Cartesian product Lag (L 0 , M ) × η ∈ Ω n (L 0 ) | L 0 η = 1 . The foliation E of Lag (L 0 , M ) canonically induces a foliation E w of Lag w (L 0 , M ). The space Lag w (L 0 , M ) is the total space of a principal Diff + (L 0 ) bundle, whose base can be identified with the set Lag w (M ) of Lagrangian submanifolds in M equipped with a top degree form of total measure 1. The foliation E w descends to a foliation F w of the base, so that Lag w (L 0 , M ) Ew (cf. Section 2.18) is the total space of a principal Diff + (L 0 ) bundle over Lag w (M ) Fw . On each leaf of E w we define a 2-form Ω, basic with respect to this principal group action, which descends to a weak symplectic structure on Lag w (M ) Fw . Finally, we show that the tangent spaces of Lag w (M ) and of leaves of F w can be identified with the tangent space descriptions in Weinstein's original construction, and that Weinstein's symplectic structure Ω W corresponds to our symplectic structure Ω. Definition 3. A weighting of a compact oriented manifold L is a top degree form ρ on L satisfying L ρ = 1. A pair (L, ρ) will be called a weighted manifold.

Notation
Let Ω n 1 (S 0 ) denote the set of n-forms on a manifold S 0 that integrate to 1 (where n = dim S 0 ), Ω n 0 (S 0 ) the set of n-forms on S 0 that integrate to 0, and Lag w (L 0 , M ) the product Lag (L 0 , M )× Ω n 1 (L 0 ). That is, Integration along L 0 defines a continuous linear functional L 0 : Ω n (L 0 ) → R on the convenient vector space Ω n (L 0 ) = Γ ( n T * L 0 ). So the kernel Ω n 0 (L 0 ) is a c ∞ -closed (convenient) subspace. The space Ω n 1 (L 0 ) is an affine translation of Ω n 0 (L 0 ), which means it is a smooth manifold modeled on Ω n 0 (L 0 ). So Lag w (L 0 , M ) is a smooth manifold modeled on the space is the atlas on Lag (L 0 , M ) defined in Proposition 1 then the charts U (i,η) , ϕ (i,η) are defined by This atlas and the subspace B 1 (L 0 ) ⊕ X(L 0 ) ⊕ Ω n 0 (L 0 ) define a foliation E w on Lag w (L 0 , M ).
Definition 4. We will call the foliation E w the isodrastic foliation of Lag w (L 0 , M ) and a leaf of E w will be called an isodrast in Lag w (L 0 , M ).
To each point (i, η) ∈ H w in an isodrast we assign a skew-symmetric bilinear form on T (i,η) H w via the expression where α X k = dh k for some h k ∈ C ∞ (L 0 , R). This assignment does not depend on the choice of primitives h k since the top degree forms ϑ k integrate to 0. Equivalently, the pointwise assignment in (3)  Proof . We will first show that the assignment Ω defines a differential 2-form on each leaf H w of E w . The assignment in (3) defines a map Ω : To check that this map is smooth, it is enough to check it in each chart. If U (i,η) , ϕ (i,η) denotes a chart on H w then Ω defines a map from U (i,η) to L 2 alt (Γ exact (i * T M ) ⊕ Ω n 0 (L 0 ) , R) (after B 1 (L 0 ) ⊕ X (L 0 ) ⊕ Ω n 0 (L 0 ) has been identified with Γ exact (i * T M ) × Ω n 0 (L 0 ) via the splitting map s i : Z 1 (L 0 ) → Γ closed (i * T M ) (see Remark 9)). This map is smooth if it maps smooth curves in U (i,η) to smooth curves in L 2 Thus to verify that Ω is smooth, it is enough to check the following statement: If (M, ω) = (T * L 0 , ω T * L 0 ), i : L 0 ֒→ T * L 0 denotes the zero section inclusion, (α t • a t , η t ) is a smooth curve in H w , (X 1 (t) , ϑ 1 (t)) and (X 2 (t) , ϑ 2 (t)) are smooth curves in Γ exact (i * T (T * L 0 )) ⊕ Ω n 0 (L 0 ) satisfying α X k (t) = dh k (t) for smooth curves h k (t) in C ∞ (L 0 , R), Z 1 (t) and Z 2 (t) are the unique time dependent vector fields on is smooth. Since this statement follows from the smoothness of all quantities in the integral, Ω is indeed a section of L 2 alt (T H w , H w × R) → H w . We will now show that Ω is basic with respect to the action of Diff which means Ω is Diff + (L 0 )-invariant. To check that Ω is horizontal, let Y ∈ X (L 0 ) be in the Lie algebra of Diff + (L 0 ). If a t is a smooth curve in Diff + (L 0 ) through the identity map with time derivative Y at t = 0, then the generating vector field at a point (i, η) ∈ Lag w (L 0 , M ) is given by Since , we conclude that Ω is also horizontal and thus basic.
Proof . For each (L, ρ) ∈ Lag w (M ), by Theorem 1 there exists a symplectic embedding λ (L,ρ) : M ⊃ W (L,ρ) → T * L of a neighbourhood of L onto a neighbourhood of the zero section in the cotangent bundle. If π T * L : T * L → L denotes the cotangent bundle projection, then the restriction of π T * L to the graph of any 1-form α ∈ Ω 1 (L) in T * L defines a diffeomorphism of that graph onto L. Define All chart changings are smooth again by the smoothness of the symplectic embeddings λ L , so the collection U (L,ρ) , ϕ (L,ρ) (L,ρ)∈Lag w (M ) defines a smooth atlas on Lag w (M ).
We will now describe the identification of tangent spaces of Lag w (M )with spaces Γ closed (i * T M) is a vector space isomorphism.
The canonical projection Lag w (M ) → Lag (M ), which forgets the weightings, pulls back F to a foliation F w on Lag w (M ). That is, the collection of subspaces B 1 (L) ⊕ Ω n 0 (L) and atlas U (L,ρ) , ϕ (L,ρ) indexed by (L, ρ) ∈ Lag w (M ) define a foliation F w on Lag w (M ).
Definition 5. The foliation F w will be called the isodrastic foliation of Lag w (M ) and a leaf I w of F w will be called an isodrast in Lag w (M ).

B. Lee
If p : Lag w (L 0 , M ) Ew → Lag w (M ) Fw denotes the projection to the quotient, then so that the collection U [(i,η)] , ψ [(i,η)] (i,η)∈Lag w (L 0 ,M ) defines a fiber bundle atlas. Since Ω defines a basic 2-form on the total space Lag w (L 0 , M ) Ew , it descends to a differential 2-form (also denoted Ω) on Lag w (M ) Fw (see Section 2.34).
We will now check closedness of Ω locally in a chart U (L,ρ) , ϕ (L,ρ) . On U (L,ρ) tangent vectors can be identified with pairs (Z, ϑ) where Z ∈ X (T * L 0 ) is a vector field on the cotangent bundle satisfying ı (Z) ω = π * dh for h ∈ C ∞ (L 0 , R), and ϑ ∈ Ω n 0 (L 0 ). Under such an identification, if (i, η) is a representative in the class (L, ρ) ∈ Lag w (M ), and if we identify M with T * L 0 using the symplectic embedding λ i : T * L 0 ⊃ W i → M , then terms like ω (X 1 , X 2 ) η = i * [ω (Z 1 , Z 2 )] η in the expression for Ω vanish since the Z k 's are tangent to the cotangent fibers. So on U (L,ρ) , It follows that dΩ = 0 since locally Ω does not depend on (N, σ). Finally, weak nondegeneracy follows from the local expression for Ω in (4) and the fact that the h k 's and ϑ k 's can be chosen independently. Indeed, if h 1 is nonzero on some open subset V ⊂ L 0 , then we can take h 2 to be zero and choose ϑ 2 such that it is supported on V and L 0 h 1 ϑ 2 is nonzero. If ϑ 1 is nonzero on an open subset V ⊂ L 0 , then we can choose ϑ 2 to be zero and choose h 2 to be supported on V so that L 0 h 2 ϑ 1 is nonzero.
Lag w (M ) can also be described as the set of equivalence classes (L, [ ρ]) where ρ is an n-form on a neighbourhood of L satisfying L ρ = 1, and ρ 1 ∼ ρ 2 if and only if ρ 1 and ρ 2 have the same pullback to L. In [23] Weinstein used this approach and heuristically viewed Lag w (M ) and each leaf I w of F w as infinite dimensional manifolds. He viewed tangent vectors as equivalence classes of paths in Lag w (M ) and I w to give the following description of their tangent spaces and wrote down a closed, weakly nondegenerate, skew-symmetric bilinear form Ω W on each isodrast I w : Theorem 2 (see Theorem 3.2 & Lemma 3.3 in [23]). The tangent space to Lag w (M ) at (L, ρ) can be identified with the set of quadruples L, ρ, X, θ , where ρ is an n-form on a neighbourhood of L satisfying L ρ = 1, X is a symplectic vector field on a neighbourhood of L, and θ is an n-form on a neighbourhood of L satisfying L L X ρ + θ = 0, subject to the following equivalence relation. L, ρ 1 , X 1 , θ 1 ∼ L, ρ 2 , X 2 , θ 2 if and only if the following conditions hold: (1) ρ 1 and ρ 2 have the same pullback to L; (2) X 1 − X 2 is tangent to L; (3) the pullbacks to L of L X 1 ρ 1 + θ 1 and L X 1 ρ 2 + θ 2 are equal.
The tangent vectors to an isodrast I w are represented by equivalence classes L, ρ, X f , θ where X f is a Hamiltonian vector field on a neighbourhood of L. I w admits a closed, weakly nondegenerate, skew-symmetric bilinear form Ω W defined by We will show that this heuristic description of the tangent spaces and bilinear structure Ω W due to Weinstein can be derived from the smooth structures on Lag w (M ) and I w defined in Proposition 8 and from the weak symplectic structure Ω on I w (see (3)).

Notation
For (L, ρ) ∈ Lag w (M ) let Q symp (L,ρ) denote the space of equivalence classes L, ρ, X, θ where X is a symplectic vector field defined on a neighbourhood of L. Let Q ham (L,ρ) denote the space of equivalence classes L, ρ, X f , θ where X f is a Hamiltonian vector field defined on a neighbourhood of L.

B. Lee
The linear map τ symp (L,ρ) has an inverse given by The isomorphism τ ham (L,ρ) : T (L,ρ) I w → Q ham (L,ρ) is described similarly. Finally if ζ 1 , ζ 2 ∈ T (L,ρ) I w with ζ k = [(X k , ϑ k )], with representatives (X k , ϑ k ) such that dλ −1 (i,η) • X k is tangent to the cotangent fibers in T * L 0 , then ω(X 1 , Example 1. Let M = S 2 and L 0 = S 1 . Since S 1 is one dimensional, all embeddings are Lagrangian and all 1-forms on S 1 are closed. So Lag S 1 , S 2 = Emb S 1 , S 2 and for every embedding i we have that T i Lag S 1 , S 2 = Γ i * T S 2 . For any point [(i, η)] = (L, ρ) in a leaf I w , if j is a compatible almost complex structure on S 2 , i.e. g (·, ·) := ω S 2 (·, j·) defines a Riemannian metric on S 2 , then for every tangent vector ξ ∈ T [(i,η)] I w there exists a unique representative (X, ϑ) ∈ ξ with X (x) ∈ jT i(x) L for every x ∈ L 0 . For such choices of representatives the expression for Ω becomes Each ϑ k can be written as r k (x) dx for some function r k on S 1 . Meanwhile, any function f on S 1 has a Fourier series expansion which reduces the expression for Ω to This expression is a countably infinite version of the standard symplectic vector space structure.
Remark 10. Weinstein's original construction was more general than we have described so far. It included the case of Lagrangian submanifolds which are neither compact nor oriented. In this case Weinstein used compactly supported densities instead of volume forms. All of our constructions also carry through for non-oriented Lagrangian submanifolds equipped with compactly supported densities.
Example 2. Let (M, ω) = R 2 , dq ∧ dp and L 0 = R. As in Ex. 1, R is one dimensional which means Lag R, R 2 = Emb R, R 2 and for every embedding i we have that T i Lag (L 0 , M ) = Γ (i * T M ). Moreover, since H 1 (L 0 ) = 0 the leaves of E consist of path connected components in Emb R, R 2 . Thus the leaves of F consist of oriented one dimensional submanifolds in R 2 diffeomorphic to R.
Though L 0 is not compact, we can still use compactly supported 1-forms as weightings. A leaf I w then of F w consists of isotopic one dimensional submanifolds in R 2 diffeomorphic to R, equipped with compactly supported 1-forms. Any tangent vector X ∈ T i Lag R, R 2 can be written in components as X = q ∂ ∂q + p ∂ ∂p . Since any 1-form η on R can be written as η (x) dx for some function η, the expression for Ω on such a leaf I w becomes

Positive weighted Lagrangian submanifolds
In this section we will consider an open subset Lag pw (M ) of Lag w (M ) consisting of Lagrangian submanifolds weighted with volume forms. All constructions involving not necessarily positive weightings from before carry over to this case. In particular there is a foliation F pw of Lag pw (M ) whose leaves have a weak symplectic structure. The space Lag pw (M ) also has a different description. By fixing a positive weighting η 0 , the space of positive weighted Lagrangian submanifolds can be identified with the quotient of Lag (L 0 , M ) by the group of diffeomorphisms that preserve η 0 . This identification is Ham (M ) equivariant and makes use of Moser's Lemma [10].
Fix L 0 to be a compact oriented manifold and (M, ω) a symplectic manifold with dim L 0 = 1 2 dim M as before.

Notation
Let Vol 1 (S 0 ) denote the set of volume forms on a compact oriented manifold S 0 that integrate to 1 and Lag pw (L 0 , M ) the product Lag (L 0 , M ) × Vol 1 (L 0 ). That is, given by (2) except that now The atlas U (i,η) , ϕ (i,η) (i,η)∈Lag pw (L 0 ,M ) and the subspace B 1 (L 0 ) ⊕ X (L 0 ) ⊕ Ω n 0 (L 0 ) define a foliation E pw on Lag pw (L 0 , M ). Definition 6. We will call E pw the isodrastic foliation of Lag pw (L 0 , M ) and a leaf of E pw will be called an isodrast in Lag pw (L 0 , M ).

B. Lee
Definition 7. We will call F pw the isodrastic foliation of Lag pw (M ) and a leaf of F pw will be called an isodrast in Lag pw (M ).
Using a result of Moser [10], we can describe isodrasts in Lag pw (M ) more explicitly. Moser's Lemma states that if Λ 0 and Λ 1 are two volume forms on a compact manifold N , such that N Λ 0 = N Λ 1 , then there exists an isotopy ψ t ∈ Diff + (N ) satisfying ψ * 1 Λ 0 = Λ 1 . Thus a positive weighting on a Lagrangian submanifold L can be moved to any other positive weighting via an isotopy of L. Any such isotopy ψ t can be lifted to a Hamiltonian isotopy of M in the following way. Choose a symplectic embedding λ : M ⊃ U → T * L of a neighbourhood U of L onto a neighbourhood of the zero section in T * L. If ψ ♯ t denotes the cotangent lift of ψ t , i.e.
We can also describe Lag pw (M ) and each leaf I w ⊂ Lag pw (M ) in a different way. Suppose that L 0 is equipped with a fixed volume form η 0 that integrates to 1.

Notation
Let Diff (S 0 , η 0 ) denote the group of diffeomorphisms of a manifold S 0 that preserve a fixed volume form η 0 , and X (S 0 , η 0 ) the set of divergence free vector fields on S 0 . That is,

Proof . Define
Then υ is injective since for some a ∈ Diff + (L 0 ). To check surjectivity suppose that (L, ρ) = [(i, η)] ∈ Lag pw (M ) is a positive weighted Lagrangian submanifold. By Moser's Lemma, since η and η 0 are volume forms on L 0 that both induce the orientation of L 0 and integrate to 1, there exists an isotopy ψ t ∈ Diff + (L 0 ) such that ψ * 1 η 0 = η. Thus We will now describe charts into spaces Γ closed (i * T M ) /X (L 0 , η 0 ). Let i ∈ Lag(L 0 , M ) and let λ i : T * L 0 ⊃ W i → M be the symplectic embedding chosen in defining the chart (U i , ϕ i ) on Lag (L 0 , M ). Given a representative X of a class [X] ∈ Γ closed (i * T M ) /X (L 0 , η 0 ), the section dλ −1 i •X ∈ Γ T (T * L 0 )| L 0 can be decomposed as dλ −1 i •X = Z α X | L 0 +Y where ı (Z α X ) ω T * L 0 = π * α X and Y ∈ X (L 0 ). For a different choice of representative, this decomposition changes only in the component Y tangent to L 0 . Thus this decomposition defines a vector space isomorphism We claim that ζ 2 is a vector space isomorphism. It is injective since To check surjectivity, choose a metric g 0 on L 0 such that the induced volume form µ (g 0 ) equals η 0 . Suppose that ϑ = ϑ ′ · η 0 ∈ Ω n 0 (L 0 ) for ϑ ′ ∈ C ∞ (L 0 , R). By the Hodge Decomposition Theorem (see e.g. [20]), there exists a function h ′ ∈ C ∞ (L 0 , R) (unique up to constants) such that △h ′ = ϑ ′ . For such an h ′ , it follows that L ∇h ′ η 0 = △h ′ · η 0 = ϑ.
The isomorphisms ζ 1 and ζ 2 combine to define a vector space isomorphism ζ from the quo- which verifies the Ham (M ) equivariance of υ.
As in the case of not necessarily positive weighted Lagrangian submanifolds, the smooth manifold Lag pw (L 0 , M ) Epw is the total space of a Diff + (L 0 ) bundle over Lag pw (M ) Fpw . We can define a basic 2-form Ω on Lag pw (L 0 , M ) Epw by the expression where α X k = dh k for h k ∈ C ∞ (L 0 , R). This then descends to a weak symplectic structure (also labeled Ω) on Lag pw (M ) Fpw . So in particular the isodrasts in Lag pw (M ) are weakly symplectic manifolds.

Embeddings into a symplectic manifold
In this section we will make precise a heuristic construction by Donaldson [3] of a symplectic structure and moment map for a diffeomorphism group action restricted to the space of embeddings.

B. Lee
Let S 0 be a fixed finite dimensional, compact, and oriented manifold equipped with a volume form η 0 , and let (M, ω) be a finite dimensional symplectic manifold. The set of embeddings Emb (S 0 , M ) of S 0 into M is an open subset of the space C ∞ (S 0 , M ) of all smooth maps. Thus Emb (S 0 , M ) is a smooth manifold modeled on spaces Γ (i * T M ) for i ∈ Emb (S 0 , M ). Assign to each point i ∈ Emb (S 0 , M ) a skew symmetric bilinear form on T i Emb (S 0 , M ) via the expression for X 1 , X 2 ∈ T i Emb (S 0 , M ).
Proof . Checking smoothness amounts to checking the following statement: If X 1 (t) and X 2 (t) are smooth time dependent vector fields on M , i t is a smooth curve in Emb (S 0 , M ), and s : R → R is a smooth function, then the map is smooth. This statement follows from the smoothness of all functions in the integrand.
We will now prove closedness by choosing special extensions of tangent vectors to vector fields on Emb (S 0 , M ). Let X 1 , X 2 , X 3 ∈ T i Emb (S 0 , M ) be tangent vectors. Let Z 1 , Z 2 and Z 3 be vector fields defined on a neighbourhood of i (S 0 ) in M such that Z k • i = X k . Let ξ 1 , ξ 2 and ξ 3 be vector fields defined on the chart neighbourhood U i ⊂ Emb (S 0 , M ) by the expression ξ k (j) := Z k • j. For these particular vector fields, Lie brackets like [ξ 1 , ξ 2 ] at a point i ∈ Emb (S 0 , M ) can be written in terms of the Lie bracket [Z 1 , Z 2 ]: As for weak nondegeneracy, suppose that X 1 ∈ Γ (i * T M ) is nonzero on a neighbourhood W of x ∈ S 0 . Let j be a compatible almost complex structure on M (i.e. g (·, ·) := ω (·, j·) is a Riemannian metric on M ). Let χ be a positive function supported on W . Define X 2 ∈ Γ (i * T M ) by X 2 (x) := χ (i (x)) · jX 1 (x). It follows that Remark 11. Donaldson originally defined the above weakly symplectic structure Ω D on the space of smooth mappings C ∞ (S 0 , M ). When S 0 = M , the 2-form Ω D restricts to a symplectic structure defined by Khesin and Lee on the open subset of orientation preserving diffeomorphisms of M (relative to the orientation induced by the symplectic volume form, which is taken to be η 0 ; see Section 3 in [8]).
Definition 8. Let G be a Lie group with Lie algebra g. By g * we will mean all bounded linear functionals on the convenient vector space g. Let ·, · : g * × g → R denote the canonical pairing between g * and g. The coadjoint representation of G, Ad * : G → GL(g * ) ⊂ L(g * , g * ), is defined by Ad * g ζ, ξ := ζ, Ad g −1 ξ for any ξ ∈ g.
Definition 9. Let (M, σ) be a weakly symplectic smooth manifold. Let G × M → M be a smooth action of a Lie group G on M , such that l * g σ = σ for all g ∈ G. This G action is called Hamiltonian if there exists a G equivariant map (called the moment map) µ ∈ C ∞ (M, g * ) such that for all ξ ∈ g, the function µ, ξ ∈ C ∞ (M, R) is a Hamiltonian for ξ M : Proof . We first note that the Diff (S 0 , η 0 ) action on Emb (S 0 , M ) is symplectic: .
This definition is independent of the choice of A since H 1 (S 0 ) = 0. The map µ is smooth by the usual local arguments. To check that µ is a moment map, let X ∈ Γ (i * T M ) be a tangent vector at i ∈ Emb (S 0 , M ). Let Z be a vector field on a neighbourhood of i (S 0 ) satisfying Z • i = X, and suppose Z generates a flow τ t on M . Let A t be a smooth curve in Ω 1 (S 0 ) satisfying dA t := (τ t • i) * ω. Then .
which verifies the moment map condition.
Finally, µ is Diff (S 0 , η 0 ) equivariant: Let us now consider the special case when S 0 η 0 = 1, the manifold S 0 is half the dimension of M , and assume that the topological conditions H 1 (S 0 ) = 0 and [i * ω] = 0 ∈ H 2 (S 0 ) in Proposition 13 hold so that we have a moment map µ on Emb (S 0 , M ). The level surface µ −1 {0} is given by The group Diff (S 0 , η 0 ) acts freely on µ −1 {0} in the usual way, with the quotient given by By Proposition 11, the set Lag (S 0 , M ) /Diff (S 0 , η 0 ) is a smooth manifold modeled on convenient spaces Γ closed (i * T M ) /X (S 0 , η 0 ) for i ∈ Lag (S 0 , M ). In fact, the manifold Lag (S 0 , M ) is the total space of a principal Diff(S 0 , η 0 ) bundle over the quotient Lag (S 0 , M ) /Diff (S 0 , η 0 ). Since the 2-form Ω D µ −1 {0} is basic, it descends to a unique 2-form Ω D red on µ −1 {0} /Diff (S 0 , η 0 ). Under the topological assumption that H 1 (S 0 ) = 0, the subspace given by the isodrastic foliation F pw at each point equals the entire tangent space to Lag pw (M ) at that point. The weak symplectic structure Ω on isodrasts in this case becomes well defined on all of Lag pw (M ). In fact, the pull back of Ω under the diffeomorphism υ in Proposition 11 is given by In other words, the "symplectic quotient" This last result is suggestive, leading one to wonder if the symplectic structure Ω D on Emb (S 0 , M ) might be related to the symplectic structure Ω on isodrasts in Lag pw (M ) via some sort of reduction procedure. In the next section we will make this relationship clear. In Proposition 13, the topological assumption H 1 (S 0 ) = 0 was essential in defining a moment map. Since the transverse spaces to the leaves of an isodrastic foliation become nontrivial exactly when H 1 (L 0 ) is nontrivial, we would like to remove such a topological condition on S 0 . This means we must use a notion of reduction that does not depend on having a moment map.
Let us begin by looking at the standard reduction of a finite dimensional symplectic manifold (P, σ) with respect to a Hamiltonian G action using a moment map µ. Suppose r is a regular value of µ. The tangent space at p to the level surface µ −1 {r} is equal to the set D p of all vectors X ∈ T p P satisfying σ (X, ξ P (p)) = 0 for all ξ ∈ g. These subspaces D p are defined for any symplectic action on P , even in the absence of a moment map, and define a distribution D on P . If G P is a free symplectic action, then this distribution can be taken as the starting point of the "optimal reduction method" of Juan-Pablo Ortega and Tudor S. Ratiu [11]. We will not describe the details here, but simply note that for a symplectic G action G (P, σ) • the optimal reduction method yields symplectic spaces (P ρ , σ ρ ) where ρ ∈ P/G D and G D is the pseudogroup of flows generated by Hamiltonian vector fields in X D (P ) corresponding to G-invariant Hamiltonian functions; • the "optimal momentum map" is given by the projection J : P → P/G D ; • each reduced symplectic space (P ρ , σ ρ ) is the quotient of an integral manifold of D (the level surface J −1 {ρ}) by the stabilizer G ρ at ρ under the action G P/G D : g·[p] := [g · p]; • if G P is a free Hamiltonian action with moment map µ, and the point r ∈ g * is a regular value of µ, and µ −1 {r} is connected, then µ −1 {r} is a G D orbit ρ and the reduced symplectic space P ρ coincides with the symplectic quotient µ −1 {r} /G r (here G r denotes the stabilizer of r ∈ g * with respect to the coadjoint action of G).
This suggests a way to define reduction in the infinite dimensional case, and motivates the following definition: Definition 10. Let (P, σ) be a weakly symplectic smooth manifold. Let G P be a smooth free action of a Lie group G on P , such that l * g σ = σ for all g ∈ G. The collection of subspaces for x ∈ P defines a distribution D on P . Let i N : N ֒→ P be a maximal integral manifold of D and let q : P → P/G denote the projection to the orbit space. Suppose that 1) q (N ) is a smooth manifold, 2) there exists a unique weak symplectic structure σ red on q (N ) such that ( q| N ) * σ red = i * N σ.
Then the weakly symplectic manifold (q (N ) , σ red ) will be called a reduction or symplectic quotient of (P, σ) with respect to the G action.

Convention
From now on, we will make no topological assumptions on i * ω or H 1 (S 0 ). For the Diff (S 0 , η 0 ) action on the symplectic manifold Emb (S 0 , M ) , Ω D , the subspaces D i can be described in very familiar terms: Proposition 14. For every i ∈ Emb (S 0 , M ), Proof . The distribution D on Emb (S 0 , M ) is defined by for i ∈ Emb(S 0 , M ). Since S 0 dh (ξ) η 0 = 0 for any function h on S 0 and all ξ ∈ X (S 0 , η 0 ), it follows that X ∈ Γ (i * T M ) | α X ∈ B 1 (S 0 ) ⊂ D i . Let X ∈ D i , i.e. S 0 α X (ξ) η 0 = S 0 α X ∧ı ξ η 0 = 0 for all divergence free ξ. If U is a coordinate neighbourhood in S 0 , η 0 = ηdx 1 ∧ · · · ∧ dx n on U , and f a function with supp (f ) ⊂ U , then define the divergence free vector field Y 12 := 1 where we have used integration by parts. If this is to vanish for all f then ∂a 2 ∂x 1 = ∂a 1 ∂x 2 . Similarly, by considering vector fields like Y 13 := 1 it follows that ∂a 3 ∂x 1 = ∂a 1 ∂x 3 , etc., which means α X is closed.
Let g 0 be a Riemannian metric on S 0 whose volume form equals η 0 . For every X ∈ D i , since α X is closed, there exists a function h on S 0 such that β X := α X − dh is harmonic. Moreover, S 0 β X (ξ) η 0 = 0 for every ξ ∈ X (S 0 , η 0 ). Define the vector field Y β X on S 0 by β X = g 0 (Y β X , ·). Let V be a coordinate neighbourhood in S 0 , and suppose β The group Ham (M ) acts freely on Emb (S 0 , M ) under the action In what follows we will show that Ham (M ) orbits through isotropic embeddings are maximal integral manifolds of D. For this purpose, we will need to make use of the following isotropic embedding theorem of Weinstein: Theorem 3 (see [22]). Let (M, ω) be a symplectic manifold and i : I ֒→ M be an isotropic submanifold, i.e. i * ω = 0. The vector bundle T * I ⊕ (T I ω /T I) admits a symplectic structure on a neighbourhood of the zero section, which is given by ω T * I + ω R 2n on the zero section. Furthermore, there exists a neighbourhood U 1 of I in M , a neighbourhood U 2 of I in T * I ⊕ (T I ω /T I), and a symplectomorphism from U 1 to U 2 fixing I. . Given X ∈ D i with α X = dh for some h ∈ C ∞ (S 0 , R), let X = X fib + X tan denote the decomposition of X into components tangent to the fibers and tangent to S. Extend X fib constantly along the fibers in T * S ⊕ N to a vector field Z defined on a neighbourhood of the zero section. It follows that Z is a Hamiltonian vector field satisfying ı (Z) ω = d(pr * 1 π * Since ω = ω T * S + ω R 2n along the zero section, ı(X tan )ω| T S = 0. Thus for each point x in the zero section, we have that ı(X tan )ω defines a linear functional (ı(X tan )ω) x on the fiber T * x S ⊕ N x . The smooth function H tan : T * S ⊕ N → R defined by H tan (x, p, v) := (ı(X tan )ω) x (p, v) is the primitive of a Hamiltonian vector field Z tan satisfying Z tan • i = X tan . It follows that D i ⊂ T i (Ham(M ) · i). The converse inclusion follows from the fact that α X H •i = di * H for any Hamiltonian vector field To check weak nondegeneracy, we first note that the 2-form Ω D red is given by the expression Again, by choosing a symplectic embedding λ : M ⊃ U → T * S ⊕ N we can assume that M = T * S ⊕ N and that i is the zero section inclusion. Given [X 2 ] ∈ Ker Ω D red , let X 2 = X fib + X tan be the decomposition of X 2 into components tangent to the fibers and to the zero section respectively. Extend X fib and X tan to Hamiltonian vector fields Z fib and Z tan respectively as before.

Convention
From now on we will assume that S 0 η 0 = 1.

B. Lee
The pull back of Ω under this map is given by So indeed, the symplectic quotient O, Ω D red is symplectomorphic to the isodrast (I w , Ω). Example 3. Let (M, ω) = R 2 , dq ∧ dp and S 0 = S 1 . Since S 1 is one dimensional Lag S 1 , R 2 = Emb S 1 , R 2 . However, since H 1 S 1 is nontrivial, we have nontrivial isodrasts in Emb S 1 , R 2 that can be described as follows.
Let β = pdq denote the canonical 1-form on the plane. Given a map i ∈ Emb S 1 , R 2 , the action integral A (i) of i is defined as the integral of β around i S 1 : An isotopic deformation is Hamiltonian if and only if the action integrals are constant along the deformation (see Proposition 2.1 in [23]) 1 . The idea is as follows. Given two nearby loops in a symplectic manifold (M, ω), we can define the difference in their action integrals to be the integral of −ω over a cylindrical surface joining the two loops. This is well defined even when ω is not exact. Lagrangian submanifolds near a given L ∈ Lag (M ) can be identified with graphs of 1-forms in T * L by Theorem 1. Two such graphs can be joined by a Hamiltonian deformation if and only if their corresponding 1-forms are cohomologous. If γ ′ is a small deformation of a loop γ in the zero section corresponding to a deformation of the zero section to a Lagrangian submanifold L ′ = Graph (α), C denotes a cylinder joining γ and γ ′ , and if β T * L denotes the canonical 1-form of the T * L, then So a small deformation of the zero section is the graph of an exact 1-form if and only if the difference in action integrals is 0 for all loops γ and γ ′ in the zero section and the deformed image respectively. It follows that two graphs of 1-forms can be joined by a Hamiltonian deformation if and only if the difference in action integrals remains constant for all loops in these Lagrangian submanifolds. Returning to our example, this means that the isodrasts in Emb S 1 , R 2 consist of circle embeddings that can be joined by an isotopy that preserves action integrals, i.e. H ⊂ Emb S 1 , R 2 is an isodrast if and only if it consists of isotopic circle embeddings and the map A : Emb S 1 , R 2 → R sending i to its action integral A (i) is constant on H.
On the circle we can take η 0 = dt 2π so that Diff S 1 , dt 2π consists of rigid rotations of the circle. Let O := q (H) be the image of an isodrast H in the orbit space. Each representative X for a tangent vector [X] ∈ T [i] O can be decomposed as X = Z + Y where Z is a normal to i S 1 and Y is tangent to i S 1 . It follows that if α X k = dh k then the reduced symplectic structure is given by

Weighted isotropic submanifolds
The results of the last section suggest a way to generalize Weinstein's construction of a symplectic structure on spaces of weighted Lagrangian submanifolds to spaces of weighted isotropic submanifolds. In this section we will pursue this idea and thereby obtain a generalization of Theorem 4 suggested by Proposition 15. Let (M, ω) be a symplectic manifold and I 0 a fixed compact oriented manifold with dim I 0 ≤  Definition 12. We will call E w the isodrastic foliation of Iso w (I 0 , M ), and each leaf of E w will be called an isodrast in Iso w (I 0 , M ). Similarly, F w will be called the isodrastic foliation of Iso w (M ), and each leaf of F w will be called an isodrast in Iso w (M ).
The pointwise assignment where α X k = dh k for h k ∈ C ∞ (I 0 , R) defines a basic 2-form on Iso w (I 0 , M ) Ew . It descends to a weak symplectic structure (also labeled Ω) on Iso w (I 0 , M ) Fw . Here closedness and nondegeneracy of Ω on I w follow from using the local model of isotropic submanifolds afforded by Theorem 3, and the fact that in such a model the symplectic form along the zero section in T * I ⊕ Γ (T I ω /T I) is given by ω T * I + ω R 2n where ω R 2n is the standard symplectic vector space structure on the fibers of Γ (T I ω /T I). The proof is completely analogous to that of Theorem 4.
To check that the map Φ : I w → g * is Ham (M ) equivariant we observe that As for the image of Φ, since the stabilizers at points (I, ρ) ∈ I w and Φ (I, ρ) ∈ g * are given by Remark 13. This last theorem makes rigourous the heuristic moment map written down by Weinstein. Also, it extends it to a map identifying isodrasts in the space of positive weighted isotropic submanifolds with coadjoint orbits in the dual of the Lie algebra of the group of Hamiltonian symplectomorphisms.

Poisson structures
In this section we will define a Poisson algebra (A, {·, ·}) for a subalgebra A ⊂ C ∞ (M ) of smooth functions on a smooth manifold M . Such a manifold will be called an A-Poisson manifold if there are enough Hamiltonian vector fields in a sense we will specify. We define what a leaf of such an A-Poisson manifold is. We then show that reductions of Emb (S 0 , M ) consisting of positive weighted isotropic submanifolds are symplectic leaves of a Poisson structure.

Notation
Let C ∞ R (Emb (S 0 , M )) denote the set of maps F : Emb (S 0 , M ) → R such that for all i 0 ∈ Emb (S 0 , M ) there exists a c ∞ -open neighbourhood U of i 0 , a map A ∈ C ∞ (R n , R), and smooth functions h 1 , . . . , h n ∈ C ∞ (W, R) defined on a neighbourhood W of i 0 (S 0 ) so that for all i ∈ U .
Remark 14. The algebra of functions C ∞ R (Emb (S 0 , M )) was chosen with the axioms of differential structures in mind (see [13]; also cf. Section 3 in [2]). A differential structure on a topological space Q is a set C ∞ (Q) of continuous functions on Q with the following properties: 1. The topology of Q is generated by sets of the form F −1 (V ) where F ∈ C ∞ (Q) and V is an open subset of R.
3. If F : Q → R is a function such that, for every x ∈ Q there is an open neighbourhood U of x in Q and a function F U ∈ C ∞ (Q) such that F | U = F U , then F ∈ C ∞ (Q).
A topological space Q together with a differential structure C ∞ (Q) is called a differential space. When Emb (S 0 , M ) is equipped with the topology T generated by sets of the form and V is open in R, then the topological space (Emb (S 0 , M ) , T ) together with C ∞ R (Emb (S 0 , M )) defines a differential space.
Proposition 16. For every F ∈ C ∞ R (Emb (S 0 , M )), the local assignments given by v F (i) := n j=1 ∂A Proof . We will first compute the exterior derivative of a map in C ∞ R (Emb (S 0 , M )) locally. Suppose that F ∈ C ∞ R (Emb (S 0 , M )) can be written as in (9) on U for some A ∈ C ∞ (R n , R) and smooth functions h 1 , . . . , h n ∈ C ∞ (W, R) defined on a neighbourhood W of i 0 (S 0 ), then the pointwise exterior derivative of F in the direction of v is given by It follows that the vector field v F ∈ X (U ), defined by v F (i) := n j=1 ∂A ∂y j X h j • i (where X h j is the Hamiltonian vector field with Hamiltonian h j ), satisfies dF = ı (v F ) Ω D on U . In fact, if U 1 and U 2 are c ∞ -open neighbourhoods in Emb (S 0 , M ) with nonempty intersection, and So by the nondegeneracy of Ω D , the local assignments v F (i) := n j=1 ∂A ∂y j X h j • i coincide on overlapping regions and so define a vector field v F on Emb (S 0 , M ) satisfying dF = ı (v F ) Ω D . Uniqueness of the Hamiltonian vector field v F also follows from nondegeneracy of Ω D . In this case the bracket [·, ·] will be called a Poisson bracket. If (A, [·, ·]) is a Poisson algebra, we will say that M is an A-Poisson manifold.

Convention
In what follows, we will use the same letter F to denote a function in C ∞ R (Emb (S 0 , M )), its restriction to Iso (S 0 , M ), as well as the corresponding map on the quotient Iso pw (M ) to avoid introducing more notation.
For Remark 15. In finite dimensions the distribution C is integrable. In infinite dimensions it might be possible to prove an analogue of Frobenius' theorem for some spaces of mappings and apply it to our spaces.
Remark 16. The first condition in the previous definition is satisfied within the subalgebra C ∞ R (Iso pw (M )) but not in C ∞ R (Emb (S 0 , M )) because of the directions generated by the Diff(S 0 , η 0 ) action.
For each F ∈ C ∞ R (Iso pw (M )) we have that dr (v F ) is the unique vector field on Iso pw (M ) satisfying dF = ı (dr (v F )) Ω D red (uniqueness follows from the nondegeneracy of Ω D red on Iso pw (M ) Fpw ). It follows that if F ∈ C ∞ R (Iso pw (M )) then dG (X F ) = {F, G} Iso pw (M ) for all G ∈ C ∞ R (Iso pw (M )). On the C ∞ R (Iso pw (M ))-Poisson manifold Iso pw (M ), the distribution C is given by the collection of vectors

Momentum weighted Lagrangian submanifolds
In this section we will discuss a kinematic interpretation of isodrasts in Lag (M ) to motivate a different choice of "weightings" to obtain a symplectic structure. Let us view points in an isodrast I ⊂ Lag (M ) as configurations of a submanifold constrained to move in I. What are the possible velocities? By Proposition 5 and the description of the isodrastic foliation F, we know that the tangent bundle of I can be described by Thus the velocities at a configuration L ∈ I correspond to functions on L modulo constants, i.e. to elements of C ∞ (L, R) /R. The conjugate momenta to configurations in I (i.e. cotangent vectors) should be linear functionals from C ∞ (L, R) /R to R, that are in 1-1 correspondence with C ∞ (L, R) /R. This expectation that the cotangent fibers should be the "same size" as C ∞ (L, R) /R reflects a physical expectation that all momenta should be accessible by motions of particles in the configuration space, and that all such motions can be assigned momenta. In any case, integration against n-forms in Ω n 0 (L) certainly fits the above description. This motivates the following definition: Definition 15. Let I be an isodrast in Lag (M ). A momentum weighting of a Lagrangian submanifold L ∈ I is a top degree form χ on L satisfying L χ = 0. Pairs (L, χ) will be called momentum weighted Lagrangian submanifolds.
As in the case of weighted Lagrangian submanifolds, the canonical projection from Lag mw (M ) to Lag (M ) pulls back the foliation F to a foliation F mw on Lag mw (M ), whose leaves Definition 16. We will call F mw the isodrastic foliation of Lag mw (M ) and each leaf of F mw will be called an isodrast in Lag mw (M ).
The tangent space to an isodrast I mw at a point [(i, χ)] is given by [ω (X 1 , where α X k = dh k , defines an exact weak symplectic structure on I mw satisfying Ω = −dΘ where Proof . We will compute the exterior derivative of Θ locally in charts. That is, be means of a symplectic embedding λ (i,χ) : T * L 0 ⊃ W (i,χ) → M chosen in defining a chart U (i,χ) , ϕ (i,χ) on Lag mw (M ), we can assume that M = T * L 0 and that each tangent vector in T [(i,χ)] I mw is represented by a pair (X, ϑ) where α X ∈ B 1 (L 0 ) and X is tangent to the cotangent fibers. Given a tangent vector ξ = (X, ϑ) ∈ T [(i,χ)] I mw , we can extend it to a vector field on I mw (also labeled ξ) in the following way. Let Z f denote the Hamiltonian vector field defined on T * L 0 satisfying ı (Z f ) ω = π * α X where α X = dh and f = π * h. Define ξ to be the vector field on I mw that assigns ξ ([i ′ , ν ′ ]) = (Z f • i ′ , ϑ) to nearby points [(i ′ , χ ′ )]. So given tangent vectors ξ 1 and ξ 2 in T [(i,χ)] I mw , extend them to vector fields (also labeled ξ 1 and ξ 2 ) as just described. Then Nondegeneracy then follows from the fact that the h's and ϑ's are independent of one another.
If k 1 = 0, then we are left with − M Tr g −1 k 2 g −1 l 1 µ (g) which means we can choose k 2 so that the integral does not vanish for the same reason.

Notation
The manifold topology on Met q mw (M ) is finer than the trace of the Whitney C ∞ -topology on Γ (L (T M, T * M )) (see Section 45.1 in [9]). Let C ∞ R (Met q mw (M )) denote the set of functions F : Met q mw (M ) → R such that for every g 0 ∈ Met q mw (M ) there exists a neighbourhood U of g 0 , a map A : R n → R, and sections r 1 , . . . , r n ∈ Γ c S 2 T * M so that F (g) = A M Tr g −1 r 1 g −1 h µ (g) , . . . , M Tr g −1 r n g −1 h µ (g) for all g ∈ U .
Remark 18. The algebra C ∞ R (Met q mw (M )) contains the constant functions.
By an argument similar to that in Proposition 16, we have the following proposition: Proposition 19. For every F ∈ C ∞ R (Met q mw (M )), the local assignments given by v F (g) := n j=1 ∂A ∂y j · X Fr j (g) on each neighbourhood U define a unique vector field v F on Met q mw (M ) satisfying dF = ı (v F ) Ω.
It follows that {F, G} := −Ω (v F , v G ) defines a Poisson bracket on Met q mw (M ).