Weighted nonlinear flag manifolds as coadjoint orbits

A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Frechet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags.


Introduction
In this article we study manifolds of weighted nonlinear flags, motivated by the fact that one can use them to describe new coadjoint orbits of the Hamiltonian group.This adds to the already known coadjoint orbits described with configuration spaces of points, weighted isotropic nonlinear Grassmannians [26,15,7], symplectic nonlinear Grassmannians [8], and manifolds of symplectic nonlinear flags [9].
Let M be a smooth manifold, and let S = (S 1 , . . ., S r ) be a collection of closed manifolds of strictly increasing dimensions.We consider the Fréchet manifold Flag S (M ) of nonlinear flags of type S, i.e., sequences of nested embedded submanifolds N 1 ⊆ • • • ⊆ N r in M , with N i diffeomorphic to S i .Considering submanifolds equipped with nowhere zero densities, one obtains the manifold Flag wt S (M ) of weighted nonlinear flags.We describe its Fréchet manifold structure in two ways: as a splitting smooth submanifold of the cartesian product of weighted nonlinear Grassmannians of type S i in M , and as a locally trivial smooth fiber bundle over Flag S (M ) associated to the principal bundle of nonlinear frames of type S in M .To each weighted nonlinear flag one associates a compactly supported distribution on M with controlled singularities, by the Diff(M ) equivariant inclusion J : Flag wt S (M ) ֒→ C ∞ (M ) * , J((N 1 , ν 1 ), . . ., (N r , ν r )), f := The Diff(M ) orbits in Flag wt S (M ) are submanifolds of finite codimension, for which we give an explicit homological description, up to connected components, using nonlinear flags decorated with cohomology classes (see Theorems 2.11 and 2.15).
Inspired by the results in [26,15,7] on weighted isotropic nonlinear Grassmannians, we consider the manifold of weighted isotropic nonlinear flags in a symplectic manifold (M, ω).The orbits for the natural action of the Hamiltonian group Ham c (M ) are submanifolds of finite

Manifolds of weighted nonlinear flags
A nonlinear flag is a sequence of nested closed submanifolds N 1 ⊆ • • • ⊆ N r in a smooth manifold M .A weighted nonlinear flag is a nonlinear flag together with a volume density ν i on each submanifold N i .Integrating against test functions f ∈ C ∞ (M ), a weighted nonlinear flag provides a compactly supported distribution on M with mild singularities, r i=1 N i f ν i .We will show that the space of all weighted nonlinear flags in M is a Fréchet manifold in a natural way.In fact, this is the total space of a locally trivial smooth bundle over the manifold of nonlinear flags discussed in [9].The natural Diff(M ) action on the base of this bundle is locally transitive [9, Proposition 2.9(a)].The main aim of this section is to describe the Diff(M ) orbits in the space of weighted nonlinear flags, see Theorem 2.11 below.

Weighted nonlinear Grassmannians
In this section we recall some basic facts about the manifolds of weighted submanifolds that appear in [26,15,7].These weighted nonlinear Grassmannians constitute a special case of the weighted nonlinear flags to be introduced in Section 2.2.We present them here in a manner that readily generalizes to the setting of nonlinear flags.
Let S be a closed manifold of dimension k, allowed to be nonconnected and nonorientable.For each manifold M , we let Gr S (M ) denote the nonlinear Grassmannian of type S in M , i.e., the space of all smooth submanifolds in M that are diffeomorphic to S.Moreover, we let Emb S (M ) denote the space of all parametrized submanifolds of type S in M , i.e., the space of all smooth embeddings of S into M .Both, Emb S (M ) and Gr S (M ), are Fréchet manifolds in a natural way.Furthermore, the Diff(M ) equivariant map Emb S (M ) → Gr S (M ), ϕ → ϕ(S), is a smooth principal bundle with structure group Diff(S), a Fréchet Lie group, see [2,19,20] and [14,Theorem 44.1].
For each closed k-dimensional manifold S, let Den(S) := Γ ∞ (|Λ| S ) = Ω k (S; O S ) denote the space of all smooth densities on S.Here O S denotes the orientation bundle of S and |Λ| S = Λ k T * S ⊗ O S , see e.g.[16].Densities on S are the geometric quantities which can be integrated over S in a coordinate independent way, without specifying an orientation or even assuming orientability.We denote by Den × (S) the space of volume densities, i.e., the space of nowhere vanishing densities.Clearly, this is an open subset in the Fréchet space Den(S).We define the weighted nonlinear Grassmannian of type S in M by Gr wt S (M ) := (N, ν) N ∈ Gr S (M ), ν ∈ Den × (N ) , that is, the space of all submanifolds of type S in M , decorated with a nowhere zero density.
We equip this space with the structure of a Fréchet manifold by declaring the natural bijection Gr wt S (M ) = Emb S (M ) × Diff(S) Den × (S), to be a diffeomorphism.Here the right hand side denotes the total space of the bundle associated to the nonlinear frame bundle in (1) and the natural Diff(S) action on Den × (S).In particular, the canonical forgetful map Gr wt S (M ) → Gr S (M ), (N, ν) → N, becomes a locally trivial smooth bundle with typical fiber Den × (S).Indeed, it corresponds to the bundle projection of the associated bundle Emb S (M ) × Diff(S) Den × (S) → Gr S (M ) via the identification in (3).
There is a canonical Diff(M ) equivariant map This map is injective, and its image consists of compactly supported distributions with mild singularities: J(N, ν) is supported on N , and its wave front set coincides with the conormal bundle of N .Let µ ∈ Den × (S) be a volume density.The space is called the nonlinear Grassmannian of weighted submanifolds of type (S, µ) in M .It consists of all weighted submanifolds (N, ν) in M such that there exists a diffeomorphism S → N taking µ to ν. Denoting the Diff(S) orbit of µ by Den(S) µ , the identification in (3) restricts to a canonical bijection It is well known [21] that the Diff(S) 0 orbit of µ is a convex subset that consists of all volume densities on S that represent the same cohomology class as µ in H k (S; O S ), the de Rham cohomology with coefficients in the orientation bundle.Hence, the Diff(S) orbit of µ coincides with the set of all volume densities on S that are in the preimage of H k (S; O S ) [µ] , the (finite) Diff(S) orbit of [µ] in H k (S; O S ), under the Diff(S) equivariant linear map More succinctly, Den(S) Hence, Den(S) µ is an open subset in a finite union of parallel closed affine subspaces with finite codimension.In particular, Den(S) µ is a splitting smooth submanifold in Den × (S) with finite codimension dim H k (S; O S ) and with tangent spaces Using (5) we conclude that Gr wt S,µ (M ) is a splitting smooth submanifold in Gr wt S (M ) with finite codimension dim H k (S; O S ).Moreover, the canonical forgetful map in (4) restricts to a locally trivial smooth fiber bundle Gr wt S,µ (M ) → Gr S (M ) with typical fiber Den(S) µ .The space of cohomologically weighted submanifolds of type S in M is defined as Using the canonical bijection we turn Gr hwt S (M ) into a smooth vector bundle of finite rank dim H k (S; O S ) over Gr S (M ).The canonical Diff(M ) equivariant map is a smooth bundle map over Gr S (M ).Indeed, via the diffeomorphisms in ( 5) and ( 9) it corresponds to the map induced by (6).
The space of cohomologically weighted submanifolds of type (S, [µ]) in M is defined by and consists of all cohomologically weighted submanifolds (N, [ν]) such that there exists a diffeomorphism S → N taking the cohomology class [µ] to [ν].As (9) restricts to a bijection we see that Gr hwt S,[µ] (M ) is a finite covering of Gr S (M ).Using (7) we conclude It is well known that the Diff c (M ) action on Emb S (M ) admits local smooth sections, see for instance [9, Lemma 2.1(c)].Furthermore, the (transitive) Diff(S) action on Den(S) µ also admits local smooth sections.The latter can be shown using Moser's method of proof in [21, Section 4], see Lemma 2.12 below.Using Lemma A.1 in the Appendix, we conclude that the natural Diff c (M ) action on Gr wt S,µ (M ) admits local smooth sections.In particular, this action is locally transitive.Hence, each connected component of Gr This is the case considered in [26,15,7].
If S is built out of two diffeomorphic connected components, then H k (S; O S ) ∼ = R 2 and any diffeomorphism swapping the two connected components acts nontrivially on this cohomology.If µ has equal total volume on the two connected components, then the orbit H k (S; O S ) [µ] is a one-point set and Den(S) µ is connected.Otherwise H k (S; O S ) [µ] consists of two points and, by (7), Den(S) µ has two connected components.

Weighted nonlinear flag manifolds
Fix natural numbers k i such that and let S = (S 1 , . . ., S r ) be a collection of closed smooth manifolds with dim S i = k i .For a smooth manifold M we let denote the space of nonlinear flags of type S in M , and we write for the space of the space of nonlinear frames of type S in M .In [ Diff(S i ).
We denote the space of weighted nonlinear flags of type S in M by Gr wt S i (M ) ∀i : This is a splitting smooth submanifold in r i=1 Gr wt S i (M ), for it coincides with the preimage of the splitting smooth submanifold Flag S (M ) under the bundle projection r i=1 Gr wt S i (M ) → r i=1 Gr S i (M ).Moreover, the canonical Diff(M ) equivariant forgetful map is a smooth fiber bundle with typical fiber The latter is a Diff(S) invariant open subset in the Fréchet space Den(S) := r i=1 Den(S i ).Furthermore, the canonical Diff(M ) equivariant bijection Flag wt S (M ) = Fr S (M ) × Diff(S) Den × (S), (17) ϕ 1 (S 1 ), (ϕ 1 ) * µ 1 , . . ., ϕ r (S r ), (ϕ r ) * µ r ↔ (ϕ 1 , . . ., ϕ r ), (µ 1 , . . ., µ r ) , is a diffeomorphism between Flag wt S (M ) and the bundle associated to the nonlinear frame bundle in (13) and the canonical Diff(S) action on Den × (S).Indeed, this is just the bundle diffeomorphism r i=1 Gr wt S i (M ) = r i=1 Emb S i (M ) × Diff(S i ) Den × (S i ) obtained by taking the product of the diffeomorphisms in (3), restricted over the submanifold Flag S (M ) in its base r i=1 Gr S i (M ).We have a canonical Diff(M ) equivariant map The image of J consists of compactly supported distributions on M with mild singularities.More precisely, the wave front set of J((N 1 , ν 1 ), . . ., (N r , ν r )) coincides with the union of the conormal bundles of N 1 , . . ., N r .
Remark 2.6.Suppose ω is a symplectic form on M , and let Flag symp S (M ) denote the manifold of symplectic nonlinear flags of type S, cf.[9,Section 4.2].Recall that this is the open subset consisting of all flags (N 1 , . . ., N r ) ∈ Flag S (M ) such that ω restricts to a symplectic form on each N i .Hence, k i must be even and ω k i /2 pulls back to a volume form on N i which in turn gives rise to a volume density Consequently, the symplectic form ω provides a Symp(M, ω) equivariant injective smooth map (section) which is right inverse to the restriction of the canonical bundle projection in (15).Composing the map in (19) with J in (18), we obtain the moment map considered in [9, Eq. ( 38)].
Remark 2.7.A Riemannian metric g on M induces a volume density on every submanifold of M .Hence, g provides a smooth section Flag S (M ) → Flag wt S (M ) of the canonical bundle projection in (15), which is Isom(M, g) equivariant, cf.[26, Section 6].

Reduction of structure group
It will be convenient to use a reduction of the structure group for the principal frame bundle in (13).To this end, we fix embeddings ι i : S i → S i+1 and put ι = (ι 1 , . . ., ι r−1 ).
We begin by recalling some facts from [9, Proposition 2.10].The space of nonlinear flags of type (S, ι) in M , open and closed subset in Flag S (M ).The space of parametrized nonlinear flags (nonlinear frames) of type (S, ι) in M , is a splitting smooth submanifold of Fr S (M ).Moreover, the map Fr S,ι (M ) → Flag S,ι (M ) obtained by restriction of ( 13) is a smooth principal bundle with structure group The latter is a splitting Lie subgroup in Diff(S) with Lie algebra We obtain a Diff(M ) equivariant commutative diagram Fr S,ι (M ) which may be regarded as a reduction of the structure group for (13) along the inclusion Diff(S; ι) ⊆ Diff(S) over Flag S,ι (M ), see [9, Proposition 2.10] for more details.
Remark 2.8.The Diff(M ) equivariant bijection is a diffeomorphism [9, Proposition 2.10(b)].Correspondingly, we have a group isomorphism where Diff(S r ; Σ) denotes the subgroup of all diffeomorphisms of S r preserving the nonlinear flag Σ = (Σ 1 , . . ., Σ r−1 ) in S r , where The latter is a splitting Lie subgroup of Diff(S r ), see [9, Proposition 2.9(b)], and ( 24) is a diffeomorphism of Lie groups [9, Proposition 2.10(a)].The Lie algebra of Diff(S; ι), can be identified in a similar way with X(S r ; Σ), the Lie algebra of vector fields on S r that are tangent to Σ 1 , . . ., Σ r−1 .
We are interested in the reduction of structure group (22) because the Diff c (M ) action on Fr S,ι (M ) admits local smooth sections.This follows from [9, Lemma 2.1(c)] and the diffeomorphism in (23).
Let Flag wt S,ι (M ) denote the preimage of Flag S,ι (M ) under the bundle projection in (15).Restricting the diffeomorphism in (17) over Flag S,ι (M ) and combining this with the diffeomorphism in (23), we obtain a Diff(M ) equivariant diffeomorphism of bundles over Flag S,ι (M ), 2. 4 The Diff(M) action on the space of weighted nonlinear flags In this section we aim at describing the Diff(M ) orbits in Flag wt S (M ), see Theorem 2.11 below.Let ι = (ι 1 , . . ., ι r−1 ) be a collection of embeddings ι i : S i → S i+1 and suppose µ = (µ 1 , . . ., µ r ) ∈ Den × (S).We define the space of weighted flags of type (S, ι, µ) in M by that is, the space of all weighted flags (N 1 , ν 1 ), . . ., (N r , ν r ) in M such that there exist diffeomorphisms S i → N i , 1 ≤ i ≤ r, intertwining ι i with the canonical inclusion N i ⊆ N i+1 , and taking µ i to ν i .
Denoting the Diff(S, ι) orbit of µ by Den(S) ι,µ , the diffeomorphism in (25) restricts to a Diff(M ) equivariant bijection Consider the finite dimensional vector space Here the left hand side denotes relative de Rham cohomology with coefficients in the orientation bundle, and we are using the convention S 0 = ∅.The Diff(S, ι) equivariant identification on the right hand side indicates Poincaré-Lefschetz duality.We have a Diff(S, ι) equivariant linear map h S,ι : Pinning down the class [µ] := h S,ι (µ) thus amounts to specifying the integrals of µ i over each connected component of S i \ ι i−1 (S i−1 ) for i = 1, . . ., r.
Remark 2.9 (Large codimensions).If the codimensions dim(S i ) − dim(S i−1 ) are all strictly larger than one, then Hence, in this case, the cohomology space H(S, ι) does not depend on the embeddings ι.
In particular, Den(S) ι,µ is a splitting smooth submanifold in Den × (S) with finite codimension dim H(S, ι) and with tangent spaces Moreover, the Diff(S, ι) action on Den(S) ι,µ admits local smooth sections.
(d) The canonical inclusion Den(S) ι,µ ⊆ r i=1 Den(S i ) µ i is a splitting smooth submanifold of finite codimension.
We postpone the proof of this proposition and proceed with the main result in this section:  5) and (26), see also (23).We will prove Proposition 2.10 by induction on the depth of the flags, using the following crucial lemma whose proof we postpone.Lemma 2.12.Let S be a closed submanifold of N such that dim(S) < dim(N ) = n, and consider the Diff(N, S) equivariant linear map Then, for each κ ∈ H n (N, S; O N ), the natural Diff(N, S) 0 action on admits local smooth sections.
Let us next establish the following infinitesimal version of Lemma 2.12: , and Z ∈ X(S).Then there exists a vector field X ∈ X(N ) such that L X µ = dγ and X| S = Z.
Proof.Let Z ∈ X(N ) be any extension of Z, i.e.Z| S = Z.Note that i Z µ ∈ Ω n−1 (N ; O N ) vanishes when pulled back to S, hence the same holds for Let us first construct γ ∈ Ω n−1 (N ; O N ) such that dγ = dβ and γ| S = 0. To this end, we fix a smooth homotopy h : Hence, the vector field X := Z + Y has the desired properties.
Proof of Lemma 2.12.Recall that the infinitesimal Diff(N, S) action on Diff(S) × Den(N ) is where X ∈ X(N, S), f ∈ Diff(S), and µ ∈ Den(S).Here, for Z ∈ T id Diff(S) = X(S) we let R Z (f ) denote the right invariant vector field on Diff(S) such that R Z (id) = Z.
Note that the vector field Z in the proof of Lemma 2.13 can be chosen to depend smoothly (and linearly) on Z. Hence, the proof of said lemma actually provides a smooth map σ : Combining this with the right trivialization of T Diff(S), we obtain a smooth map , and ξ ∈ T µ Den × (N ) ∩ h −1 (κ) .As Diff(N, S) is a regular Lie group, we may apply Lemma A.2 to conclude that the Diff(N, S) 0 action on (31) admits local smooth sections.
This completes the proof of Theorem 2.11.
Remark 2.14.In view of Remark 2.3, we expect that the isotropy subgroup Diff(S; ι, µ) := (g 1 , . . ., g r ) ∈ Diff(S, ι) ∀i : is a splitting Lie subgroup of Diff(S; ι) with Lie algebra and the surjective map provided by the action, Diff(S, ι) → Den(S) ι,µ , is a locally trivial smooth principal fiber bundle with structure group Diff(S; ι, µ).This would follow in a rather straigthforward manner, via induction on the depth of the flags, if one could show that the isotropy group {f ∈ Diff(N, S) : f | S = id, f * µ = µ} is a splitting Lie subgroup in Diff(N, S), whenever S is a closed submanifold of N and µ is a volume density on N .The proof in [11, Theorem III.2.5.3 on page 203] covers the case S = ∅.However, the adaptation of said proof to nontrivial S is not entirely straigthforward, and we will not attempt to prove this here.Note that via the diffeomorphism in (24) the group Diff(S; ι, µ) corresponds to the subgroup of Diff(S r ; Σ) consisting of all diffeomorphisms that preserve µ r and whose restriction to Similarly, the Lie algebra X(S; ι, µ) can be identified to the corresponding subalgebra of X(S r ; Σ).If the expectation formulated in the preceding paragraph were indeed true, then the surjective and Diff(M ) equivariant map would be a locally trivial smooth principal fiber bundle with structure group Diff(S; ι, µ), see (26).Moreover, generalizing Remark 2.4, the isotropy group of a weighted flag (N , ν),

A homological description
In this section we give a more explicit description of the manifold Flag wt S,ι,µ (M ).If N = (N 1 , . . ., N r ) is a nonlinear flag of type S in M , we put We define the space of homologically weighted flags of type S in M by Note that we have a Diff(M ) equivariant forgetful map as well as a Diff(M ) equivariant map Let Flag hwt S,ι (M ) denote the preimage of the open subset Flag S,ι (M ) under the projection in (38).Using the canonical Diff(M ) equivariant identifications we equip Flag hwt S (M ) with the structure of a smooth vector bundle of finite (possibly nonconstant) rank over Flag S (M ) and with projection (38).The map in (39) is a smooth bundle map over Flag S (M ).Indeed, via the identifications in ( 25) and (40), the map h S,ι in (28) induces a bundle map Flag wt S,ι (M ) → Flag hwt S,ι (M ) which coincides with the restriction of (39).We define the space of homologically weighted flags of type (S, ι, that is, the space of all homologically weighted flags (N , [ν]) in M such that there exist diffeomorphisms Proof.This follows from the identity (29) in Proposition 2.10(c), using ( 26) and (41).
A description for nested spheres in the same vein can be found in Example 2.19.
Example 2.17 (Nested tori).If (S, ι) denotes the standard (meridional) embeddings between tori, Example 2.18 (Nested projective spaces).If (S, ι) denotes the standard embeddings between projective spaces, Example 2.19 (Nested spheres [13]).If (S, ι) denotes the standard equatorial embeddings between spheres, r+1) .The 2(r + 1) numbers assigned to [µ] ∈ H(S, ι) by this isomorphism are where S i + and S i − denote the northern and southern hemispheres of S i , respectively.Considering reflections on hyperplanes, we see that for each 0 ≤ i ≤ r there exists a diffeomorphism in Diff(S, ι) swapping S i + with S i − , but leaving all other hemispheres S k ± invariant.Such a diffeomorphism interchanges a + i with a − i , but leaves all other numbers a ± k unchanged.Hence, the Diff(S, ι) orbit H(S, ι) [µ] has 2 s elements, where s is the number of 0 ≤ i ≤ r with a + i = a − i .Actually the Diff(S, ι) action on H(S, ι) ∼ = R 2(r+1) factorizes through an (Z 2 ) 2(r+1) action by switching or not the numbers a + i and a − i .We obtain a description similar to the one in (43): 3 Coadjoint orbits of the Hamiltonian group Throughout this section (M, ω) denotes a symplectic manifold.The nonlinear Grassmannian of all isotropic submanifolds of type S r in M , denoted by Gr iso Sr (M ), is a splitting smooth submanifold in Gr Sr (M ) which is invariant under the Hamiltonian group.In fact the Ham(M ) orbits provide a smooth foliation of finite codimension in Gr iso Sr (M ) which is called the isodrastic foliation [26,15].Suppose L is an isodrastic leaf in Gr iso Sr (M ) and let Flag wt iso S,ι,µ (M )| L denote the preimage of L under the canonical bundle projection Flag wt S,ι,µ (M ) → Gr Sr (M ).We will show that the natural Ham c (M ) action on Flag wt iso S,ι,µ (M )| L admits local smooth sections.In particular, each connected component of the latter space is an orbit of Ham c (M ).
The space Flag wt iso S,ι,µ (M )| L comes equipped with a canonical weakly non-degenerate symplectic form, and the restriction of ( 18) provides an Ham(M ) equivariant injective moment map for the Ham c (M ) action, This moment map J maps each connected component of Flag wt iso S,ι,µ (M )| L one-to-one onto the corresponding coadjoint orbit, see Theorem 3.14.Thereby, we identify coadjoint orbits of the Hamiltonian group Ham c (M ) that can be modeled on weighted nonlinear flags.
The material in this section is inspired by the results in [26,15,7] on weighted isotropic nonlinear Grassmannians.

Isodrasts as Ham(M) orbits
In view of the tubular neighborhood theorem for isotropic embeddings [23,24], the space Gr iso S (M ) of all isotropic submanifolds of type S in M is a splitting smooth submanifold of Gr S (M ), see for instance [15,Section 8].The tangent space at an isotropic submanifold N is where T N ⊥ := T M | N /T N denotes the normal bundle to N .Weinstein's [26] isodrastic distribution D on Gr iso S (M ) is given by and has finite codimension dim H 1 (S; R).This is an integrable distribution [26,15] whose leaves are orbits of Ham(M ).It gives rise to a smooth foliation of Gr iso S (M ) called the isodrastic foliation.In particular, the Ham(M ) orbits in Gr iso S (M ) are splitting smooth submanifolds.Restricting the fundamental frame bundle in (1), we obtain a principal Diff(S) bundle Emb iso S (M ) → Gr iso S (M ) with total space Emb iso S (M ), the splitting smooth submanifold of Emb S (M ) consisting of all isotropic embeddings.On Emb iso S (M ) we consider the pullback of the isodrastic distribution D: This is an integrable distribution with the same codimension, dim H 1 (S; R), and the leaves of D are connected components in the preimage of a leaf of D. According to [26,15], the group Ham(M ) acts transitively on the leaves of D. We need the subsequent slightly stronger statement in Proposition 3.1.The Lie algebra of compactly supported Hamiltonian vector fields will be denoted by We let Ham c (M ) denote the group of diffeomorphisms obtained by integrating time dependent vector fields in ham c (M ).For our purpose it will not be necessary to consider Ham c (M ) as an infinite dimensional Lie group, cf.[14,Section 43.13].By a smooth map (section) into Ham c (M ) we will simply mean a smooth map into Diff c (M ) that takes values in Ham c (M ).Proof.To show infinitesimal transitivity, suppose ϕ ∈ Emb iso S (M ) and u ϕ ∈ D ϕ , cf. (46).Hence, there exists f ∈ C ∞ (S) such that d f = ϕ * i uϕ ω.We extend f to a function vanishes on vectors tangent to ϕ(S) ⊆ M .Hence, β can be seen as a fiberwise linear function on the normal bundle T ϕ(S) ⊥ whose differential along the zero section ϕ(S) coincides with β itself.Thus, with the help of a tubular neighborhood of ϕ(S) in M and a suitable bump function, we get We conclude that u ϕ = X f • ϕ is the infinitesimal generator at ϕ for the Hamiltonian vector field X f ∈ ham c (M ).
Using tubular neighborhoods constructed with the help of a Riemannian metric, say, we see that the function f may be chosen to depend smoothly on ϕ und u ϕ , for ϕ in a sufficiently small open neighborhood of a fixed isotropic embedding ϕ 0 ∈ E. Hence, we may apply Lemma A.2 and conclude that the Ham c (M ) action admits local smooth sections.
Corollary 3.2.The Ham c (M ) action on each leaf L ⊆ Gr iso S (M ) of the isodrastic foliation D admits local smooth sections.
Example 3.3.Every embedded closed curve in the plane is a Lagrangian submanifold of (R 2 , ω), where ω is the canonical area form, thus an element of Gr iso S 1 (R 2 ).The isodrastic distribution D has codimension one.The enclosed area a singles out one isodrast L a ⊆ Gr iso S 1 (R 2 ), i.e. one orbit of Ham c (R 2 ).
A similar phenomena happens for Lagrangian k-tori in R 2k , i.e. elements of Gr iso T k (R 2k ), where T k := (S 1 ) k .To any ϕ ∈ Emb iso T k (R 2k ) we assign the symplectic area a i of the surface in R 2k enclosed by the ith meridian ϕ i (θ) = ϕ(1, . . ., θ, . . ., 1) of the embedded k-torus.These numbers are independent of the choice of the meridian in its homotopy class and of the surface having the meridian as boundary.The k-tuple (a 1 , . . ., a k ) is an invariant under isodrastic deformations.Actually a i is the action integral of the ith meridian, as defined in [26].
Let E a 1 ,...,a k ⊆ Emb iso T k (R 2k ) be the space of all isotropic embeddings having symplectic areas a 1 , . . ., a k .It is a union of isodrastic leaves of Lagrangian embeddings, but it is not necessarily Diff(T k ) saturated.The Diff(T k ) action on the k-tuples (a 1 , . . ., a k ) factorizes through an GL(k, Z) action.Let [a 1 , . . ., a k ] denote the orbit of (a 1 , . . ., a k ).We define L [a 1 ,...,a k ] ⊆ Gr iso T k (R 2k ) to be the image of E a 1 ,...,a k under the principal Diff(T k ) bundle projection (1).
There is also a direct description of , where γ i are loops in N with action integrals a i .We observe that the GL(k, Z) orbit [a 1 , . . ., a k ] is independent of the choices and L [a 1 ,...,a k ] is the space of all Lagrangian tori in Gr S T k (R 2k ) such that the orbit of these action integrals is [a 1 , . . ., a k ].
We will also need the following observation.Lemma 3.4.If L is an isotropic submanifold in M , then the canonical inclusion Gr S (L) ⊆ Gr iso S (M ) is a splitting smooth submanifold.Moreover, each connected component of Gr S (L) is a splitting smooth submanifold in an isodrastic leaf in Gr iso S (M ).
Proof.Suppose N ∈ Gr S (L), i.e., N ∼ = S is a closed submanifold in L. By the tubular neighborhood theorem, we may w.l.o.g.assume that L is the total space of a vector bundle p : L → N , the normal bundle of N in L, and identify N with the zero section in L. We have a canonical short exact sequence 0 → p * L → T L → p * T N → 0 of vector bundles over L. Choosing a linear connection on L, we obtain a splitting of this sequence and thus an isomorphism T L ∼ = p * T N ⊕p * L of vector bundles over L. Dualizing, we obtain an isomorphism T * L ∼ = p * T * N ⊕ p * L * of vector bundles over L. We regard this as a diffeomorphism that maps the zero section L ⊆ T * L identically onto the summand L on the right hand side.
Via this isomorphism we have , where π 1 and π 2 denote the projections from By the tubular neighborhood theorem for isotropic embeddings [23,24], we may assume , where E is a vector bundle over N , π1 and π2 denote the projections from T * L ⊕ p * E onto T * L and p * E, respectively, and ρ is a closed 2-form on the total space of p * E. Combining this with the diffeomorphism in the previous paragraph, we obtain a diffeomorphism mapping L ⊆ M identically onto the summand L on the right hand side, and such that where π 1 and π 2 denote the projections from T * N ⊕ L ⊕ L * ⊕ E onto T * N and L ⊕ L * ⊕ E, respectively, and σ is a closed 2-form on the total space of L ⊕ L * ⊕ E.
The diffeomorphism in (48) provides a (standard) chart for the smooth structure on Gr S (M ) In this chart, the inclusions Gr S (L) ⊆ Gr iso S (M ) ⊆ Gr S (M ) become As L is isotropic, σ vanishes when pulled back to L. Hence, by the Poincaré lemma, σ = dβ for a 1-form β on the total space of where Z 1 (N ) denotes the space of closed 1-forms on N .Clearly, both inclusions admit complementary subspaces.In particular, Gr S (L) is a splitting smooth submanifold in Gr iso S (M ).The second assertion follows from the fact that the isodrastic leaf through N corresponds to the subspace is closed and Diff(S) invariant.Moreover, its kernel is spanned by the infinitesimal generators of the Diff(S) action.
By Lemma 3.6, the leafwise differential 2-form Ω in (51) descends to a leafwise symplectic form Ω on (Gr wt iso S (M ), D).Thus every leaf G of the isodrastic distribution D in Gr wt iso S (M ) is endowed with a symplectic form, the restriction of Ω, which we denote by the same letter.By Proposition 3.5, G is an orbit for the Ham c (M ) action on the weighted isotropic nonlinear Grassmannian Gr wt iso S (M ).This action is Hamiltonian with injective and Ham(M ) equivariant moment map [15] J : G ⊆ Gr wt iso Here we use the Symp(M ) equivariant isomorphism ham c (M ) = C ∞ 0 (M ) where the latter denotes the Lie algebra of all compactly supported functions on M for which the integral with respect to the Liouville form vanishes on all closed connected components of M .
In this generality the theorem follows from the discussion above and the following folklore result (see for instance the Appendix in [9]): Proposition 3.8.Suppose the action of G on (M, Ω) is transitive with injective equivariant moment map J : M → g * .Then J is one-to-one onto a coadjoint orbit of G.Moreover, it pulls back the Kostant-Kirillov-Souriau symplectic form ω KKS on the coadjoint orbit to the symplectic form Ω.
The same coadjoint orbits of the Hamiltonian group, under the additional restriction H 1 (S; R) = 0 on the closed connected manifold S, can be obtained via symplectic reduction in the Marsden-Weinstein ideal fluid dual pair, as shown in [7].

Manifolds of isotropic nonlinear flags
In this section we extend the constructions of the previous section to nonlinear flags.
As in Section 2.3, we let ι = (ι 1 , . . ., ι r−1 ) denote a collection of fixed embeddings ι i : S i → S i+1 .We let Flag iso S,ι (M ) denote the preimage of Gr iso Sr (M ) under the canonical bundle projection Flag S,ι (M ) → Gr Sr (M ), cf.[9,Remark 2.11].This is a splitting smooth submanifold in Flag S,ι (M ) called the manifold of isotropic nonlinear flags of type (S, ι) in M .Using Lemma 3.4, and proceeding by induction on the depth of the flag, one readily shows that the canonical inclusion is a splitting smooth submanifold.Let Flag wt iso S,ι (M ) denote the preimage of Flag iso S,ι (M ) under the canonical bundle projection Flag wt S,ι (M ) → Flag S,ι (M ).This is a splitting smooth submanifold in Flag wt S,ι (M ) called the manifold of weighted isotropic nonlinear flags of type (S, ι) in M .The diffeomorphism in (25) restricts to a diffeomorphism of bundles over Flag iso S,ι (M ), Flag wt iso S,ι (M ) = Emb iso Sr (M ) × Diff(S;ι) Den × (S).
The canonical inclusion Flag wt iso S,ι (M ) ⊆ r i=1 Gr wt iso S i (M ) is a splitting smooth submanifold in view of (53).
Using (53) and proceeding as in the proof of Theorem 2.11(c), one readily shows that the canonical inclusion Flag wt iso S,ι,µ (M ) ⊆ r i=1 Gr wt iso S i ,µ i (M ) is a splitting smooth submanifold.Recall the Diff(S; ι) equivariant linear map h S,ι : Den(S) → H(S, ι) in (28) with kernel ker T h S,ι = {(dγ 1 , . . ., dγ r ) : , whose restriction to Den × (S) we also denote by h S,ι .The product with the isodrastic distribution gives the integrable distribution D × ker T h S,ι on Emb iso Sr (M ) × Den × (S), of finite codimension dim H 1 (S r ; R) + dim H(S, ι).This Diff(S; ι) invariant distribution descends to an integrable distribution D = D × Diff(S;ι) ker T h S,ι of the same codimension on Flag wt iso S,ι (M ).The image of D under the forgetful map is an integrable distribution on Flag iso S,ι (M ) of codimension dim H 1 (S r ; R), which coincides with the distribution that descends from the Diff(S; ι) invariant isodrastic distribution D on Emb iso Sr (M ) by the principal bundle projection.Using Proposition 2.10(c) and (54), we see that each leaf F of D is a connected component in the preimage of an isodrast L in Gr iso Sr (M ) under the bundle projection Flag wt iso S,ι,µ (M ) → Gr iso Sr (M ), for some volume density µ on S. Hence, each leaf F of D is a splitting smooth submanifold of codimension dim H 1 (S r ; R) in Flag wt iso S,ι,µ (M ).Remark 3.9.If H 1 (S r ; R) = 0, then the leaves of the isodrastic foliation D are the connected components of Flag wt iso S,ι,µ (M ).

Weighted isotropic nonlinear flag manifolds as coadjoint orbits
In this section we describe coadjoint orbits of the Hamiltonian group consisting of weighted isotropic nonlinear flags.
We aim at defining a leafwise symplectic form Ω on (Flag wt iso S,ι (M ), D).We start with leafwise differential 2-forms Ω i on (Emb iso S i (M ) × Den × (S i ), D i × ker T h S i ), for i = 1, . . ., r, defined as in Lemma 3.6.Let j i := ι r−1 • • • • • ι i ∈ Emb(S i , S r ).We consider which maps the distribution D × ker T h S,ι to D i × ker T h S i . Then is a closed leafwise differential 2-form on (Emb iso Sr (M ) × Den × (S), D × ker T h S,ι ).Remark 3.12.The image under T q i of the distribution D × ker T h S,ι is, in general, strictly included in the distribution D i × ker T h S i .The reason is that the projection on the ith factor pr i : ker T h S,ι → ker T h S i is not surjective in general.This happens for instance in the setting of Example 3.18, where ι : {1, . . ., k} → S 1 with k ≥ 2: the projection on the second factor is d{γ ∈ C ∞ (S 1 ) : γ • ι = 0}, strictly included in ker T h S 1 = dC ∞ (S 1 ).The Diff(S; ι) action on Emb iso Sr (M ) × Den × (S), has infinitesimal generators of the form They belong to the integrable distribution D×ker T h S,ι .Notice that q i = j * i ×pr i is equivariant over the homomorphism g ∈ Diff(S; ι) → g i ∈ Diff(S i ), because j i • g i = g r • j i for all i.Now, from the Diff(S i ) invariance of Ω i and (56), we deduce the Diff(S; ι) invariance of Ω.
The next lemma is a generalization of the Lemma 3.6: Lemma 3.13.The kernel of the leafwise differential form Ω is spanned by the infinitesimal generators of the Diff(S; ι) action (57) on Emb iso Sr (M ) × Den × (S).Proof.The formula for the leafwise 2-form in Lemma 3.6 provides a similar formula for Ω: The contraction of Ω with an infinitesimal generator ζ Z , given in (58), vanishes for all Z ∈ X(S; ι).This follows from the analogous statement for Ω i and the infinitesimal generators ζ Z i on Emb iso S i (M ) × Den × (S i ), together with the fact that the infinitesimal generators ζ Z and ζ Z i are q i related.
We need to find Z = (Z i ) ∈ X(S; ι) such that u ϕ = T ϕ • Z r and dλ i = di Z i α i .In particular the rth component Z r has to be tangent to the nonlinear flag Σ = (Σ 1 , . . ., Σ r−1 ) in S r , where Σ i := j i (S i ).
We assume by contradiction that u ϕ ∈ D ϕ is not everywhere tangent to the isotropic submanifold ϕ(S r ) ⊆ M .The subset S r \ Σ r−1 is dense in S r (because dim S r−1 < dim S r ), so there exists x ∈ S r \ Σ r−1 such that u ϕ (x) is not tangent to ϕ(S r ).We choose another tangent vector v ϕ ∈ D ϕ taking values in the symplectic orthogonal to ϕ(S r ) in M , i.e. ϕ * i vϕ ω = 0, supported in an open neighborhood of x in S r \ Σ r−1 .The isotropic embedding theorem of Weinstein [24] allows to choose v ϕ such that ω(u ϕ , v ϕ ) ∈ C ∞ (S r ) is a nonzero nowhere negative function.Plugging v ϕ in (61), all terms are zero except Sr ω(u ϕ , v ϕ )α r > 0, leading to a contradiction.Thus there exists Z r ∈ X(S r ) with u ϕ = T ϕ • Z r .
The identity (61), rewritten with our u ϕ = T ϕ • Z r and arbitrary v ϕ ∈ D ϕ , so ϕ * i vϕ ω = dh with arbitrary h ∈ C ∞ (S), becomes: We need to show that Z r ∈ X(S r ) is tangent to all the submanifolds of the nonlinear flag Σ 1 ⊆ • • • ⊆ Σ r−1 ⊆ S r .We do it successively, starting with Σ r−1 and ending with Σ 1 .We assume by contradiction that Z r is not tangent to Σ r−1 ⊆ S r .Since dim Σ r−2 < dim Σ r−1 , we can find a point x ∈ Σ r−1 \ Σ r−2 such that Z r (x) is not tangent to Σ r−1 .We choose h ∈ C ∞ (S) supported in a tubular neighorhod of Σ r−1 with the following properties: h vanishes on Σ r−1 (so that j * i dh = 0 for i = 1, . . ., r − 1) and i Zr dh restricted to Σ r−1 is a positive bump function at x that vanishes on Σ r−2 (so that j * i (i Zr dh) = 0 for i = 1, . . ., r − 2).Inserting it in (62) only three terms survive, giving The integral over S r−1 on the left doesn't change when cutting h with a function supported in a tubular ε-neighborhood of Σ r−1 which is constant in an ε/2-neighborhood of Σ r−1 , while the absolute value of the integral over S r on the right can be made as small as we need by making ε small enough.This leads to a contradiction.Thus Z r must be tangent to Σ r−1 , so there exists Z r−1 ∈ X(S r−1 ) which is j r−1 related to Z r .
In a similar way, step by step, we address all the remaining submanifolds Σ r−2 , . . ., Σ 1 of the nonlinear flag Σ, finding Z i ∈ X(S i ) that are j i related to Z r .Knowing this, the identity (62) becomes by Remark 3.9 each connected component of Flag wt iso S,ι,µ (M ) is a coadjoint orbit of Ham c (M ).Similarly to (44) we get Flag wt iso S,ι,µ (M ) = (N , ν) ∈ Flag wt iso S,ι (M ) Example 3.18.The coadjoint orbits of Ham c (R 2 ) consisting of pointed vortex loops, studied in [3], are the lowest dimensional examples of coadjoint orbits of weighted nonlinear flags as described in Theorem 3.14.
(64) In particular, the invariants are the area a enclosed by the loop C, the point vorticities Γ i , and the net vorticities w i = x i+1 x i ν 1 between two consecutive points on the loop.
Example 3.19.Let L [a 1 ,a 2 ] be a union of isodrastic leaves of Lagrangian 2-tori in R 4 , as in Example 3.3, with [a 1 , a 2 ] the GL(2, Z) orbit of the pair of action integrals (a 1 , a 2 ) ∈ R 2 .Let S 1 be a disjoint union of k circles, S 2 = T 2 the 2-torus, and ι : S 1 → S 2 the embedding that maps the i-th circle to the circle {t i } × S 1 ⊆ T 2 , with consecutive points t 1 , . . ., t k ∈ S 1 .For fixed density µ ∈ Den × (S), we aim at describing the coadjoint orbits of Ham c (R 4 ) that are connected components of Flag wt iso S,ι,µ (R 4 )| L [a 1 ,a 2 ] .To this end we need the isomorphism given by integration of µ 1 over the components of S 1 and of µ 2 over the torus surface between two successive embedded circles.As in the previous example, the Diff(S, ι) action on H(S, ι) factorizes through the dihedral group.One can now express these coadjoint orbits of Ham c (R 4 ) as in (64), with the help of the dihedral group.Thus, the invariants are: the GL(2, Z) orbit of the two action integrals for the embedded 2-torus, the total weights of the k isotopic loops on the torus, and the partial weights between two such consecutive loops on the torus.

A Transitive actions on associated bundles
The manifolds and Lie groups in this appendix may be infinite dimensional and are assumed to be modelled on convenient vector spaces as in [14].
Recall that a smooth G action on M is said to admit local smooth sections if every point x 0 in M admits an open neighborhood U and a smooth map σ : U → G such that σ(x)x 0 = x, for all x ∈ U .Clearly, such an action is locally and infinitesimally transitive.Due to the lack of a general implicit function theorem, one can not expect the converse implication to hold for general Fréchet manifolds.
Lemma A.1.Let P → B be a principal G-bundle endowed with the action of a Lie group H on P that commutes with the principal G action.Suppose the structure group G acts on another manifold Q, and consider the canonically induced H action on the associated bundle P × G Q. If the H action on P and the G action on Q both admit local smooth sections, then the H action on P × G Q admits local smooth sections too.
Proof.Suppose ξ 0 ∈ P × G Q. As the canonical projection P × Q → P × G Q is a locally trivial smooth bundle [14, Theorem 37.12], there exist an open neighborhood U of ξ 0 as well as smooth maps π : U → P and ρ : U → Q such that for all ξ ∈ U we have [π(ξ), ρ(ξ)] = ξ.
Hence, σ is the desired local smooth section of the H action on P × G Q.
We will denote the fundamental vector fields of a smooth G action on M by

Remark 2 . 1 .
wt S,µ (M ) is a Diff c (M ) 0 orbit.Consequently, each Diff c (M ) or Diff(M ) orbit in Gr wt S,µ (M ) is a union of connected components.Poincaré duality provides a canonical Diff(S) equivariant isomorphism H k (S; O S ) = H 0 (S; R).Hence, specifying a cohomology class [µ] ∈ H k (S; O S ) amounts to specifying the total volume of µ on each connected component of S.Example 2.2.If S is connected, then H k (S; O S ) = R and the Diff(S) action is trivial on this cohomology.Hence, the orbit H k (S; O S ) [µ] is a one-point set, and Den(S) µ = α ∈ Den × (S) : S α = S µ (11) is connected.Correspondingly, Gr wt S,µ (M ) = (N, ν) ∈ Gr wt S (M ) : N ν = S µ .

Remark 2 . 4 .
Suppose (N, ν) ∈ Gr wt S (M ) and let Gr wt S (M ) (N,ν) denote its Diff c (M ) orbit.Combining the preceeding remark with the fact that the Diff c (M ) action on Emb S (M ) admits local smooth sections [9, Lemma 2.1(c)], we see that the map provided by the action, Diff c (M ) → Gr wt S (M ) (N,ν) , f → f (N ), f * ν is a smooth principal bundle with structure group Diff c (M, N, ν), the group of diffeomorphisms preserving N and ν.The latter is a splitting Lie subgroup in Diff c (M ), for it coincides with the preimage of Diff(N, ν) under the canonical bundle projection Diff c (M, N ) → Diff(N ), see [9, Lemma 2.1(d)].Hence, each orbit may be regarded as a homogeneous space, Gr wt S (M ) (N,ν) = Diff c (M )/ Diff c (M, N, ν).

Theorem 2 . 11 .
In this situation the following hold true: (a) The space Flag wt S,ι,µ (M ) is a splitting smooth submanifold in Flag wt S,ι (M ) with finite codimension dim H(S, ι).(b)The canonical Diff(M ) equivariant forgetful map Flag wt S,ι,µ (M ) → Flag S,ι (M ) is a locally trivial smooth fiber bundle with typical fiber Den(S) ι,µ .(c) The canonical inclusion Flag wt S,ι,µ (M ) ⊆ r i=1 Gr wt S i ,µ i (M ) is a splitting smooth submanifold.(d) The Diff c (M ) action on Flag wt S,ι,µ (M ) admits local smooth sections.In particular, each connected component of Flag wt S,ι,µ (M ) is a Diff c (M ) 0 orbit.Furthermore, every Diff(M ) or Diff c (M ) orbit in Flag wt S,ι,µ (M ) is a union of connected components.Proof.Parts (a) and (b) follow by combining (25) and (26) with Proposition 2.10(c).Part (c) follows from Proposition 2.10(d) and the reduction of structure groups in (22) via the diffeomorphisms in ( Let us finally turn to part (d).By Proposition 2.10(c), the (transitive) Diff(S; ι) action on Den(S) ι,µ admits local smooth sections.The Diff c (M ) action on Emb Sr (M ) admits local smooth sections too, cf.[9, Lemma 2.1(c)].Using Lemma A.1, we conclude that the Diff c (M ) action on Flag wt S,ι,µ (M ) admits local smooth sections, cf.(25).

Proposition 3 . 1 .
The Ham c (M ) action on each leaf E ⊆ Emb iso S (M ) of the isodrastic foliation D admits local smooth sections.
denotes the projection and C denotes the linear connection on L, viewed as a fiberwise linear map C : T L → p * L over L.

Proposition 3 . 10 .
The Ham c (M ) action on each leaf F ⊆ Flag wt iso S,ι (M ) of the isodrastic foliation D admits local smooth sections.Proof.Suppose L is an isodrastic leaf in Gr iso Sr (M ) and let Flag wt iso S,ι,µ (M )| L denote its preimage under the canonical projection Flag wt S,ι,µ (M ) → Gr Sr (M ).It suffices to show that the Ham c (M ) action on Flag wt iso S,ι,µ (M )| L admits local smooth sections.The diffeomorphism in (54) restricts to a diffeomorphism Flag wt iso S,ι,µ (M )| L = Emb iso Sr (M )| L × Diff(S;ι) Den(S) ι,µ .(55) By Proposition 3.1, the Ham c (M ) action on Emb iso Sr (M )| L admits local smooth sections.According to Proposition 2.10(c), the Diff(S; ι) action on Den(S) ι,µ admits local smooth sections too.With the help of Lemma A.1, we conclude that the Ham c (M ) action on the associated bundle (55) admits local smooth sections.Corollary 3.11.The Ham c (M ) action on each leaf of the isodrastic foliation on Flag iso S,ι (M ) admits local smooth sections.

ζ
X (x) := ∂ ∂t t=0 exp(tX)x, where X ∈ g and x ∈ M .Lemma A.2. Let G be a regular Lie group acting on a smooth manifold M .Suppose every point x 0 in M admits an open neighborhood U and a smooth map σ :T M | U → g such that ζ σ(X) (x) = X,(65)for all x ∈ U and X ∈ T x M .Then the G action on M admits local smooth sections.