1 Introduction
The nonlinear Grassmannian
$\operatorname {Gr}_S(M)$
, consisting of all smooth submanifolds in a manifold M which are diffeomorphic to a closed manifold S, has a natural Fréchet manifold structure. Nonlinear Grassmannians, a.k.a. differentiable Chow manifolds or shape spaces, play an important role in computer vision [Reference Bauer, Bruveris and Michor1, Reference Michor20] and continuum mechanics [Reference Michor, Segev and Epstein21]. They have also been used to describe coadjoint orbits of diffeomorphism groups [Reference Haller and Vizman8]. Further coadjoint orbits of diffeomorphism groups can be described using weighted nonlinear Grassmannians, i.e., spaces of submanifolds equipped with volume densities. For instance, weighted nonlinear Grassmannians of isotropic submanifolds in a symplectic manifold have been used to describe coadjoint orbits of the Hamiltonian group [Reference Gay-Balmaz and Vizman7, Reference Lee15, Reference Weinstein26], and weighted nonlinear Grassmannians of isotropic submanifolds in a contact manifold have been identified with coadjoint orbits of the contact group [Reference Haller and Vizman10].
Nonlinear Grassmannians have been generalized to spaces of nonlinear flags in [Reference Haller and Vizman9]. Given a collection of closed manifolds
${\mathcal S}=(S_1,\dotsc ,S_r)$
, a nonlinear flag of type
${\mathcal S}$
in M is a sequence of nested embedded submanifolds
$N_1\subseteq \cdots \subseteq N_r$
in M, with
$N_i$
diffeomorphic to
$S_i$
, for all i. The space of all nonlinear flags of type
${\mathcal S}$
in M is a Fréchet manifold in a natural way, denoted by
$\operatorname {Flag}_{{\mathcal S}}(M)$
. Manifolds of low-dimensional nonlinear flags have appeared as shape spaces in [Reference Ciuclea, Tumpach, Vizman, Nielsen and Barbaresco3, Reference Jermyn, Kurtek, Laga and Srivastava12, Reference Strait, Kurtek, Bartha and MacEachern23]. Symplectic nonlinear flags have been used to describe coadjoint orbits of the Hamiltonian group [Reference Haller and Vizman9].
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. Considering submanifolds
$N_i$
equipped with nowhere zero densities
$\nu _i$
, one obtains the manifold
$\operatorname {Flag}_{{\mathcal S}}^{\operatorname {wt}}(M)$
of weighted nonlinear flags. We describe its Fréchet manifold structure in two ways: as a splitting smooth submanifold in the Cartesian product of weighted nonlinear Grassmannians of type
$S_i$
in M, and as a locally trivial smooth fiber bundle over
$\operatorname {Flag}_{{\mathcal S}}(M)$
associated with the principal bundle of nonlinear frames of type
${\mathcal S}$
in M. To each weighted nonlinear flag, one associates a compactly supported distribution on M by the
$\operatorname {Diff}(M)$
equivariant inclusion

We aim at describing the
$\operatorname {Diff}(M)$
orbits in
$\operatorname {Flag}_{{\mathcal S}}^{\operatorname {wt}}(M)$
. Under the
$\operatorname {Diff}(M)$
action, the diffeomorphism type of the inclusions
$N_1\subseteq \cdots \subseteq N_r$
remains invariant, and the total volumes of the connected components of
$N_i\setminus N_{i-1}$
with respect to
$\nu _i$
remain invariant as well, for all i. This suggests to fix embeddings
$S_1\xrightarrow {\iota _1}S_2\xrightarrow {\iota _2}\cdots \xrightarrow {\iota _{r-1}} S_r$
and volume densities
$\mu _i$
on
$S_i$
, and to consider the
$\operatorname {Diff}(M)$
invariant subspace
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}}(M)$
consisting of all weighted nonlinear flags diffeomorphic to the model
$({\mathcal S},\iota ,\mu )$
, where
$\iota =(\iota _1,\dots ,\iota _{r-1})$
and
$\mu =(\mu _1,\dots ,\mu _r)$
. Using a Moser-type argument, we will show that these are exactly the weighted nonlinear flags for which the volumes of corresponding connected components in
$N_i\setminus N_{i-1}$
and in
$S_i\setminus \iota _{i-1}(S_{i-1})$
coincide. A more precise statement, using homologically weighted nonlinear flags, is formulated in Theorem 2.15. This description permits to show that
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}}(M)$
is a splitting smooth submanifold of finite codimension in
$\operatorname {Flag}_{{\mathcal S}}^{\operatorname {wt}}(M)$
(see Theorem 2.11). Furthermore, the
$\operatorname {Diff}(M)$
action on
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}}(M)$
admits local smooth sections. In particular, every
$\operatorname {Diff}(M)$
orbit is a union of connected components in
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}}(M)$
.
For a symplectic manifold M, we are interested in the orbits of the Hamiltonian group
$\operatorname {Ham}_c(M)$
acting on
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}}(M)$
. Clearly, the open subset of weighted symplectic nonlinear flags
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}\operatorname {symp}}(M)$
is invariant under this action. Although the Hamiltonian group acts locally transitive on unweighted symplectic nonlinear flags [Reference Haller and Vizman9, Proposition 4.4], the action on
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}\operatorname {symp}}(M)$
is not locally transitive, not even in the Grassmannian case [Reference Gay-Balmaz and Vizman7, Section 4]. Some orbits turn out to be nice submanifolds, e.g., when the weights are provided by the Liouville volume forms (cf. Remark 2.6), but the general orbit is considerably more singular.
In this paper, we will consider the action of the Hamiltonian group on the invariant set of weighted isotropic nonlinear flags
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}\operatorname {iso}}(M)$
. In view of the tubular neighborhood theorem for isotropic embeddings [Reference Weinstein25], this is a splitting smooth submanifold in
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}}(M)$
. If
$H^1(S_r;\mathbb {R})=0$
, the action is locally transitive. In general, the orbits of the Hamiltonian group provide a smooth foliation of codimension
$\dim H^1(S_r;\mathbb {R})$
in
$\operatorname {Flag}_{{\mathcal S},\iota ,\mu }^{\operatorname {wt}\operatorname {iso}}(M)$
, the analogue of Weinstein’s isodrastic foliation in the Grassmannian case [Reference Weinstein26]. Each isodrastic leaf of weighted nonlinear flags comes equipped with a canonical symplectic form, and the map J in (1.1) restricts to an equivariant moment map for the
$\operatorname {Ham}_c(M)$
action, thus identifying the leaf with a coadjoint orbit of the Hamiltonian group (see Theorem 3.15). Moreover, this coadjoint orbit is a splitting symplectic submanifold in a product of coadjoint orbits of weighted submanifolds of type
$S_i$
in M (see Remark 3.16). The lowest-dimensional examples are the coadjoint orbits of
$\operatorname {Ham}_c(\mathbb {R}^2)$
consisting of pointed weighted vortex loops, treated in [Reference Ciuclea and Vizman4]. We give more examples with nested spheres or tori, and provide explicit descriptions of the corresponding coadjoint orbits of the Hamiltonian group.
The results on weighted nonlinear flags sketched above generalize well-known results on weighted nonlinear Grassmannians, which correspond to
$r=1$
. Many of the proofs presented in this article proceed by induction on the depth r of the nonlinear flags. Often a relative version of the corresponding result on nonlinear Grassmannians is required for the induction step. These relative versions are not readily available in the literature and often require new arguments. Lemma 2.12, for instance, is crucial in the proof of Proposition 2.10, and Lemma 3.7 is used in the proof of Proposition 3.14. Similar relative versions also play an important role in the study of unweighted nonlinear flags (see [Reference Haller and Vizman9, Lemma 2.1] used, e.g., in the proof of [Reference Haller and Vizman9, Proposition 2.9]). In Remark 2.14, we indicate the significance of a relative version of a well-known result on the group of volume-preserving diffeomorphisms [Reference Hamilton11] which appears naturally in this context. To the best of our knowledge, the latter relative version is still open.
2 Manifolds of weighted nonlinear flags
A nonlinear flag is a sequence of nested closed submanifolds
$N_1\subseteq \cdots \subseteq N_r$
in a smooth manifold M. A weighted nonlinear flag is a nonlinear flag together with a volume density
$\nu _i$
on each submanifold
$N_i$
. Integrating against test functions
${f\in C^\infty (M)}$
, a weighted nonlinear flag provides a compactly supported distribution on M with mild singularities,
$\sum _{i=1}^r\int _{N_i}f\nu _i$
.
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 [Reference Haller and Vizman9]. The natural
$\operatorname {Diff}(M)$
action on the base of this bundle is locally transitive [Reference Haller and Vizman9, Proposition 2.9(a)]. The main aim of this section is to describe the
$\operatorname {Diff}(M)$
orbits in the space of weighted nonlinear flags (see Theorem 2.11).
2.1 Weighted nonlinear Grassmannians
In this section, we recall some basic facts about the manifolds of weighted submanifolds that appear in [Reference Gay-Balmaz and Vizman7, Reference Lee15, Reference Weinstein26]. 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
$\operatorname {Gr}_S(M)$
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
$\operatorname {Emb}_S(M)$
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,
$\operatorname {Emb}_S(M)$
and
$\operatorname {Gr}_S(M)$
, are Fréchet manifolds in a natural way. Furthermore, the
$\operatorname {Diff}(M)$
equivariant map

is a smooth principal bundle with structure group
$\operatorname {Diff}(S)$
, a Fréchet Lie group (see [Reference Binz and Fischer2, Reference Gay-Balmaz and Vizman6, Reference Michor18, Reference Michor19] and [Reference Kriegl and Michor14, Theorem 44.1]).
For each closed k-dimensional manifold S, let

denote the space of all smooth densities on S. Here,
${\mathcal O}_S$
denotes the orientation bundle of S and
$|\Lambda |_S=\Lambda ^{k}T^*S\otimes {\mathcal O}_S$
(see, e.g., [Reference Lee16]). 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
$\operatorname {Den}_\times (S)$
the space of volume densities, i.e., the space of nowhere vanishing densities. Clearly, this is an open subset in the Fréchet space
$\operatorname {Den}(S)$
.
We define the weighted nonlinear Grassmannian of type S in M by

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

to be a diffeomorphism. Here, the right-hand side denotes the total space of the bundle associated with the nonlinear frame bundle in (2.1) and the natural
$\operatorname {Diff}(S)$
action on
$\operatorname {Den}_\times (S)$
. In particular, the canonical forgetful map

becomes a locally trivial smooth bundle with typical fiber
$\operatorname {Den}_\times (S)$
. Indeed, it corresponds to the bundle projection of the associated bundle
$\operatorname {Emb}_S(M)\times _{\operatorname {Diff}(S)}\operatorname {Den}_\times (S)\to \operatorname {Gr}_S(M)$
via the identification in (2.3).
There is a canonical
$\operatorname {Diff}(M)$
equivariant map

This map is injective, and its image consists of compactly supported distributions with mild singularities:
$J(N,\nu )$
is supported on N, and its wave front set coincides with the conormal bundle of N.
Let
$\mu \in \operatorname {Den}_\times (S)$
be a volume density. The space

is called the nonlinear Grassmannian of weighted submanifolds of type
$(S,\mu )$
in M. It consists of all weighted submanifolds
$(N,\nu )$
in M such that there exists a diffeomorphism
$S\to N$
taking
$\mu $
to
$\nu $
. Denoting the
$\operatorname {Diff}(S)$
orbit of
$\mu $
by
$\operatorname {Den}(S)_\mu $
, the identification in (2.3) restricts to a canonical bijection

It is well known [Reference Moser22] that the
$\operatorname {Diff}(S)_0$
orbit of
$\mu $
is a convex subset that consists of all volume densities on S that represent the same cohomology class as
$\mu $
in
$H^k(S;{\mathcal O}_S)$
, the de Rham cohomology with coefficients in the orientation bundle. Hence, the
$\operatorname {Diff}(S)$
orbit of
$\mu $
coincides with the set of all volume densities on S that are in the preimage of
$H^k(S;\mathcal O_S)_{[\mu ]}$
, the (finite)
$\operatorname {Diff}(S)$
orbit of
$[\mu ]$
in
$H^k(S;{\mathcal O}_S)$
, under the
$\operatorname {Diff}(S)$
equivariant linear map

More succinctly,

Hence,
$\operatorname {Den}(S)_\mu $
is an open subset in a finite union of parallel closed affine subspaces with finite codimension. In particular,
$\operatorname {Den}(S)_\mu $
is a splitting smooth submanifold in
$\operatorname {Den}_\times (S)$
with finite codimension
$\dim H^k(S;\mathcal O_S)$
and with tangent spaces

Using (2.5), we conclude that
$\operatorname {Gr}^{\operatorname {wt}}_{S,\mu }(M)$
is a splitting smooth submanifold in
$\operatorname {Gr}^{\operatorname {wt}}_S(M)$
with finite codimension
$\dim H^k(S;\mathcal O_S)$
. Moreover, the canonical forgetful map in (2.4) restricts to a locally trivial smooth fiber bundle
$\operatorname {Gr}^{\operatorname {wt}}_{S,\mu }(M)\to \operatorname {Gr}_S(M)$
with typical fiber
$\operatorname {Den}(S)_\mu $
.
The space of homologically weighted submanifolds of type S in M is defined as

Using the canonical bijection

we turn
$\operatorname {Gr}_{S}^{\operatorname {hwt}}(M)$
into a smooth vector bundle of finite rank
$\dim H^k(S;{\mathcal O}_S)$
over
$\operatorname {Gr}_S(M)$
. The canonical
$\operatorname {Diff}(M)$
equivariant map

is a smooth bundle map over
$\operatorname {Gr}_S(M)$
. Indeed, via the diffeomorphisms in (2.5) and (2.9), it corresponds to the map induced by (2.6).
The space of homologically weighted submanifolds of type
$(S,[\mu ])$
in M is defined by

and consists of all homologically weighted submanifolds
$(N,[\nu ])$
such that there exists a diffeomorphism
$S\to N$
taking the cohomology class
$[\mu ]$
to
$[\nu ]$
. As (2.9) restricts to a bijection

we see that
$\operatorname {Gr}_{S,[\mu ]}^{\operatorname {hwt}}(M)$
is a finite covering of
$\operatorname {Gr}_S(M)$
. Using (2.7), we conclude

It is well known that the
$\operatorname {Diff}_c(M)$
action on
$\operatorname {Emb}_S(M)$
admits local smooth sections (see, for instance, [Reference Haller and Vizman9, Lemma 2.1(c)]). Furthermore, the (transitive)
$\operatorname {Diff}(S)$
action on
$\operatorname {Den}(S)_\mu $
also admits local smooth sections. The latter can be shown using Moser’s method of proof in [Reference Moser22, Section 4] (see Lemma 2.12). Using Lemma A.1 in the Appendix, we conclude that the natural
$\operatorname {Diff}_c(M)$
action on
$\operatorname {Gr}^{\operatorname {wt}}_{S,\mu }(M)$
admits local smooth sections. In particular, this action is locally transitive. Hence, each connected component of
$\operatorname {Gr}^{\operatorname {wt}}_{S,\mu }(M)$
is a
$\operatorname {Diff}_c(M)_0$
orbit. Consequently, each
$\operatorname {Diff}_c(M)$
or
$\operatorname {Diff}(M)$
orbit in
$\operatorname {Gr}^{\operatorname {wt}}_{S,\mu }(M)$
is a union of connected components.
Remark 2.1 Poincaré duality provides a canonical
$\operatorname {Diff}(S)$
equivariant isomorphism

Hence, specifying a cohomology class
$[\mu ]\in H^k(S;{\mathcal O}_S)$
amounts to specifying the total volume of
$\mu $
on each connected component of S.
Example 2.2 If S is connected, then
$H^k(S;{\mathcal O}_S)=\mathbb {R}$
and the
$\operatorname {Diff}(S)$
action is trivial on this cohomology. Hence, the orbit
$H^k(S;{\mathcal O}_S)_{[\mu ]}$
is a one-point set, and

is connected. Correspondingly,

This is the case considered in [Reference Gay-Balmaz and Vizman7, Reference Lee15, Reference Weinstein26].
If S is built out of two diffeomorphic connected components, then
$H^k(S;{\mathcal O}_S)\cong \mathbb {R}^2$
and any diffeomorphism swapping the two connected components acts nontrivially on this cohomology. If
$\mu $
has equal total volume on the two connected components, then the orbit
$H^k(S;{\mathcal O}_S)_{[\mu ]}$
is a one-point set and
$\operatorname {Den}(S)_\mu $
is connected. Otherwise,
$H^k(S;{\mathcal O}_S)_{[\mu ]}$
consists of two points and, by (2.7),
$\operatorname {Den}(S)_\mu $
has two connected components.
Remark 2.3 Suppose
$\mu \in \operatorname {Den}_\times (S)$
. It is well known that
$\operatorname {Diff}(S,\mu )$
, the group of diffeomorphisms preserving
$\mu $
, is a splitting Lie subgroup in
$\operatorname {Diff}(S)$
(see [Reference Hamilton11, Theorem III.2.5.3 on page 203]). Moreover, the map provided by the action,
$\operatorname {Diff}(S)\to \operatorname {Den}(S)_\mu $
,
$f\mapsto f_*\mu $
, is a smooth principal bundle with structure group
$\operatorname {Diff}(S,\mu )$
. Via (2.5), this implies that the surjective and
$\operatorname {Diff}(M)$
equivariant map

is smooth principal bundle with structure group
$\operatorname {Diff}(S,\mu )$
.
Remark 2.4 Suppose
$(N,\nu )\in \operatorname {Gr}_{S}^{\operatorname {wt}}(M)$
and let
$\operatorname {Gr}_{S}^{\operatorname {wt}}(M)_{(N,\nu )}$
denote its
$\operatorname {Diff}_c(M)$
orbit. Combining the preceding remark with the fact that the
$\operatorname {Diff}_c(M)$
action on
$\operatorname {Emb}_S(M)$
admits local smooth sections [Reference Haller and Vizman9, Lemma 2.1(c)], we see that the map provided by the action,

is a smooth principal bundle with structure group
$\operatorname {Diff}_c(M,N,\nu )$
, the group of diffeomorphisms preserving N and
$\nu $
. The latter is a splitting Lie subgroup in
$\operatorname {Diff}_c(M)$
, for it coincides with the preimage of
$\operatorname {Diff}(N,\nu )$
under the canonical bundle projection
$\operatorname {Diff}_c(M,N)\to \operatorname {Diff}(N)$
(see [Reference Haller and Vizman9, Lemma 2.1(d)]). Hence, each orbit may be regarded as a homogeneous space,

2.2 Weighted nonlinear flag manifolds
Fix natural numbers
$k_i$
such that

and let
${\mathcal S}=(S_1,\dotsc ,S_r)$
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
${\mathcal S}$
in M, and we write

for the space of nonlinear frames of type
${\mathcal S}$
in M. In [Reference Haller and Vizman9, Proposition 2.3], it has been shown that
$\operatorname {Flag}_{{\mathcal S}}(M)$
and
$\operatorname {Fr}_{{\mathcal S}}(M)$
are splitting smooth submanifolds of
$\prod _{i=1}^r\operatorname {Gr}_{S_i}(M)$
and
$\prod _{i=1}^r\operatorname {Emb}_{S_i}(M)$
, respectively. Moreover, the canonical
$\operatorname {Diff}(M)$
equivariant map

is a smooth principal fiber bundle with structure group

We denote the space of weighted nonlinear flags of type
${\mathcal S}$
in M by

This is a splitting smooth submanifold in
$\prod _{i=1}^r\operatorname {Gr}^{\operatorname {wt}}_{S_i}(M)$
, for it coincides with the preimage of the splitting smooth submanifold
$\operatorname {Flag}_{{\mathcal S}}(M)$
under the bundle projection
$\prod _{i=1}^r\operatorname {Gr}^{\operatorname {wt}}_{S_i}(M)\to \prod _{i=1}^r\operatorname {Gr}_{S_i}(M)$
. Moreover, the canonical
$\operatorname {Diff}(M)$
equivariant forgetful map

is a smooth fiber bundle with typical fiber

The latter is a
$\operatorname {Diff}({\mathcal S})$
invariant open subset in the Fréchet space
$\operatorname {Den}(\mathcal S):=\prod _{i=1}^r\operatorname {Den}(S_i)$
. Furthermore, the canonical
$\operatorname {Diff}(M)$
equivariant bijection

is a diffeomorphism between
$\operatorname {Flag}_{\mathcal S}^{\operatorname {wt}}(M)$
and the bundle associated with the nonlinear frame bundle in (2.13) and the canonical
$\operatorname {Diff}({\mathcal S})$
action on
$\operatorname {Den}_\times ({\mathcal S})$
. Indeed, this is just the bundle diffeomorphism
$\prod _{i=1}^r\operatorname {Gr}^{\operatorname {wt}}_{S_i}(M)=\prod _{i=1}^r\operatorname {Emb}_{S_i}(M)\times _{\operatorname {Diff}(S_i)}\operatorname {Den}_\times (S_i)$
obtained by taking the product of the diffeomorphisms in (2.3), restricted over the submanifold
$\operatorname {Flag}_{{\mathcal S}}(M)$
in its base
$\prod _{i=1}^r\operatorname {Gr}_{S_i}(M)$
.
We have a canonical
$\operatorname {Diff}(M)$
equivariant map

The image of J consists of compactly supported distributions on M with mild singularities. More precisely, the wave front set of
$J((N_1,\nu _1),\dotsc ,(N_r,\nu _r))$
coincides with the union of the conormal bundles of
$N_1,\dotsc ,N_r$
.
Lemma 2.5 The map in (2.18) is injective.
Proof Suppose
$J((N_1,\nu _1),\dotsc ,(N_r,\nu _r))=J((N_1',\nu _1'),\dotsc ,(N_r',\nu _r'))$
. Proceeding by induction on r, it suffices to show
$N_r=N_r'$
and
$\nu _r=\nu _r'$
.
To show
$N_r=N_r'$
, we assume by contradiction that there exists
$x\in N_r$
with
$x\notin N_r'$
. Using (2.12), we see that
$N_r\setminus N_{r-1}$
is dense in
$N_r$
. Thus, we may w.l.o.g. assume
$x\notin N_{r-1}$
. Moreover,
$\nu _r(x)\neq 0$
as
$\nu _r$
does not vanish on
$N_r$
. Hence, if f is a smooth bump function supported on a sufficiently small neighborhood of x, then
$\langle J((N_1,\nu _1),\dotsc ,(N_r,\nu _r)),f\rangle =\int _{N_r}f\nu _r\neq 0$
and
$\langle J((N_1',\nu _1'),\dotsc ,(N_r',\nu _r')),f\rangle =0$
. Since this contradicts our assumption, we conclude
$N_r=N_r'$
.
To show
$\nu _r=\nu _r'$
, we assume by contradiction that there exists
$x\in N_r=N_r'$
with
$\nu _r(x)\neq \nu _r'(x)$
. As before, we may w.l.o.g. assume
$x\notin N_{r-1}$
and
$x\notin N_{r-1}'$
. Hence, if f is a smooth bump function supported in a sufficiently small neighborhood of x, then
$\langle J((N_1,\nu _1),\dotsc ,(N_r,\nu _r)),f\rangle =\int _{N_r}f\nu _r\neq \int _{N_r'}f\nu _r'=\langle J((N_1',\nu _1'),\dotsc ,(N_r',\nu _r')),f\rangle $
. Since this contradicts our assumption, we conclude
$\nu _r=\nu _r'$
.
Remark 2.6 Suppose
$\omega $
is a symplectic form on M, and let
$\operatorname {Flag}_{{\mathcal S}}^{\operatorname {symp}}(M)$
denote the manifold of symplectic nonlinear flags of type
${\mathcal S}$
(cf. [Reference Haller and Vizman9, Section 4.2]). Recall that this is the open subset consisting of all flags
$(N_1,\dotsc ,N_r)\in \operatorname {Flag}_{{\mathcal S}}(M)$
such that
$\omega $
restricts to a symplectic form on each
$N_i$
. Hence,
$k_i$
must be even and
$\omega ^{k_i/2}$
pulls back to a volume form on
$N_i$
which in turn gives rise to a volume density
$\nu _i=|\iota _{N_i}^*\omega ^{k_i/2}|$
on
$N_i$
. Consequently, the symplectic form
$\omega $
provides a
$\operatorname {Symp}(M,\omega )$
equivariant injective smooth map (section)

which is right inverse to the restriction of the canonical bundle projection in (2.15). Composing the map in (2.19) with J in (2.18), we obtain the moment map considered in [Reference Haller and Vizman9, equation (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
$\operatorname {Flag}_{{\mathcal S}}(M)\to \operatorname {Flag}_{{\mathcal S}}^{\operatorname {wt}}(M)$
of the canonical bundle projection in (2.15), which is
$\operatorname {Isom}(M,g)$
equivariant (cf. [Reference Weinstein26, Section 6]).
2.3 Reduction of structure group
It will be convenient to use a reduction of the structure group for the principal frame bundle in (2.13). To this end, we fix embeddings
$\iota _i\colon S_i\to S_{i+1}$
and put
$\iota =(\iota _1,\dotsc ,\iota _{r-1})$
.
We begin by recalling some facts from [Reference Haller and Vizman9, Proposition 2.10]. The space of nonlinear flags of type
$({\mathcal S},\iota )$
in M,

consists of all nonlinear flags
$(N_1,\dotsc ,N_r)$
in M such that there exist diffeomorphisms
$S_i\to N_i$
,
$1\leq i\leq r$
intertwining
$\iota _i$
with the canonical inclusion
$N_i\subseteq N_{i+1}$
. This is a
$\operatorname {Diff}(M)$
invariant open and closed subset in
$\operatorname {Flag}_{{\mathcal S}}(M)$
. The space of parametrized nonlinear flags (nonlinear frames) of type
$({\mathcal S},\iota )$
in M,

is a splitting smooth submanifold of
$\operatorname {Fr}_{{\mathcal S}}(M)$
. Moreover, the map
$\operatorname {Fr}_{{\mathcal S},\iota }(M)\to \operatorname {Flag}_{{\mathcal S},\iota }(M)$
obtained by restriction of (2.13) is a smooth principal bundle with structure group

The latter is a splitting Lie subgroup in
$\operatorname {Diff}({\mathcal S})$
with Lie algebra

We obtain a
$\operatorname {Diff}(M)$
equivariant commutative diagram

which may be regarded as a reduction of the structure group for (2.13) along the inclusion
$\operatorname {Diff}(\mathcal S;\iota )\subseteq \operatorname {Diff}({\mathcal S})$
over
$\operatorname {Flag}_{\mathcal S,\iota }(M)$
(see [Reference Haller and Vizman9, Proposition 2.10] for more details).
Remark 2.8 The
$\operatorname {Diff}(M)$
equivariant bijection

is a diffeomorphism [Reference Haller and Vizman9, Proposition 2.10(b)]. Correspondingly, we have a group isomorphism

where
$\operatorname {Diff}(S_r;\Sigma )$
denotes the subgroup of all diffeomorphisms of
$S_r$
preserving the nonlinear flag
$\Sigma =(\Sigma _1,\dotsc ,\Sigma _{r-1})$
in
$S_r$
, where
$\Sigma _i:=(\iota _{r-1}\circ \cdots \circ \iota _i)(S_i)$
. The latter is a splitting Lie subgroup of
$\operatorname {Diff}(S_r)$
(see [Reference Haller and Vizman9, Proposition 2.9(b)]), and (2.24) is a diffeomorphism of Lie groups [Reference Haller and Vizman9, Proposition 2.10(a)]. The Lie algebra of
$\operatorname {Diff}({\mathcal S};\iota )$
can be identified in a similar way with
$\mathfrak X(S_r;\Sigma )$
, the Lie algebra of vector fields on
$S_r$
that are tangent to
$\Sigma _1,\dotsc ,\Sigma _{r-1}$
.
We are interested in the reduction of structure group (2.22) because the
$\operatorname {Diff}_c(M)$
action on
$\operatorname {Fr}_{{\mathcal S},\iota }(M)$
admits local smooth sections. This follows from [Reference Haller and Vizman9, Lemma 2.1(c)] and the diffeomorphism in (2.23).
Let
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota }(M)$
denote the preimage of
$\operatorname {Flag}_{{\mathcal S},\iota }(M)$
under the bundle projection in (2.15). Restricting the diffeomorphism in (2.17) over
$\operatorname {Flag}_{{\mathcal S},\iota }(M)$
and combining this with the diffeomorphism in (2.23), we obtain a
$\operatorname {Diff}(M)$
equivariant diffeomorphism of bundles over
$\operatorname {Flag}_{{\mathcal S},\iota }(M)$
,

2.4 The
$\operatorname {Diff}(M)$
action on the space of weighted nonlinear flags
In this section, we aim at describing the
$\operatorname {Diff}(M)$
orbits in
$\operatorname {Flag}_{{\mathcal S}}^{\operatorname {wt}}(M)$
(see Theorem 2.11).
Let
$\iota =(\iota _1,\dotsc ,\iota _{r-1})$
be a collection of embeddings
$\iota _i:S_i\to S_{i+1}$
and suppose
$\mu =(\mu _1,\dotsc ,\mu _r)\in \operatorname {Den}_\times ({\mathcal S})$
. We define the space of weighted flags of type
$({\mathcal S},\iota ,\mu )$
in M by

that is, the space of all weighted flags
$\left ((N_1,\nu _1),\dotsc ,(N_r,\nu _r)\right )$
in M such that there exist diffeomorphisms
$S_i\to N_i$
,
$1\leq i\leq r$
, intertwining
$\iota _i$
with the canonical inclusion
$N_i\subseteq N_{i+1}$
, and taking
$\mu _i$
to
$\nu _i$
.
Denoting the
$\operatorname {Diff}({\mathcal S},\iota )$
orbit of
$\mu $
by
$\operatorname {Den}({\mathcal S})_{\iota ,\mu }$
, the diffeomorphism in (2.25) restricts to a
$\operatorname {Diff}(M)$
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=\emptyset $
. The
$\operatorname {Diff}({\mathcal S},\iota )$
equivariant identification on the right-hand side indicates Poincaré–Lefschetz duality. We have a
$\operatorname {Diff}({\mathcal S},\iota )$
equivariant linear map

Pinning down the class
$[\mu ]:=h_{{\mathcal S},\iota }(\mu )$
thus amounts to specifying the integrals of
$\mu _i$
over each connected component of
$S_i\setminus \iota _{i-1}(S_{i-1})$
for
$i=1,\dotsc ,r$
.
Remark 2.9 (Large codimensions)
If the codimensions
$\dim (S_i)-\dim (S_{i-1})$
are all strictly larger than one, then
$H^{k_i}\left (S_i,\iota _{i-1}(S_{i-1});{\mathcal O}_{S_i}\right )=H^{k_i}\left (S_i;{\mathcal O}_{S_i}\right )=H_0(S_i;\mathbb {R})$
, and

Hence, in this case, the cohomology space
$H({\mathcal S},\iota )$
does not depend on the embeddings
$\iota $
.
Proposition 2.10 In this situation, the following hold true:
-
(a) The
$\operatorname {Diff}(\mathcal S,\iota )_0$ orbit of
$\mu $ coincides with the convex set
$\operatorname {Den}_\times (\mathcal S)\cap h_{{\mathcal S},\iota }^{-1}([\mu ])$ . In particular, this orbit is a splitting smooth submanifold in
$\operatorname {Den}_\times (\mathcal S)$ with finite codimension
$\dim H({\mathcal S},\iota )$ .
-
(b) The
$\operatorname {Diff}(\mathcal S,\iota )_0$ action on
$\operatorname {Den}_\times (\mathcal S)\cap h_{{\mathcal S},\iota }^{-1}([\mu ])$ admits local smooth sections.
-
(c) Denoting the (finite)
$\operatorname {Diff}(\mathcal S;\iota )$ orbit of
$[\mu ]$ by
$H(\mathcal S,\iota )_{[\mu ]}$ , the
$\operatorname {Diff}({\mathcal S};\iota )$ orbit of
$\mu $ is
(2.29)In particular,$$ \begin{align} \operatorname{Den}(\mathcal S)_{\iota,\mu}=\operatorname{Den}_\times(\mathcal S)\cap h_{{\mathcal S},\iota}^{-1}\left(H(\mathcal S,\iota)_{[\mu]}\right). \end{align} $$
$\operatorname {Den}(\mathcal S)_{\iota ,\mu }$ is a splitting smooth submanifold in
$\operatorname {Den}_\times (\mathcal S)$ with finite codimension
$\dim H({\mathcal S},\iota )$ and with tangent spaces
(2.30)Moreover, the$$ \begin{align} T_\alpha\operatorname{Den}(\mathcal S)_{\iota,\mu} =\ker h_{{\mathcal S},\iota} =\left\{(d\gamma_1,\dotsc,d\gamma_r):\gamma_i\in\Omega^{k_i-1}(S_i;\mathcal O_{S_i}),\iota_{i-1}^*\gamma_i=0\right\}. \end{align} $$
$\operatorname {Diff}(\mathcal S,\iota )$ action on
$\operatorname {Den}(\mathcal S)_{\iota ,\mu }$ admits local smooth sections.
-
(d) The canonical inclusion
$\operatorname {Den}(\mathcal S)_{\iota ,\mu }\subseteq \prod _{i=1}^r\operatorname {Den}(S_i)_{\mu _i}$ is a splitting smooth submanifold of finite codimension.
We postpone the proof of this proposition and proceed with the main result in this section.
Theorem 2.11 In this situation, the following hold true:
-
(a) The space
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)$ is a splitting smooth submanifold in
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota }(M)$ with finite codimension
$\dim H({\mathcal S},\iota )$ .
-
(b) The canonical
$\operatorname {Diff}(M)$ equivariant forgetful map
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)\to \operatorname {Flag}_{{\mathcal S},\iota }(M)$ is a locally trivial smooth fiber bundle with typical fiber
$\operatorname {Den}({\mathcal S})_{\iota ,\mu }$ .
-
(c) The canonical inclusion
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)\subseteq \prod _{i=1}^r\operatorname {Gr}_{S_i,\mu _i}^{\operatorname {wt}}(M)$ is a splitting smooth submanifold.
-
(d) The
$\operatorname {Diff}_c(M)$ action on
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)$ admits local smooth sections. In particular, each connected component of
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)$ is a
$\operatorname {Diff}_c(M)_0$ orbit. Furthermore, every
$\operatorname {Diff}(M)$ or
$\operatorname {Diff}_c(M)$ orbit in
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)$ is a union of connected components.
Proof Parts (a) and (b) follow by combining (2.25) and (2.26) with Proposition 2.10(c).
Part (c) follows from Proposition 2.10(d) and the reduction of structure groups in (2.22) via the diffeomorphisms in (2.5) and (2.26) (see also (2.23)).
Let us finally turn to part (d). By Proposition 2.10(c), the (transitive)
$\operatorname {Diff}(\mathcal S;\iota )$
action on
$\operatorname {Den}(\mathcal S)_{\iota ,\mu }$
admits local smooth sections. The
$\operatorname {Diff}_c(M)$
action on
$\operatorname {Emb}_{S_r}(M)$
admits local smooth sections too (cf. [Reference Haller and Vizman9, Lemma 2.1(c)]). Using Lemma A.1, we conclude that the
$\operatorname {Diff}_c(M)$
action on
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)$
admits local smooth sections (cf. (2.25)).
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
$\operatorname {Diff}(N,S)$
equivariant linear map

Then, for each
$\kappa \in H^n(N,S;\mathcal O_N)$
, the natural
$\operatorname {Diff}(N,S)_0$
action on

admits local smooth sections.
Proof of Proposition 2.10
We proceed by induction on r using Lemma 2.12. Let us denote the truncated sequences by
$\mathcal S':=(S_1,\dotsc ,S_{r-1})$
,
$\mu ':=(\mu _1,\dotsc ,\mu _{r-1})$
, and
$\iota ':=(\iota _1,\dotsc ,\iota _{r-2})$
. By induction, the
$\operatorname {Diff}(\mathcal S',\iota ')_0$
action on
$\operatorname {Den}_\times (\mathcal S')\cap h_{{\mathcal S}',\iota '}^{-1}([\mu '])$
admits local smooth sections. Hence, there exist an open neighborhood
$U'$
of
$\mu '$
in
$\operatorname {Den}_\times (\mathcal S')\cap h_{{\mathcal S}',\iota '}^{-1}([\mu '])$
and a smooth map

such that for all
$\tilde \mu '\in U'$
, we have

Recall that
$\operatorname {Diff}(\mathcal S',\iota ')$
is a splitting Lie subgroup in
$\operatorname {Diff}(S_{r-1})$
(cf. [Reference Haller and Vizman9, Propositions 2.9(b) and 2.10(a)]). Using [Reference Haller and Vizman9, Lemma 2.1(d)], this implies that
$\operatorname {Diff}(\mathcal S,\iota )$
is a splitting Lie subgroup in
$\operatorname {Diff}(S_r,\iota _{r-1}(S_{r-1}))$
. Hence, restricting a local smooth section as in Lemma 2.12, we see that the
$\operatorname {Diff}(\mathcal S,\iota )_0$
action on
$\operatorname {Diff}(\mathcal S',\iota ')\times \left (\operatorname {Den}_\times (S_r)\cap h^{-1}([\mu _r])\right )$
admits local smooth sections. In other words, there exist an open neighborhood V of the identity in
$\operatorname {Diff}(\mathcal S';\iota ')$
, an open neighborhood
$U"$
of
$\mu _r$
in
$\operatorname {Den}_\times (S_r)\cap h^{-1}([\mu _r])$
, and a smooth map

such that for all
$g\in V$
and
$\tilde \mu _r\in U"$
, we have

Hence,
$U:=(\sigma ')^{-1}(V)\times U"$
is an open neighborhood of
$\mu $
in
$\operatorname {Den}_\times (\mathcal S)\cap h_{{\mathcal S},\iota }^{-1}([\mu ])$
, and

is a local smooth section for the
$\operatorname {Diff}(\mathcal S;\iota )_0$
action on
$\operatorname {Den}_\times (\mathcal S)\cap h_{{\mathcal S},\iota }^{-1}([\mu ])$
, i.e.,

for all
$\tilde \mu \in U$
. By convexity,
$\operatorname {Den}_\times (\mathcal S)\cap h_{{\mathcal S},\iota }^{-1}([\mu ])$
is connected. Therefore, the
$\operatorname {Diff}(\mathcal S;\iota )_0$
action is transitive on
$\operatorname {Den}_\times (\mathcal S)\cap h_{{\mathcal S},\iota }^{-1}([\mu ])$
. This shows (a) and (b). Part (c) follows immediately.
To see (d), let A denote the preimage of
$\prod _{i=1}^rH^{k_i}(S_i;\mathcal O_{S_i})_{[\mu _i]}$
under the canonical linear surjection
$H({\mathcal S},\iota )\to \prod _{i=1}^rH^{k_i}(S_i;\mathcal O_{S_i})$
. Hence, A is a finite union of affine subspaces in
$H({\mathcal S},\iota )$
. In view of (2.7), we have
$\prod _{i=1}^r\operatorname {Den}(S_i)_{\mu _i}=\operatorname {Den}_\times ({\mathcal S})\cap h_{{\mathcal S},\iota }^{-1}(A)$
. Combining this with (2.29), we conclude that
$\operatorname {Den}({\mathcal S})_{\iota ,\mu }$
is a splitting smooth submanifold in
$\prod _{i=1}^r\operatorname {Den}(S_i)_{\mu _i}$
with finite codimension
$\dim A$
.
Let us next establish the following infinitesimal version of Lemma 2.12.
Lemma 2.13 Let S be a closed submanifold of N such that
$\dim (S)<\dim (N)=n$
. Suppose
$\mu \in \operatorname {Den}_\times (N)$
,
$\gamma \in \Omega ^{n-1}(N,S;{\mathcal O}_N):=\{\alpha \in \Omega (N;{\mathcal O}_N):\iota _S^*\alpha =0\}$
, and
$Z\in \mathfrak X(S)$
. Then there exists a vector field
$X\in \mathfrak X(N)$
such that
$L_X\mu =d\gamma $
and
$X|_S=Z$
.
Proof Let
$\tilde Z\in \mathfrak X(N)$
be any extension of Z, i.e.,
$\tilde Z|_S=Z$
. Note that
$i_{\tilde Z}\mu \in \Omega ^{n-1}(N;{\mathcal O}_N)$
vanishes when pulled back to S; hence, the same holds for
$\beta :=\gamma -i_{\tilde Z}\mu \in \Omega ^{n-1}(N;{\mathcal O}_N)$
.
Let us first construct
$\tilde \gamma \in \Omega ^{n-1}(N;{\mathcal O}_N)$
such that
$d\tilde \gamma =d\beta $
and
$\tilde \gamma |_S=0$
. To this end, we fix a smooth homotopy
$h\colon N\times [0,1]\to N$
such that
$h_1=\operatorname {id}_N$
,
$h_t|_S=\operatorname {id}_S$
, and such that
$h_0$
maps a neighborhood of S into S. Consider the corresponding chain homotopy
$\phi \colon \Omega ^*(N;{\mathcal O}_N)\to \Omega ^{*-1}(N;{\mathcal O}_N)$
defined by
$\phi (\alpha ):=\int _0^1\iota _t^*i_{\partial _t}h^*\alpha \,dt$
, where
$\iota _t\colon N\to N\times [0,1]$
denotes the inclusion at t, that is,
$\iota _t(x):=(x,t)$
. Then
$\phi (\alpha )|_S=0$
and
$d(\phi (\alpha ))+\phi (d\alpha )=h_1^*\alpha -h_0^*\alpha $
, for all forms
$\alpha \in \Omega ^*(N;{\mathcal O}_N)$
. In particular,
$d\alpha =d\left (h_0^*\alpha +\phi (d\alpha )\right )$
. Defining
$\tilde \gamma :=h_0^*\beta +\phi (d\beta )$
, we obtain
$d\tilde \gamma =d\beta $
. Moreover,
$h_0^*\beta |_S=0$
because
$\beta $
vanishes when pulled back to S, and hence
$\tilde \gamma |_S=0$
, as desired.
Defining a vector field
$Y\in \mathfrak X(N)$
by
$i_Y\mu :=\tilde \gamma $
, we obtain
$Y|_S=0$
and
$L_Y\mu =di_Y\mu =d\tilde \gamma =d(\gamma -i_{\tilde Z}\mu )=d\gamma -L_{\tilde Z}\mu $
. Hence, the vector field
$X:=\tilde Z+Y$
has the desired properties.
Proof of Lemma 2.12
Recall that the infinitesimal
$\operatorname {Diff}(N,S)$
action on
$\operatorname {Diff}(S)\times \operatorname {Den}(N)$
is

where
$X\in \mathfrak X(N,S)$
,
$f\in \operatorname {Diff}(S)$
, and
$\mu \in \operatorname {Den}(S)$
. Here, for
$Z\in T_{\operatorname {id}}\operatorname {Diff}(S)=\mathfrak X(S)$
, we let
$R_Z(f)$
denote the right invariant vector field on
$\operatorname {Diff}(S)$
such that
$R_Z(\operatorname {id})=Z$
.
Note that the vector field
$\tilde Z$
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

such that
$\zeta _{\tilde \sigma (Z,d\gamma )}(\operatorname {id},\mu )=(Z,d\gamma )$
for all
$Z\in \mathfrak X(S)=T_{\operatorname {id}}\operatorname {Diff}(S)$
,
$\mu \in \operatorname {Den}_\times (N)\cap h^{-1}(\kappa )$
, and
$d\gamma \in d\Omega ^{n-1}(N,S;\mathcal O_N)=T_\mu \left (\operatorname {Den}_\times (N)\cap h^{-1}(\kappa )\right )$
. Combining this with the right trivialization of
$T\operatorname {Diff}(S)$
, we obtain a smooth map

such that
$\zeta _{\sigma (Z,\xi )}(f,\mu )=(Z,\xi )$
for all
$f\in \operatorname {Diff}(S)$
,
$Z\in T_f\operatorname {Diff}(S)$
,
$\mu \in \operatorname {Den}_\times (N)\cap h^{-1}(\kappa )$
, and
$\xi \in T_\mu \left (\operatorname {Den}_\times (N)\cap h^{-1}(\kappa )\right )$
. As
$\operatorname {Diff}(N,S)$
is a regular Lie group, we may apply Lemma A.2 to conclude that the
$\operatorname {Diff}(N,S)_0$
action on (2.31) 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

is a splitting Lie subgroup of
$\operatorname {Diff}({\mathcal S};\iota )$
with Lie algebra

and the surjective map provided by the action,
$\operatorname {Diff}({\mathcal S},\iota )\to \operatorname {Den}({\mathcal S})_{\iota ,\mu }$
, is a locally trivial smooth principal fiber bundle with structure group
$\operatorname {Diff}({\mathcal S};\iota ,\mu )$
. This would follow in a rather straightforward manner, via induction on the depth of the flags, if one could show that the isotropy group
$\{f\in \operatorname {Diff}(N,S):f|_S=\operatorname {id},f^*\mu =\mu \}$
is a splitting Lie subgroup in
$\operatorname {Diff}(N,S)$
, whenever S is a closed submanifold of N and
$\mu $
is a volume density on N. The proof in [Reference Hamilton11, Theorem III.2.5.3 on page 203] covers the case
$S=\emptyset $
. However, the adaptation of said proof to nontrivial S is not entirely straightforward, and we will not attempt to prove this here. Note that via the diffeomorphism in (2.24), the group
$\operatorname {Diff}(\mathcal S;\iota ,\mu )$
corresponds to the subgroup of
$\operatorname {Diff}(S_r;\Sigma )$
consisting of all diffeomorphisms that preserve
$\mu _r$
and whose restriction to
$\Sigma _i$
preserves
$(\iota _{r-1}\circ \cdots \circ \iota _i)_*\mu _i$
, for
$1\leq i\leq r-1$
. Similarly, the Lie algebra
$\mathfrak X(\mathcal S;\iota ,\mu )$
can be identified to the corresponding subalgebra of
$\mathfrak X(S_r;\Sigma )$
.
If the expectation formulated in the preceding paragraph were indeed true, then the surjective and
$\operatorname {Diff}(M)$
equivariant map


would be a locally trivial smooth principal fiber bundle with structure group
$\operatorname {Diff}({\mathcal S};\iota ,\mu )$
(see (2.26)). Moreover, generalizing Remark 2.4, the isotropy group of a weighted flag
$(\mathcal {N},\nu )$
,

would be a splitting Lie subgroup of
$\operatorname {Diff}_c(M)$
, for it coincides with the preimage of
$\operatorname {Diff}({\mathcal S};\iota ,\mu )$
under the bundle projection
$\operatorname {Diff}_c(M;\mathcal N)\to \operatorname {Diff}(\mathcal {N},\iota _{\mathcal {N}})$
(cf. [Reference Haller and Vizman9, Lemma 2.1(d) and Propositions 2.9(b) and 2.10(a)]). Furthermore, the orbit map
$\operatorname {Diff}_c(M)\to \operatorname {Flag}^{\operatorname {wt}}_{\mathcal S}(M)_{(\mathcal N,\nu )}$
provided by the action would be a locally trivial smooth principal bundle with structure group
$\operatorname {Diff}_c(M;\mathcal N,\nu )$
. Hence, the
$\operatorname {Diff}_c(M)$
orbit of
$(\mathcal {N},\nu )$
could be regarded as a homogeneous space

2.5 A homological description
In this section, we give a more explicit description of the manifold
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)$
.
If
$\mathcal N=(N_1,\dotsc ,N_r)$
is a nonlinear flag of type
${\mathcal S}$
in M, we put

with the convention that
$N_0=\emptyset $
.
We define the space of homologically weighted flags of type
${\mathcal S}$
in M by

Note that we have a
$\operatorname {Diff}(M)$
equivariant forgetful map

as well as a
$\operatorname {Diff}(M)$
equivariant map

Let
$\operatorname {Flag}^{\operatorname {hwt}}_{{\mathcal S},\iota }(M)$
denote the preimage of the open subset
$\operatorname {Flag}_{{\mathcal S},\iota }(M)$
under the projection in (2.38). Using the canonical
$\operatorname {Diff}(M)$
equivariant identifications

we equip
$\operatorname {Flag}^{\operatorname {hwt}}_{{\mathcal S}}(M)$
with the structure of a smooth vector bundle of finite (possibly nonconstant) rank over
$\operatorname {Flag}_{{\mathcal S}}(M)$
and with projection (2.38). The map in (2.39) is a smooth bundle map over
$\operatorname {Flag}_{{\mathcal S}}(M)$
. Indeed, via the identifications in (2.25) and (2.40), the map
$h_{{\mathcal S},\iota }$
in (2.28) induces a bundle map
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota }(M)\to \operatorname {Flag}^{\operatorname {hwt}}_{{\mathcal S},\iota }(M)$
which coincides with the restriction of (2.39).
We define the space of homologically weighted flags of type
$({\mathcal S},\iota ,[\mu ])$
in M by

that is, the space of all homologically weighted flags
$(\mathcal N,[\nu ])$
in M such that there exist diffeomorphisms
$S_i\to N_i$
,
$1\leq i\leq r$
, intertwining
$\iota _i$
with the canonical inclusion
$N_i\subseteq N_{i+1}$
and taking
$[\mu ]\in H({\mathcal S},\iota )$
to
$[\nu ]\in H(\mathcal N)$
. The diffeomorphism in (2.40) restricts to a bijection

Hence, since the orbit
$H({\mathcal S},\iota )_{[\mu ]}$
is finite,
$\operatorname {Flag}^{\operatorname {hwt}}_{\mathcal S,\iota ,[\mu ]}(M)$
is a
$\operatorname {Diff}(M)$
invariant closed submanifold in
$\operatorname {Flag}^{\operatorname {hwt}}_{\mathcal S,\iota }(M)$
of codimension
$\dim H({\mathcal S},\iota )$
by (2.40). Moreover, the forgetful map

is a
$\operatorname {Diff}(M)$
equivariant covering map with finite fibers.
We complement Theorem 2.11 with the following.
Theorem 2.15 In this situation, we have

Remark 2.16 If the action of
$\operatorname {Diff}({\mathcal S},\iota )$
on
$H({\mathcal S},\iota )$
is trivial, then the
$\operatorname {Diff}({\mathcal S},\iota )_0$
and
$\operatorname {Diff}({\mathcal S},\iota )$
orbits in
$\operatorname {Den}_\times ({\mathcal S})$
coincide in view of Proposition 2.10, and the forgetful (covering) map in (2.42) is a diffeomorphism,
$\operatorname {Flag}^{\operatorname {hwt}}_{\mathcal S,\iota ,[\mu ]}(M)=\operatorname {Flag}_{\mathcal S,\iota }(M)$
. This happens in particular when all
$S_i\setminus \iota (S_{i-1})$
are connected, as in Examples 2.17 and 2.18. Under the latter connectedness assumption,
$H({\mathcal S},\iota )=\mathbb {R}^r$
via the isomorphism
$[\mu ]\mapsto \left (\int _{S_1}\mu _1,\dotsc ,\int _{S_r}\mu _r\right )$
and

Moreover, Theorem 2.15 ensures that

This also applies in the situation of Remark 2.9, provided each model manifold
$S_i$
is connected. A description for nested spheres in the same vein can be found in Example 2.19.
Example 2.17 (Nested tori)
If
$({\mathcal S},\iota )$
denotes the standard (meridional) embeddings between tori,
$\mathbb T^0\subseteq \mathbb T^1\subseteq \cdots \subseteq \mathbb T^r$
, then
$H({\mathcal S},\iota )=\mathbb {R}^{r+1}$
via the isomorphism
${[\mu ]\mapsto (\int _{\mathbb T^i}\mu _i)}$
.
Example 2.18 (Nested projective spaces)
If
$({\mathcal S},\iota )$
denotes the standard embeddings between projective spaces,
$\mathbb {P}^0\subseteq \mathbb {P}^1\subseteq \cdots \subseteq \mathbb {P}^r$
, then
$H({\mathcal S},\iota )=\mathbb {R}^{r+1}$
via the isomorphism
$[\mu ]\mapsto (\int _{\mathbb {P}^i}\mu _i)$
.
Example 2.19 (Nested spheres [Reference Jung, Dryden and Marron13])
If
$({\mathcal S},\iota )$
denotes the standard equatorial embeddings between spheres,
$S^0\subseteq S^1\subseteq \cdots \subseteq S^r$
, then
$H({\mathcal S},\iota )=\mathbb {R}^{2(r+1)}$
. The
$2(r+1)$
numbers assigned to
$[\mu ]\in H({\mathcal S},\iota )$
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
$0\leq i\leq r$
there exists a diffeomorphism in
$\operatorname {Diff}({\mathcal S},\iota )$
swapping
$S^i_+$
with
$S^i_-$
, but leaving all other hemispheres
$S^k_\pm $
invariant. Such a diffeomorphism interchanges
$a_i^+$
with
$a_i^-$
, but leaves all other numbers
$a_k^\pm $
unchanged. Hence, the
$\operatorname {Diff}({\mathcal S},\iota )$
orbit
$H({\mathcal S},\iota )_{[\mu ]}$
has
$2^s$
elements, where s is the number of
$0\leq i\leq r$
with
$a_i^+\neq a_i^-$
. Actually, the
$\operatorname {Diff}({\mathcal S},\iota )$
action on
${H({\mathcal S},\iota )\cong \mathbb {R}^{2(r+1)}}$
factorizes through a
$(\mathbb Z_2)^{2(r+1)}$
action by switching or not the numbers
$a_i^+$
and
$a_i^-$
. We obtain a description similar to the one in (2.43):

3 Coadjoint orbits of the Hamiltonian group
Throughout this section,
$(M,\omega )$
denotes a symplectic manifold. The nonlinear Grassmannian of all isotropic submanifolds of type
$S_r$
in M, denoted by
$\operatorname {Gr}_{S_r}^{\operatorname {iso}}(M)$
, is a splitting smooth submanifold in
$\operatorname {Gr}_{S_r}(M)$
which is invariant under the Hamiltonian group. In fact, the
$\operatorname {Ham}(M)$
orbits provide a smooth foliation of finite codimension in
$\operatorname {Gr}_{S_r}^{\operatorname {iso}}(M)$
which is called the isodrastic foliation [Reference Lee15, Reference Weinstein26].
Suppose
${\mathcal L}$
is an isodrastic leaf in
$\operatorname {Gr}_{S_r}^{\operatorname {iso}}(M)$
, and let
$\operatorname {Flag}^{\operatorname {wt}\operatorname {iso}}_{{\mathcal S},\iota ,\mu }(M)|_{\mathcal L}$
denote the preimage of
${\mathcal L}$
under the canonical bundle projection
$\operatorname {Flag}^{\operatorname {wt}}_{{\mathcal S},\iota ,\mu }(M)\to \operatorname {Gr}_{S_r}(M)$
. We will show that the natural
$\operatorname {Ham}_c(M)$
action on
$\operatorname {Flag}^{\operatorname {wt}\operatorname {iso}}_{{\mathcal S},\iota ,\mu }(M)|_{\mathcal L}$
admits local smooth sections. In particular, each connected component of the latter space is an orbit of
$\operatorname {Ham}_c(M)$
.
The space
$\operatorname {Flag}^{\operatorname {wt}\operatorname {iso}}_{{\mathcal S},\iota ,\mu }(M)|_{\mathcal L}$
comes equipped with a canonical weakly non-degenerate symplectic form, and the restriction of (2.18) provides a
$\operatorname {Ham}(M)$
equivariant injective moment map for the
$\operatorname {Ham}_c(M)$
action,

This moment map J maps each connected component of
$\operatorname {Flag}^{\operatorname {wt}\operatorname {iso}}_{\mathcal S,\iota ,\mu }(M)|_{\mathcal L}$
one-to-one onto the corresponding coadjoint orbit (see Theorem 3.15). Thereby, we identify coadjoint orbits of the Hamiltonian group
$\operatorname {Ham}_c(M)$
that can be modeled on weighted nonlinear flags.
The material in this section is inspired by the results in [Reference Gay-Balmaz and Vizman7, Reference Lee15, Reference Weinstein26] on weighted isotropic nonlinear Grassmannians.
3.1 Isodrasts as
$\operatorname {Ham}(M)$
orbits
In view of the tubular neighborhood theorem for isotropic embeddings [Reference Weinstein24, Reference Weinstein25], the space
$\operatorname {Gr}_S^{\operatorname {iso}}(M)$
of all isotropic submanifolds of type S in M is a splitting smooth submanifold of
$\operatorname {Gr}_S(M)$
(see, for instance, [Reference Lee15, Section 8]). The tangent space at an isotropic submanifold N is

where
$TN^\perp :=TM|_N/TN$
denotes the normal bundle to N.
Weinstein’s [Reference Weinstein26] isodrastic distribution
$ \underline {\mathcal {D}}$
on
$\operatorname {Gr}_S^{\operatorname {iso}}(M)$
is given by

and has finite codimension
$\dim H^1(S;\mathbb {R})$
. This is an integrable distribution [Reference Lee15, Reference Weinstein26] whose leaves are orbits of
$\operatorname {Ham}(M)$
. It gives rise to a smooth foliation of
$\operatorname {Gr}^{\operatorname {iso}}_S(M)$
called the isodrastic foliation. In particular, the
$\operatorname {Ham}(M)$
orbits in
$\operatorname {Gr}^{\operatorname {iso}}_S(M)$
are splitting smooth submanifolds.
Restricting the fundamental frame bundle in (2.1), we obtain a principal
$\operatorname {Diff}(S)$
bundle
$\operatorname {Emb}^{\operatorname {iso}}_{S}(M)\to \operatorname {Gr}_S^{\operatorname {iso}}(M)$
with total space
$\operatorname {Emb}^{\operatorname {iso}}_{S}(M)$
, the splitting smooth submanifold of
$\operatorname {Emb}_{S}(M)$
consisting of all isotropic embeddings. On
$\operatorname {Emb}_S^{\operatorname {iso}}(M)$
, we consider the pullback of the isodrastic distribution
$ \underline {\mathcal {D}}$
:

This is an integrable distribution with the same codimension,
$\dim H^1(S;\mathbb {R})$
, and the leaves of
$\mathcal {D}$
are connected components in the preimage of a leaf of
$ \underline {\mathcal {D}}$
. According to [Reference Lee15, Reference Weinstein26], the group
$\operatorname {Ham}(M)$
acts transitively on the leaves of
$\mathcal {D}$
. We need the subsequent slightly stronger statement in Proposition 3.1.
The Lie algebra of compactly supported Hamiltonian vector fields will be denoted by
$\operatorname {\mathfrak {ham}}_c(M)=\{X_f:f\in C^\infty _c(M)\}$
. We let
$\operatorname {Ham}_c(M)$
denote the group of diffeomorphisms obtained by integrating time dependent vector fields in
$\operatorname {\mathfrak {ham}}_c(M)$
. For our purpose, it will not be necessary to consider
$\operatorname {Ham}_c(M)$
as an infinite-dimensional Lie group (cf. [Reference Kriegl and Michor14, Section 43.13]). By a smooth map (section) into
$\operatorname {Ham}_c(M)$
, we will simply mean a smooth map into
$\operatorname {Diff}_c(M)$
that takes values in
$\operatorname {Ham}_c(M)$
.
Proposition 3.1 The
$\operatorname {Ham}_c(M)$
action on each leaf
${\mathcal E}\subseteq \operatorname {Emb}^{\operatorname {iso}}_S(M)$
of the isodrastic foliation
$\mathcal {D}$
admits local smooth sections.
Proof To show infinitesimal transitivity, suppose
$\varphi \in \operatorname {Emb}_S^{\operatorname {iso}}(M)$
and
$u_\varphi \in \mathcal D_\varphi $
(cf. (3.2)). Hence, there exists
$\bar f\in C^{\infty }(S)$
such that
$d\bar f=\varphi ^*i _{u_{\varphi }}\omega $
. We extend
$\bar f$
to a function
$f_1\in C^{\infty }_c(M)$
such that
$\bar f=f_1{\circ } \varphi $
. The 1-form
$\beta $
on M along S defined by

vanishes on vectors tangent to
$\varphi (S)\subseteq M$
. Hence,
$\beta $
can be seen as a fiberwise linear function on the normal bundle
$T\varphi (S)^\perp $
whose differential along the zero section
$\varphi (S)$
coincides with
$\beta $
itself. Thus, with the help of a tubular neighborhood of
$\varphi (S)$
in M and a suitable bump function, we get
$f_2\in C^{\infty }_c(M)$
such that
$\beta =d f_2{\circ } \varphi $
. It follows from (3.3) that
$df{\circ } \varphi =i _{u_{\varphi }}\omega $
for
$f=f_1-f_2$
. We conclude that
$u_{\varphi }=X_{f}{\circ } \varphi $
is the infinitesimal generator at
$\varphi $
for the Hamiltonian vector field
$X_f\in \operatorname {\mathfrak {ham}}_c(M)$
.
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
$\varphi $
and
$u_\varphi $
, for
$\varphi $
in a sufficiently small open neighborhood of a fixed isotropic embedding
$\varphi _0\in {\mathcal E}$
. Hence, we may apply Lemma A.2 and conclude that the
$\operatorname {Ham}_c(M)$
action admits local smooth sections.
Corollary 3.2 The
$\operatorname {Ham}_c(M)$
action on each leaf
${\mathcal L}\subseteq \operatorname {Gr}^{\operatorname {iso}}_S(M)$
of the isodrastic foliation
$ \underline {\mathcal {D}}$
admits local smooth sections.
Example 3.3 Every embedded closed curve in the plane is a Lagrangian submanifold of
$(\mathbb {R}^2,\omega )$
, where
$\omega $
is the canonical area form, thus an element of
$\operatorname {Gr}^{\operatorname {iso}}_{S^1}(\mathbb {R}^2)$
. The isodrastic distribution
$ \underline {\mathcal {D}}$
has codimension one. The enclosed area a singles out one isodrast
${\mathcal L}_a\subseteq \operatorname {Gr}^{\operatorname {iso}}_{S^1}(\mathbb {R}^2)$
, i.e., one orbit of
$\operatorname {Ham}_c(\mathbb {R}^2)$
.
A similar phenomena happens for Lagrangian k-tori in
$\mathbb {R}^{2k}$
, i.e., elements of
$\operatorname {Gr}^{\operatorname {iso}}_{\mathbb T^k}(\mathbb {R}^{2k})$
, where
$\mathbb T^k:=(S^1)^k$
. To any
$\varphi \in \operatorname {Emb}_{\mathbb T^k}^{\operatorname {iso}}(\mathbb {R}^{2k})$
, we assign the symplectic area
$a_i$
of the surface in
$\mathbb {R}^{2k}$
enclosed by the ith meridian
$\varphi _i(\theta )=\varphi (1,\dots ,\theta ,\dots ,1)$
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
$(a_1,\dots ,a_k)$
is an invariant under isodrastic deformations. Actually,
$a_i$
is the action integral of the ith meridian, as defined in [Reference Weinstein26].
Let
${\mathcal E}_{a_1,\dots ,a_k}\subseteq \operatorname {Emb}^{\operatorname {iso}}_{\mathbb T^k}(\mathbb {R}^{2k})$
be the space of all isotropic embeddings having symplectic areas
$a_1,\dots ,a_k$
. It is a union of isodrastic leaves of Lagrangian embeddings, but it is not necessarily
$\operatorname {Diff}(\mathbb T^k)$
saturated. The
$\operatorname {Diff}(\mathbb T^k)$
action on the k-tuples
$(a_1,\dots ,a_k)$
factorizes through a
$\operatorname {GL}(k,\mathbb Z)$
action. Let
$[a_1,\dots ,a_k]$
denote the orbit of
$(a_1,\dots ,a_k)$
. We define
${\mathcal L}_{[a_1,\dots ,a_k]}\subseteq \operatorname {Gr}^{\operatorname {iso}}_{\mathbb T^k}(\mathbb {R}^{2k})$
to be the image of
${\mathcal E}_{a_1,\dots ,a_k}$
under the principal
$\operatorname {Diff}(\mathbb T^k)$
bundle projection (2.1). Thus,
${\mathcal L}_{[a_1,\dots ,a_k]}$
is a union of isodrastic leaves of Lagrangian k-tori in
$\mathbb {R}^{2k}$
.
There is also a direct description of
${\mathcal L}_{[a_1,\dots ,a_k]}$
. Given a Lagrangian torus
$N\in \operatorname {Gr}_{\mathbb T^k}^S(\mathbb {R}^{2k})$
, we choose a base
$\{[\gamma _1],\dots ,[\gamma _k]\}$
of
$H_1(N,\mathbb Z)$
, where
$\gamma _i$
are loops in N with action integrals
$a_i$
. We observe that the
$\operatorname {GL}(k,\mathbb Z)$
orbit
$[a_1,\dots ,a_k]$
is independent of the choices and
${\mathcal L}_{[a_1,\dots ,a_k]}$
is the space of all Lagrangian tori in
$\operatorname {Gr}_{\mathbb T^k}^S(\mathbb {R}^{2k})$
such that the orbit of these action integrals is
$[a_1,\dots ,a_k]$
.
We will also need the following observation.
Lemma 3.4 If L is an isotropic submanifold in M, then the canonical inclusion
$\operatorname {Gr}_S(L)\subseteq \operatorname {Gr}_S^{\operatorname {iso}}(M)$
is a splitting smooth submanifold. Moreover, each connected component of
$\operatorname {Gr}_S(L)$
is a splitting smooth submanifold in an isodrastic leaf in
$\operatorname {Gr}_S^{\operatorname {iso}}(M)$
.
Proof Suppose
$N\in \operatorname {Gr}_S(L)$
, i.e.,
$N\cong S$
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
$p\colon L\to N$
, the normal bundle of N in L, and identify N with the zero section in L. We have a canonical short exact sequence
$0\to p^*L\to TL\to p^*TN\to 0$
of vector bundles over L. Choosing a linear connection on L, we obtain a splitting of this sequence and thus an isomorphism
$TL\cong p^*TN\oplus p^*L$
of vector bundles over L. Dualizing, we obtain an isomorphism
$T^*L\cong p^*T^*N\oplus p^*L^*$
of vector bundles over L. We regard this as a diffeomorphism

that maps the zero section
$L\subseteq T^*L$
identically onto the summand L on the right-hand side. Via this isomorphism, we have

where
$\pi _1$
and
$\pi _2$
denote the projections from
$T^*N\oplus L\oplus L^*$
onto
$T^*N$
and
$L\oplus L^*$
, respectively,
$\theta _L\in \Omega ^1(T^*L)$
and
$\theta _N\in \Omega ^1(T^*N)$
denote the tautological 1-forms, and
$\kappa \in \Omega ^1(L\oplus L^*)$
. Indeed, a straightforward computation yields
$\kappa (\xi )=\ell '(C(Tq\cdot \xi ))$
where
$x\in N$
,
$\ell \in L_x$
,
$\ell '\in L_x^*$
,
$\xi \in T_{(\ell ,\ell ')}(L\oplus L^*)$
,
$q\colon L\oplus L^*\to L$
denotes the projection, and C denotes the linear connection on L, viewed as a fiberwise linear map
$C\colon TL\to p^*L$
over L.
By the tubular neighborhood theorem for isotropic embeddings [Reference Weinstein24, Reference Weinstein25], we may assume
$M=T^*L\oplus p^*E$
and
$\omega =\tilde \pi _1^*d\theta _L+\tilde \pi _2^*\rho $
, where E is a vector bundle over N,
$\tilde \pi _1$
and
$\tilde \pi _2$
denote the projections from