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arXiv:2210.17306v1 [math.DG] 31 Oct 2022
SINGULAR RIEMANNIAN FOLIATIONS
AND I-POISSON MANIFOLDS
HADI NAHARI AND THOMAS STROBL
Abstract. We recall the notion of a singular foliation (SF) on a manifold M, viewed
as an appropriate submodule of X(M), and adapt it to the presence of a Riemannian
metric g, yielding a module version of a singular Riemannian foliation (SRF). Follow-
ing Gamendia-Zambon on Hausdorff Morita equivalence of SFs, we define the Morita
equivalence for SRFs (both in the module sense as well as in the more traditional geo-
metric one) and show that the leaf spaces of Morita equivalent SRFs are isomrophic as
pseudo-metric spaces.
In a second part, we introduce the category of I-Poisson manifolds. Its objects are just
Poisson manifolds (P, {·,·}) together with appropriate ideals I—generalizing coisotropic
submanifolds to the singular setting—but its morphisms are a generalization of Poisson
maps. This permits one to consider an algebraic generalization of coisotropic reduc-
tion. I-Poisson maps are now precisely those maps which induce morphisms of Poisson
algebras between the corresponding reductions.
Every SF on Mgives rise to an I-Poisson manifold on P=T∗Mand genhances this to
an SRF iff the induced Hamiltonian on Plies in the normalizer of I. This perspective
provides: i) an almost tautological proof of the fact that every module SRF gives rise to
a geometric SRF and ii) a construction of an algebraic invariant of singular foliations:
Hausdorff Morita equivalent SFs have isomorphic I-Poisson reductions.
Contents
1. Introduction 1
2. Background on singular foliations and their Morita equivalence 5
3. Singular Riemannian foliations and their Morita equivalence 8
4. I-Poisson manifolds 15
5. Singular (Riemannian) foliations through I-Poisson manifolds 19
6. The functor Φ and reduction 23
Appendix A. Almost Killing Lie algebroids 32
References 34
1. Introduction
While the study of foliations is of interest in differential topology and thus on a global level,
singular foliations are particularly interesting in the neighborhood of singular leaves. In
the literature there exist different definitions of what one would call a singular foliation, all
having in common that it consists of a well-behaved decomposition of a smooth manifold
Date: October, 2022.
1
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 2
into injectively immersed submanifolds called leaves, where “well-behaved” means that
every tangent vector to a leaf can be extended to a vector field tangent to the leaves.
A large number of examples of such leaf decompositions is provided by—in increasing
generality—Poisson manifolds, Lie algebroids, and Lie infinity algebroids. For a detailed
review of the debates on finding a suitable definition of singular foliations see [L18]. In
this article, we use the definition suggested by I. Androulidakis and G. Skandalis [AS09]:
Definition 1.1. Asingular foliation (SF) on Mis defined as a C∞(M)-submodule Fof
the module of compactly supported vector fields on M, which is locally finitely generated
and closed with respect to the Lie bracket of vector fields.
This definition induces a leaf decomposition on M(see [H62]) and, in the case where all the
leaves have the same dimension, it agrees with the definition of regular foliations. An SF
can be equivalently defined as an involutive and locally finitely generated subsheaf of the
sheaf of smooth vector fields on Mclosed under multiplication by C∞(M) [LGLS20] (see
also [GZ19]). This has the advantage that one can replace C∞(M) by an arbitrary sheaf
of rings Oon M. Definition 1.1, however, is more convenient for the present purposes.
One purpose of this article is to introduce and study a notion of singular Riemannian
foliations that is adapted to Definition 1.1 and that provides an appropriate compatibility
between such an SF (M, F) and a Riemannian structure gon M. In the context of
regular foliations, Riemannian foliations have been studied in differential geometry since
the 1950s (see for example [H58] and [R59]). They are defined as regular foliations on
a Riemannian manifold with the property that every geodesic perpendicular to one leaf
stays perpendicular to all the leaves it meets. Singular Riemannian foliations in the
traditional setting were introduced in [M98], by requiring the above property of geodesics
for the now possibly singular leaf decomposition. Examples of such constructions are
given by isometric Lie group actions on Riemannian manifolds and, more generally, orbit
decompositions induced by Riemannian groupoids [dHF18]. One might consider using this
definition for the induced leaf decomposition of an SF, but this seems inadequate because
the SF carries more information. Inspired by [KS16, KS19], stripping off unnecessary
additional data, we instead propose:
Definition 1.2. Asingular Riemannian foliation (SRF) on a Riemannian manifold
(M, g)is defined as an SF Fon (M, g)such that for every vector field X∈ F we have:
(1) LXg∈Ω1(M)⊙g♭(F).
Here g♭:X→Ω1(M), X7→ g(X, ·) is the musical isomorphism induced by the metric and
⊙stands for the symmetric tensor product. Every SRF in the above sense turns out to
have a leaf decomposition that is an SRF in the sense of [M98]—while the converse is not
always true. For regular SFs, on the other hand, the two notions coincide.
Our notion of SRFs behaves well under the pullback operation of [AS09], which is crucial
for the definition of a Morita equivalence between SRFs, which we will provide. It extends
the notion of Hausdorff Morita equivalence between SFs as defined in [GZ19]. This permits
us to prove a likewise extension of a theorem proved in that work about the existence of
a homeomorphism between the leaf spaces of Hausdorff Morita equivalent SFs:
Theorem A. Let (N1, g1,F1)and (N2, g2,F2)be Morita equivalent SRFs. Then their
leaf spaces are isometric as pseudo-metric spaces.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 3
A second purpose of this article is to introduce the category of I-Poisson manifolds IPois.
For its objects, the intention is to generalize coisotropic submanifolds (see, e.g., [MR86])
to the singular setting, following the ideas of [AS09, LGLS20] for the definition of SFs:
Definition 1.3. An I-Poisson manifold is a triple (P, {·,·} ,I)where Iis a subsheaf of
smooth functions on a Poisson manifold (P, {·,·})which is closed under multiplication by
smooth functions, locally finitely generated, and for every open subset U⊂P,I(U)⊂
C∞(U)is a Poisson subalgebra, i.e.
{I(U),I(U)} ⊂ I (U).
To describe dynamics, one needs in addition a compatible Hamiltonian, i.e. a function
H∈N(I) where
N(I) := {f∈C∞(P): {f|U,I(U)} ⊂ I(U)for every open subset U}.
We then call (P, {·,·} ,I, H) a dynamical I-Poisson manifold and the likewise extended
category dynIPois.
Such as local finiteness in Definition 1.1 plays a crucial role for the existence of the leaf
decomposition induced by a singular foliation, in Definition 1.3 it is essential for showing
that the flow of any H∈N(I), if complete, preserves the sheaf I(see Proposition 4.1 in
the main text for more details).
Definition 1.4. A smooth map ϕ:P1→P2between (P1,{·,·}1,I1)and (P2,{·,·}2,I2)
is a morphism of I-Poisson manifolds, iff the two obvious conditions ϕ∗(I2(P2)) ⊂ I1(P1)
and ϕ∗N(I2)⊂N(I1)are complemented by
(2) {ϕ∗f, ϕ∗g}1−ϕ∗{f, g}2∈ I1(P1)∀f, g ∈N(I2).
For dynamical I-Poisson manifolds we add the condition ϕ∗H2−H1∈ I1.
Like this the category Pois of Poisson manifolds is a full subcategory of IPois for the
choice of the zero ideal. Note however, despite that the objects of our category contain
ordinary Poisson manifolds, the morphisms are not necessarily Poisson maps—and this
becomes important also in applications to SRFs, as we will explain below.
The condition (2) is optimal to ensure that ϕ∗decends to a Poisson morphism on the
level of reductions: In fact, every I-Poisson manifold (P, {·,·} ,I) induces a Poisson alge-
bra structure on N(I)/I(P). In the case of coisotropic reductions [MR86], this algebra
coincides with the algebra of smooth functions on the reduced Poisson manifold; however,
the algebraic formulation here is also applicable in the more general context of I-Poisson
manifolds as defined above, where, e.g., the constraint surface, i.e. the vanishing set of
the ideal I(P), does not need to be a submanifold. The conditions in Definition 1.4 en-
sure that there is a canonical—albeit neither faithful nor full—contravariant functor from
IPois to PoisAlg, the category of Poisson algebras.
Starting from an SF (M, F) and viewing every vector field in Fas a smooth function on
T∗M, we construct a corresponding I-Poisson manifold (T∗M, {·,·}T∗M,IF). Moreover,
every metric gon Mdefines a compatible Hamiltonian (making the I-Poisson manifold
dynamical) if and only if the metric satisfies condition (1)—which provides a conceptual
motivation for Definition 1.2 of an SRF. Thus there is a canonical map from SFs and SRFs
to the objects of IPois and dynIPois, respectively. As we will see, this construction is
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 4
not only conceptually illuminating, but it also has technical advantages, in particular for
proving several properties of SFs and SRFs—e.g. to show that Definition 1.2 automatically
induces an SRF in the sense of [M98].
To complete the above map on objects to a functor, one would need a proper definition
of the categories SF and SRF of singular (Riemannian) foliations. Surprisingly, already
for SFs, in the literature there is not yet any satisfactory proposal for what a morphism
between general SFs should be. However, the situation changes if one restricts to sub-
mersions and Riemannian submersions in the case of SFs and SRFs, respectively, because
in these cases the above mentioned pullback operations are defined. For example, a Rie-
mannian submersion π: (N, h)→(M, g) between two SRFs (N, h, FN) and (M, g, FM)
which satisfies π−1FM=FNshould definitely be considered as a morphism. Let us call
SF0and SRF0the two (sub)categories with such restricted morphisms. We will show
Theorem B. There is a canonical functor Φ: SRF0→dynIPois.
As a side result, we will find that for FM= 0, Φ(π) becomes an ordinary Poisson map if
and only if the horizontal distribution (ker dπ)⊥is integrable—correcting [BWY21], where
this map has been considered as well, but claimed to always be Poisson. Note also that
it seems natural to expect that the functor Φ should extend to a functor from all of an
appropriately defined category SRF to dynIPois. Since there is no doubt about what
the morphisms in the latter category should be, this poses some reasonable restriction on
the definition of the morphisms in SRF.
As argued above, every SF (M, F) gives rise to an I-Poisson manifold. Applying the
canonical functor IPois →PoisAlg to it, we obtain the (reduced) Poisson algebra
A(F) := N(IF)/IF(T∗M). This algebra provides an invariant of Hausdorff Morita equiv-
alence, since we will prove:
Theorem C. Let (M1,F1)and (M1,F1)be Hausdorff Morita equivalent singular folia-
tions. Then the reduced Poisson algebras A(F1)and A(F2)are isomorphic.
In general, the above isomorphism depends on additional choices. For SRFs, however,
there is a canonical one.
The structure of this paper is as follows:
Section 2 contains a short review of the definitions and main properties of SFs related to
the goal of this paper, in particular the notion of Hausdorff Morita equivalence of SFs.
In Section 3 we compare different possible notions of SRFs, arguing for Definition 1.2
as the most adequate one in the spirit of SFs provided by [AS09]. In particular, we
show that every SRF which is finally generated (this is always the case locally) admits
an almost Lie algebroid structure with connection to turn the SRF into an almost Killing
Lie algebroid [KS19]. We then define Morita equivalence of SRFs, show that it defines an
honest equivalence relation, and prove Theorem A.
Section 4 introduces the category IPois, the functor to PoisAlg, and provides several
examples and some properties of I-Poisson manifolds.
In Section 5 we show how SFs and SRFs give rise to particular I-Poisson and dynamical
I-Poisson manifolds, respectively.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 5
In Section 6, finally, we prove Theorems B and C.
The definition of almost Killing Lie algebroids as well as part of the proof of Theorem 3.1
(in the form of Proposition A.1) are deferred to Appendix A.
Acknowledgements. We are grateful to Anton Alekseev, Camille Laurent-Gengoux,
Ricardo Mendes, Leonid Ryvkin and, in particular, to Marco Zambon and Mateus de
Melo for stimulating discussions related in one way or another to the present subject.
This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit´e
de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated
by the French National Research Agency (ANR). We also acknowledge having profited
from the marvellous environment provided within the program ”Higher structures and
Field Theory” at the ERWIN SCHR ¨
ODINGER INSTITUTE in Vienna.
2. Background on singular foliations and their Morita equivalence
In what follows, Mis assumed to be a smooth manifold and Xc(M) denotes the C∞(M)-
module of compactly supported vector fields on M. For more details and examples of
singular foliations see [AS09] or [LGLS20].
Definition 2.1. AC∞(M)-submodule F ⊂ Xc(M)is called locally finitely generated if
for every point q∈Mthere exist an open neighborhood U⊂Maround qsuch that the
submodule ι−1
UF ⊂ Xc(U)defined as
ι−1
UF:= {X∈ F : supp(X)⊂U}
is finitely generated; i.e. there exist finitely many vector fields X1,...,XN⊂X(U)for
some positive integer N, such that
ι−1
UF=hX1,...,XNiC∞
c(U).
Example 1. Let M=Rand Fbe the C∞(M)-submodule of compactly supported vector
fields on the real line which vanish on R−. Then Fis not locally finitely generated.
Definition 2.2. Asingular foliation on M—SF for short—is defined as C∞(M)-submodule
Fof Xc(M)which is locally finitely generated and closed with respect to the Lie bracket
of vector fields. The pair (M, F)is then called a foliated manifold.
Remark 2.1. One can equivalently define SFs as an involutive and locally finitely gen-
erated subsheaf of the sheaf of vector fields X. This is equivalent to Definition 2.2 in the
smooth setting, but it has advantages if we wish to work with the sheaves of algebraic,
real analytic or holomorphic functions (See [LGLS20] or [GZ19]). In particular, since the
mentioned sheaves of rings are Noetherian, the condition of being locally finitely generated
is automatically satisfied and therefore can be dropped.
A classical theorem of R. Hermann [H62] implies that an SF defined as above partitions
Minto smooth, connected, and injectively immersed submanifolds (of possibly different
dimensions) called leaves.
Let Lqbe the leaf passing through the point q∈Min a foliated manifold (M, F). Then,
by definition of the leaves, TqLqcan be identified with {X|q:X∈ F } ⊂ TqM, which
motivates:
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 6
Definition 2.3. For every point q∈Min a foliated manifold (M, F), the tangent of F
at qis defined as
Fq:= {X|q:X∈ F} ⊂ TqM .
If q7→ dim(Fq) is constant on M, we obtain regular foliations as particular singular ones.
In this case, by the Frobenius theorem, there is a one-to-one correspondence between the
leaf decomposition of the foliation and the module of vector fields generating it. This is
no more the case if the SF is non-regular; there always exist different modules generating
the same leaf decomposition then (for an example, see Example 3 below). Note also that
in the non-regular case all the vector fields tangent to the leaves of a given SF do not
necessarily define an SF anymore: the module of Example 1, despite not being an SF
since not finitely generated, induces a leaf decomposition, which can be obtained also by
an honest SF with the single generator χd
dx . Here χ∈C∞(R) can be chosen, e.g., as the
function
(3) χ(x) = (exp(−1
x2)x > 0
0x≤0·
Remark 2.2. The function M→Ngiven by q7→ dim(Fq)is lower semi-continuous. As
a result, the subset U⊂Mof the continuity set of dim(Fq)is open and dense, and F |U
induces a regular foliation over each connected component of U[AS09].
The following example shows the importance of being locally finitely generated:
Example 2. On M=R2, consider the module Ggenerated by the vector fields ∂xand
X(x)∂y, where χis the function defined in (3), together with all their multiple commutator
Lie brackets. Then, by construction, Gis closed under the Lie bracket. However, it is
not locally finitely generated as a C∞(M)-module since with each derivative on Xwe
obtain a new, independent coefficient in front of ∂y. As a consequence, we loose the well-
behavedness of a leaf-decomposition: although every two points in R2can be connected
by a sequence of flows of vector fields in G(so that, in this sense, there would be only
one leaf that is R2itself), the tangent of Gat every point in the left half-plane is only
one-dimensional.
As mentioned above, Definition 2.2 contains more information than a well-behaved de-
composition of Minto leaves:
Example 3. Let M=Rand let Fbe an SF generated by vector fields vanishing at the
origin at least of order k∈N. The leaf decomposition induced by Fis R=R−∪ {0} ∪ R+
for every choice of k, but Fis invariantly characterized by this integer.
To capture some of this additional information contained in the definition of an SF, we
extract some more data from the module Fby the following definition of [AS09].
Definition 2.4. Let (M, F)be a foliated manifold. For every point q∈M, the fiber of
Fat qis defined as:
Fq:= F/Iq·F
where Iq:= {f∈C∞(M) : f(q) = 0}is the vanishing ideal of qin C∞(M).
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 7
Remark 2.3. The function M→Nsending q→dim(Fq)is upper semi-continuous, and
dim(Fq)gives the minimal number of vector fields locally generating Faround q∈M
[AS09].
Note that for every point q∈Mthe evaluation map evq:Fq→Fq, [X]→X|qis a
homomorphism of vector spaces and induces the following short exact sequence:
0→ker(evq)→ Fq→Fq→0
here [X] denotes the equivalence class of the vector field X∈ F.
It is not difficult to see that the Lie bracket on Finduces a Lie bracket on the finite-
dimensional vector space ker(evq)⊂ Fq.
Definition 2.5. The vector space gF
q:= ker(evq)together with the bracket inherited by
Fqdefines the isotropy Lie algebra of Fat q.
In the case of regular foliations, the map evq:Fq→Fqis a vector space isomorphism
and gF
q= 0. So one can say that the isotropy Lie algebra gF
qcharacterizes in part the
singularity of Fat q∈M.
In Example 3 all fibers and isotropy Lie algebras at the origin are isomorphic. This
changes, if we increase the dimension of M:
Example 4. Let M=Rn,n≥2, and let Fbe the SF generated by vector fields vanishing
at the origin at least of order k∈N. There are always only two leaves M\ {0}and {0}, but
the fiber at the origin has different dimensions for different choices of k,dim F0=k+n−1
n−1.
Definition 2.6. Let (M, F)be an SF and π:N→Ma submersion, then the C∞(N)-
module generated by vector fields on Nprojectable to Fdefines the pullback foliation
(N, π−1F).
Here a vector field Von Nis called projectable to Fif there exists a vector field X∈ F
such that for every point q∈Nwe have:
dqπ(W|q) = X|π(q).
As shown in [AS09], Propositions 1.10 and 1.11, the pullback foliation is indeed finitely
generated and involutive, i.e. it is an SF. This notion behaves well under composition of
submersions: For submersions πP:P→Mand πM:M→N, one has
(πM◦πP)−1F=π−1
P(π−1
MF).
As an example, if Uis an open subset of a foliated manifold (M, F), then for the inclusion
map ιU:U ֒−→ M, the SF ι−1
UFis compatible with Definition 2.1.
Definition 2.7 ([GZ19]).Two foliated manifolds (M1,F1)and (M2,F2)are Hausdorff
Morita equivalent if there exits a smooth manifold Nand surjective submersions with
connected fibers πi:N→Mi,i= 1,2such that:
π−1
1F1=π−1
2F2.
In this case we write (M1,F1)∼ME (M2,F2).
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 8
It is shown in [GZ19] that the SFs underlying Morita equivalent Lie algebroids [G01]
or Morita equivalent Poisson manifolds [X91] are Hausdorff Morita equivalent. Also the
Morita equivalence of regular foliations [M98] is a special case. Hausdorff Morita equiv-
alence defines an equivalence relation on foliated manifolds—something that holds true
for Poisson manifolds only upon restriction to those integrating to a symplectic groupoid.
The main fact about Hausdorff Morita equivalent foliated manifolds is that they have
Morita equivalent Holonomy groupoids (as open topological groupoids) defined in [AS09].
Theorem 2.1 ([GZ19]).Let (M1,F1)and (M2,F2)be foliated manifolds which are Haus-
dorff Morita equivalent by means of (N, π1, π2). Then:
(i) The map sending the leaf passing through q∈M1to the leaf of F2containing π2(π−1
1(q))
is a homeomorphism between the leaf spaces. It preserves the codimension of leaves and
the property of being an embedded leaf.
(ii) Let q1∈N1and q2∈N2be points in corresponding leaves. Choose transversal slices
Sq1at q1and Sq2at q2. Then the foliated manifolds (Sq1, ι−1
Sq1F1)and (Sq2, ι−1
Sq2F2)are
diffeomorphic and the isotropy Lie algebras gF1
q1and gF2
q2are isomorphic.
Example 5. For smooth, connected manifolds Mand N,(M, X(M)) and (N, X(N)) are
always Hausdorff Morita equivalent. On the other hand, (M, 0) and (N, 0) are Hausdorff
Morita equivalent only if Mand Nare diffeomorphic.
3. Singular Riemannian foliations and their Morita equivalence
In what follows, (M, g) denotes a Riemannian manifold. We first recall the traditional
notion of a singular Riemannian foliations (SRF) motivated by [M98], to which we will
add the suffix ”geometric” so as to distinguish it from a second one that we will introduce
directly below.
Definition 3.1. Let Fbe an SF on (M, g). We call the triple (M, g, F)ageometric
SRF, if every geodesic orthogonal to a leaf at one point is orthogonal to all the leaves it
meets.
In this text, we focus mainly on the following definition of SRFs, streamlining the one
given in [KS19]1:
Definition 3.2. Let Fbe an SF on (M, g). We call the triple (M, g, F)amodule SRF,
if for every vector field X∈ F we have:
(4) LXg∈Ω1(M)⊙g♭(F).
Here ⊙stands for the symmetric tensor product and g♭is the map on sections induced by
the musical isomorphism g♭:T M →T∗M, (q, v)7→ gq(v, ·). Let (g♭)−1: Ω1(M)→ X (M)
denote the corresponding inverse map and g−1∈Γ(S2T M ) the 2-tensor inducing it. Then,
by means of LX(g♭)−1=−(g♭)−1◦(LXg♭)◦(g♭)−1, we can express the defining property
of a module SRF also in the following form
1For the relation of module SRFs with the notion defined in [KS19] see Appendix A as well as Theorem
3.1 below.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 9
Lemma 3.1. The triple (M, g, F)is a module SRF if and only if
LXg−1∈X(M)⊙ F
for every vector field X∈ F.
As a consequence of the following lemma and proposition, it is enough to check Equation
(4) locally for a family of generators:
Lemma 3.2. Let (M, F)be a foliated manifold such that F=hX1,...,XNiC∞
c(M)for
some positive integer N. Then the triple (M, g, F)is a module SRF if and only if there
exist ωb
a∈Ω1(M)for a, b = 1,...,N such that
LXag=
N
X
b=1
ωb
a⊙g♭(Xb).
Proof. First assume that (M, g, F) is a module SRF. Choose a partition of unity {ρi}∞
i=1
subbordinate to a locally finite cover {Ui}∞
i=1 of M. For every a= 1,...,N we have
LXag=
∞
X
i=1
ρiLXag=
∞
X
i=1
(LρiXag−(dρi)⊙g♭(Xa))
=
∞
X
i=1 N
X
b=1
ηb
i,a ⊙g♭(Xb)−(dρi)⊙g♭(Xa)!
=
N
X
b=1
ωb
a⊙g♭(Xb),
for some 1-forms ηb
i,a on Mand ωb
a:= P∞
i=1 ηb
i,a −δb
adρi. For the converse, let Xbe a vector
field in F. By assumption, there exist f1,...,fN∈C∞
c(M) such that X=PN
a=1 faXa.
It follows that
LXg=
N
X
a=1
LfaXag=
N
X
a=1
faLXag+ (dfa)⊙g♭(Xa)
=
N
X
a=1
ωb
a⊙g♭(faXb) + (dfa)⊙g♭(Xa)∈Ω1(M)⊙g♭(F),
An important property of the definition of a geometric SRF is that the defining condition
is local. This is less trivial in the case of module SRFs.
Proposition 3.1. The triple (M, g, F)is a module SRF if and only if for every point
q∈Mthere exist an open neighborhood U⊂Maround qsuch that (U, gU, ι−1
UF)is a
module SRF, where gUis the restriction to Uof g.
Proof. If (M, g, F) is a module SRF, then restricting both sides of Equation (4) to any
open subset U∈Mimplies that (U, gU, ι−1
UF) is a module SRF. It remains to prove the
converse. Choose a partition of unity {ρi}∞
i=1 subbordinate to a locally finite cover {Ui}∞
i=1
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 10
of M, with open subsets Uismall enough such that ι−1
UiF=hXi,1,...,Xi,NiiC∞
c(Ua)for some
positive integer Niand vector fields Xi,1,...,Xi,Ni∈X(Ui). Then for every vector field
X∈ F,
X=
∞
X
i=1
ρiX .
Moreover, for every positive integer i, there exist functions fi,1, ..., f i,Ni∈C∞
c(Ui) such
that
ρiX=
Ni
X
a=1
fi,aXi,a ,
and consequently
X=
∞
X
i=1
Ni
X
a=1
fi,aXi,a .
This together with Lemma 3.2 now permit us to prove that (M, g, F) is a module SRF:
LXg=
∞
X
i=1
Ni
X
a=1
fi,aLXi,a gUi+ (dfi,a)⊙(g)♭(Xi,a),
which proves LXg∈Ω1(M)⊙g♭(F) since Xis compactly supported and only finitely
many fi,a are nonzero on supp(X).
Every finitely generated SF is image of the anchor map of an almost Lie algebroid
[LGLS20] (see Appendix A). For module SRFs, one has furthermore:
Theorem 3.1.
•For every module SRF (M, g, F)with Ffinitely generated, there exists an almost
Lie algebroid (A, ρ, [·,·]A)over Mequipped with a connection ∇: Γ(A)→Γ(T∗M⊗
A)such that F:= ρ(Γc(A)) and
A∇g= 0,(5)
where A∇is the A-connection induced by ∇, see Equation (38) in Appendix A.
•Let (A, ρ, [·,·]A)be an almost Lie algebroid over a Riemannian manifold (M, g),
such that the triple (M, g, F:= ρ(Γc(A))) is a module SRF. Then there exists a
connection ∇on Asuch that (5) holds true.
Proof. The proof of the first part of the Theorem can be performed by a straightforward
adaptation of the proof of Proposition A.1 in the Appendix. In particular, the almost
Lie algebroid Athen can be chosen to be trivial, A=M×Rr, where ris the number of
generators of F.
We prove the second part of the Theorem, where now one is given a particular, not
necessarily trivial almost Lie algebroid Ainducing F, as follows: There exists a vector
bundle V→Msuch that ( ˜
A:= A⊕V)→Mis a trivial vector bundle of rank N.
Consequently there exist sections e1,...,eN∈Γ(A) and v1,...,vN∈Γ(V) such that
e1+v1,...,eN+vNis a global frame for ˜
A. Now we define the almost Lie algebroid
(˜
A, ˜ρ, [·,·]˜
A), where the bracket and the anchor map are the trivial prolongation of [·,·]A
and ρto ˜
A(since in an almost Lie algebroid one does not need to satisfy the Jacobi
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 11
identity for the bracket, this extension does not pose any problems here). By assumption
˜ρ(Γc(˜
A)) = ρ(Γc(A)) defines a module SRF on (M, g). According to Lemma 3.2, this is
equivalent to the existence of 1-forms ωb
a∈Ω1(M) such that
(6) LXag=
N
X
b=1
ωb
a⊙ιXbg∀a= 1,...,N.
Here Xa:= ˜ρ(ea+va) = ρ(ea). Now define a connection ˜
∇on ˜
Aby
˜
∇(ea+va) =
N
X
b=1
ωb
a⊗(eb+vb),
which induces a connection on Aas follows: Let s∈Γ(A)⊂Γ( ˜
A), then
∇Xs:= PrA◦(˜
∇Xs)∀X∈X(M),
where PrA:˜
A→Ais the projection to the first component. In particular, for every ea,
there exist unique functions fb
a∈C∞(M) for b= 1,...,N such that ea=PN
b=1 fb
a(eb+vb)
and we have
∇Xea=P rA◦(˜
∇X(
N
X
b=1
fb
a(eb+vb))) =
N
X
b=1
X(fb
a)eb+
N
X
b,c=1
(fb
aιXωc
b)ec.
Now for every vector field X∈X(M), we have:
2g(ρ(∇Xea), X) = 2g ρ N
X
b=1
X(fb
a)eb+
N
X
b,c=1
(fb
aιXωc
b)ec!, X!
= 2
N
X
b=1
fb
a((
N
X
c=1
(ιXωc
b)g(Xc, X)) + 2
N
X
b=1
X(fb
a)g(Xb, X)
=
N
X
b=1
fb
a N
X
b=1
ωc
b⊙ιXcg!(X, X) +
N
X
b=1
(dfb
a⊙ιXbg)(X, X)
=
N
X
b=1 fb
aLXbg+ dfb
a⊙ιXbg(X, X)
= (LXag) (X, X),
and, by Lemma A.1 in the Appendix below, the statement then follows.
So locally one can define SFs also as an equivalence class of almost Lie algebroids and
module SRFs as an equivalence class of almost Lie algebroids over a Riemannian base with
an appropriately compatible connection. (For some related cohomology see also [HS22]).
Using the language of almost Lie algebroids, the following proposition is Theorem 7 in
[KS19]. It will be proven in an alternative, more direct way in the present paper, using
the techniques of I-Poisson geometry:
Proposition 3.2. Every module SRF is a geometric SRF.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 12
Note that the converse is not true, at least not for every choice of the module F: Consider
F=h(x2+y2)(x∂y−y∂x)iC∞
c(R2)on M=R2equipped with the standard metric. The
leaves are circles centered at the origin, which is a geometric SRF, but it does not satisfy
Equation (4).
Remark 3.1. One can pose the following question as well: Assume that a leaf decomposi-
tion of a Riemannian manifold is given, such that the compatibility condition of Definition
3.1 is satisfied. Is there an SF generating a module SRF with the given leaf decomposition?
A counter-example for the polynomial or analytic setting is the singular octonionic Hopf
fibration [NS]: albeit there do exist such (real analytic or polynomial) SFs generating the
leaf decomposition, the condition (4) is not satisfied for any of them.
For SFs there is a pullback under submersions, see Definition 2.6 and the text following
it. To adapt this to the context of SRFs, we consider the following:
Definition 3.3. Let π: (N, h)→(M, g)be a smooth submersion between Riemannian
manifolds. It is called a Riemannian submersion if, for every q∈N, the restriction
dqπ:Hq→Tπ(q)Mof dqπto Hq= (ker dqπ)⊥g⊂TqNis an isometry. The smooth
distribution H= (Hq)q∈Nof rank dim(M)is called the horizontal distribution of π.
Lemma 3.3. Let π: (N, h)→(M, g)be a Riemannian submersion and (M, F)an SF.
Then the pullback SF can be generated as follows
(7) π−1F=hFH+ Γ (ker dπ)iC∞
c(N),
where FHis the horizontal lift of F.
Proof. By definition 2.6 the inclusion hFH+ Γ (ker dπ)iC∞
c(N)⊂π−1Fis evident. Now
let Wbe a projectable vector field on Nprojecting to F, i.e. there exists a vector field
X∈ F such that dqπ(W|q) = X|π(q). On the other hand, if we decompose Winto its
horizontal part WHand its vertical part WV, we have dqπ(WH|q) = X|π(q), which gives
XH=VH. This means that generators of π−1Fbelongs to FH+ Γ (ker dπ), consequently
π−1F=hFH+ Γ (ker dπ)iC∞
c(N).
Proposition 3.3. Let π: (N, h)→(M, g)be a Riemannian submersion and let (M, g, F)
be a module SRF. Then (N, h, π−1F)is a module SRF as well. The same statement holds
true for geometric SRFs.
Proposition 3.3 will be proven in Section 6 below. As a consequence, and by the fact that
(regular) Riemannian foliations are locally modeled on Riemannian submersions [M98],
we obtain:
Proposition 3.4. Let (M, F)be a regular foliation on a Riemannian manifold (M, g).
Then (M, g, F)is a geometric SRF if and only if it is a module SRF.
Example 6. Let Gbe a Lie group acting by isometries on (M, g). Then after Lemma
3.2 the C∞(M)-submodule F ⊂ Xc(M)generated by fundamental vector fields is a module
SRF on (M, g), since every fundamental vector field Xis a Killing vector field: LXg= 0.
Example 7. The proof of Theorem 1 in [KS19] shows that the geometric SRF induced on
the manifold of objects of a Riemannian Groupoid—as defined in [dHF18]—is a module
SRF.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 13
Definition 3.4. Two module SRFs (M1, g1,F1)and (M2, g2,F2)are Morita equivalent
if there exists a Riemannian manifold (N, h)together with two surjective Riemannian
submersions with connected fibers πi: (N, h)→(Mi, gi)for i= 1,2such that
π−1
1F1=π−1
2F2
and we write (N1, g1,F1)∼M E (N2, g2,F2).
Remark 3.2. This notion of Morita equivalence can be defined for geometric SRFs as well
as for module ones. Consequently, if two module SRFs are Morita equivalent then they
are also Morita equivalent as geometric SRFs. Moreover, if we forget about Riemannian
metrics, we obtain Hausdorff Morita equivalent foliated manifolds.
While for Hausdorff Morita equivalence of SFs transitivity of the equivalence relation
is relatively easy to show, this is more involved in case of the additional Riemannian
structure due to the presence of the metric.
Proposition 3.5. The Morita equivalence of module SRFs defines an equivalence relation.
Proof. Reflexivity is evident from the definition and for the self-equivalence the identity
map defines a Morita equivalence between a module SRF and itself. Now we prove the
transitivity as follows: Assume that (M1, g1,F1)∼M E (M2, g2,F2) given by πi: (U, gU)→
(Mi, gi) for i= 1,2 and (M2, g2,F2)∼M E (M3, g3,F3) given by ηi: (W, gW)→(Mi, gi) for
i= 2,3. Now consider the smooth manifold Uπ2×η2Wdefined as
Uπ2×η2W:= {(u, w)∈U×W|π2(u) = η2(w)∈M2}
with canonical projections pU:Uπ2×η2W→Uand pW:Uπ2×η2W→W. Note that the
tangent space at (u, w)∈Uπ2×η2Wis given by
T(u,w)(Uπ2×η2W) = {(X, Y )∈TuU×TwW|duπ2(X) = dwη2(Y)}
since every smooth curve on Uπ2×η2Wcan be expressed as (γU, γW) where γUand γW
are smooth curves on Uand W, respectively, such that π2(γU) = η2(γW). We now define
a Riemannian metric gon Uπ2×η2Was follows:2
(8) g((X1, Y1),(X2, Y2)) := gU(X1, X2) + gW(Y1, Y2)−g2(duπ2(X1),duπ2(X2))
where (Xi, Yi)∈T(u,w)(Uπ2×η2W) for i= 1,2, and note that duπU(X1) = dwηW(Yi) for
i= 1,2. It is clearly smooth and symmetric. In addition we have
g((X, Y ),(X, Y )) = kXk2+kYk2− kduπ2(X)k2=kXk2+kYk2− kdwη2(Y)k2≥0
for every (X, Y )∈T(u,w)(Uπ2×η2W) since πUand ηWare Riemannian submersions, and
it is zero if and only if both Xand Yare zero vectors. Hence (Uπ2×η2W, g) defines a
Riemannian manifold. Now we claim that the projections pUand pWare Riemannian
submersions. We have
ker(d(u,w)pU) = {(0, Y )∈TuU×TwW|dwη2(Y) = 0},
2We were informed that this idea has been used already in [W83] and [dHF18].
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 14
so its orthogonal complement is given by
H(u,w)=(X, Y )∈T(u,w)(Uπ2×η2W)|gW(Y, Z) = 0 ∀Z∈ker(dwη2).
Using the fact that ηWis a Riemannian submersion, for every two vectors (X1, Y1) and
(X2, Y2) in H(u,w)we have:
g((X1, Y1),(X2, Y2)) = gU(X1, X2) + gW(dwη2(Y1),dwη2(Y2)) −g2(dwη2(Y1),dwη2(Y2))
=gU(X1, X2) = gU(d(u,w)pU(X1, Y1),d(u,w)pU(X2, Y2))
which proves that pUis a Riemannian submersion. It has connected fibers since for
every u∈U, we have p−1
U(u) = {u} × η−1
2(π2(u)), which is connected. Similarly it is
shown that pWis a Riemannian submersion with connected fibers. These two Riemannian
submersions are surjective by construction. So the Riemannian manifold (Uπ2×η2W, g)
and the surjective Riemannian submersions with connected fibers π1◦pUand π2◦pW
define a Morita equivalence between (N1, g1,F1) and (N3, g3,F3). This completes the
proof.
Although the leaf space of an SRF may not be topologically well-behaved, it inherits a
pseudo-metric space structure from the Riemannian metric. For every two leaves L1and
L2of an SRF (M, g, F), the distance between them is defined by
dM/F(L1, L2) := inf (N
X
i=1
Lg(γi)).
Here the infimum is taken over all discrete paths (γ1,··· , γN) joining L1and L2, i.e. a
family of piecewise smooth curves γ1,··· , γN: [0,1] →Mfor some positive integer N,
such that γ1(0) ∈L1,γN(1) ∈L2and γi(1) and γi+1(0) belong to the same leaf for each
i= 1,··· , n −1.
As a corollary of Remark 3.2 and Theorem 2.1, there exists a homeomorphism between the
leaf spaces of Morita equivalent module SRFs. The following theorem is the Riemannian
counterpart of part (i) of Theorem 2.1:
Theorem 3.2. Let (N1, g1,F1)and (N2, g2,F2)be Morita equivalent module SRFs. Then
the homeomorphism between the leaf spaces given in Theorem 2.1 is distance preserving.
Proof. Assume that (M1, g1,F1)∼M E (M2, g2,F2) is given by πi: (N, h)→(Mi, gi)
for i= 1,2. Let L1and L′
1be two leaves in (M1, g1,F1) and let L2and L′
2be their
corresponding leaves in (M2, g2,F2). Consider a discrete path (γ1,··· , γn) joining L1
and L′
1. By lifting each γiinto finitely many piecewise smooth horizontal paths, one
obtains a discrete path (η1,...,ηn′) for some n′≥non njoining π−1
1(L1) and π−1
1(L′
1)
with the same length as (γ1,··· , γn)—since the lifts are horizontal with respect to the
Riemannian submersion π1. Since π2is a Riemannian submersion, (π2(η1),··· , π2(ηn′))
is a discrete path joining L2and L′
2with a length which is smaller than or equal to the
length of (γ1,··· , γn)—since the lifts are not necessarily horizontal with respect to π2.
Consequently
dM1/F1(L1, L′
1)≥dM2/F2(L2, L′
2).
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 15
Similarly dM2/F2(L2, L′
2)≥dM1/F1(L1, L′
1), which implies dM1/F1(L1, L′
1) = dM1/F2(L2, L′
2).
This proves the statement.
To define a category SRF of module SRFs one needs to specify their morphisms. We are
not going to do this in the present article. But since any good notion of such morphisms
should include Riemannian submersions which satisfy that the pullback of the SF on
the base agrees with the SF on the total space, we define the following full subcategory
SRF0:
Definition 3.5. The category SRF0has module SRFs as its objects and Riemannian
submersions π: (N, h, FN)→(M, g, FM)satisfying π−1FM=FNas its morphisms.
4. I-Poisson manifolds
In what follows (P, {·,·}) stands for a Poisson manifold and C∞denotes the sheaf of
smooth functions on P. For every H∈C∞(P), the Hamiltonian vector field of Hand
its Hamiltonian flow are denoted by XHand Φt
H, respectively. See [dSW99, LGPV13,
CFM21] for an introduction to Poisson geometry.
Definition 4.1. A subsheaf Iof the sheaf C∞of smooth functions on Pwhich is closed
under multiplication by smooth functions is called locally finitely generated if for every
q∈Pthere exist an open neighborhood U⊂Pcontaining qand finitely many functions
g1, ..., gN∈C∞(U)such that I(V) = hg1|V, ..., gN|ViC∞(V)for every open subset V⊂U.
Definition 4.2. An I-Poisson manifold is a triple (P, {·,·} ,I)where Iis a subsheaf
of smooth functions on Pwhich is closed under multiplication by smooth functions, lo-
cally finitely generated, and for every open subset U⊂P,I(U)⊂C∞(U)is a Poisson
subalgebra, i.e.
{I(U),I(U)} ⊂ I (U).
Example 8. Let (P, {·,·})be a Poisson manifold and C⊂Ma coisotropic submanifold.
Then the triple (P, {·,·} ,IC)where IC(U) := {f∈C∞(U) : f|C∩U≡0}for every open
subset U⊂Pdefines an I-Poisson manifold. Note that the Hamiltonian vector fields
of functions in ICare tangent to Cand they are closed under the Lie bracket, hence
defining an SF on C. If this SF is regular and the quotient map π:C→Cred to the leaf
space Cred is a smooth submersion, then Cred inherits a Poisson bracket {·,·}red such that
π∗{f, g}red ={F, G} |C, where Fand Gare smooth functions on Psatisfying F|C=π∗f
and G|C=π∗g. This process is called the coisotropic reduction [MR86].
Remark 4.1. The notion of an I-Poisson manifold is motivated by generalizing the previ-
ous example to a potentially singular setting, where the quotient Cred does not need to exist
as a manifold and the reduction is performed algebraically, using the following definitions.
Definition 4.3. The Poisson normalizer of an I-Poisson manifold (P, {·,·} ,I), denoted
by N(I), is defined as
N(I) := {f∈C∞(P): {f|U,I(U)} ⊂ I(U)for every open subset U}.
As a consequence of the Jacobi identity, the Poisson nomalizer of an I-Poisson manifold
is a Poisson algebra. Due to the fact that I(P)⊂N(I)⊂C∞(P) is a Poisson subalge-
bra, the quotient N(I)/I(P) is a Poisson algebra as well. This motivates the following
definition:
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 16
Definition 4.4. The reduced Poisson algebra of the I-Poisson manifold (P, {·,·} ,I)is
defined to be the Poisson algebra R(I) := N(I)/I(P).
Remark 4.2. This is a straightforward generalization of the set of Dirac observables
[D50]. The algebra R(I)also appears in [SW83] as an algebraic method of reducing
Hamiltonian G-spaces with singular moment maps.
Example 9. If Cred in Example 8 is smooth, then R(IC)is isomorphic to the Poisson
algebra C∞(Cred).
Example 10. Let Gbe a Lie group acting on a Poisson manifold (P, {·,·})by Poisson
diffeomorphisms and an equivariant moment µ:P→g∗. Following [SW83], the subsheaf
I ⊂ C∞generated by smooth functions hµ, giis a Poisson subalgebra. Then we have
R(I)∼
=(C∞(P)/I)G.
Moreover if Gis compact, then Proposition 5.12 in [AGJ90] states that
R(I)∼
=C∞(P)G/IG.
Example 11. Let P=T∗Rn,n > 1, with coordinates (q1,...,qn, p1,...,pn)and I ⊂ C∞
the subsheaf generated by the n(n−1)/2functions qipj−qjpifor 1≤i < j ≤n. This is
a special case of Example 10 for the diagonal action of G=SO(n)on T∗Rn. We have
R(I)∼
=W∞(D),
where D⊂T∗Rnis defined by
D:= (x1, x2, x3)∈R3|x2
1+x2
2=x2
3and x3≥0
and W∞(D)stands for the smooth functions on Din the sense of Whitney, i.e. the
restriction of C∞(R3)to D. For more details and proofs see Theorem 5.6and Example
5.11(a)of [AGJ90].
The Poisson bracket on W∞(D)can be understood as follows: Identify R3with the Pois-
son manifold so(2,1)∗and, simultaneously, with 2+1 dimensional Minkowski space. The
symplectic leaves of so(2,1)∗then consist of spacelike vectors of a fixed Minkowski norm
(one-sheeted hyperboloids), null vectors decompose into the origin, the forward light cone,
and the backward light cone as three distinct leaves, and finally timelike vectors of a fixed
norm yield two leaves each (two-sheeted hyperboloids). Then restriction to Dcorresponds
precisely to restricting to the forward lightcone and the origin in this Minkowski space.
This bracket does not depend on the extension of a function on Dto the ambient space
since Dis the collection of (two) symplectic leaves.
Remark 4.3. If in the previous example one restricts to the polynomial functions, such
that I ⊂ R[q1,...,qn, p1, . . . , pn], one finds
R(I)∼
=S•(so(2,1)) /hx2
1+x2
2−x2
3i,
i.e. the polynomial functions on so(2,1)∗modulo the ideal generated by the quadratic
Casimir. So one looses the restriction x3≥0that one finds in the smooth setting.
Definition 4.5. Adynamical I-Poisson manifold denoted by (P, {·,·} ,I, H)consists of
an I-Poisson manifold (P, {·,·} ,I)and a Hamiltonian function H∈N(I). Its reduction
is defined to be the pair (R(I),[H]) where [H]∈ R(I)is the equivalence class of H.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 17
The following proposition reveals the main property of dynamical I-Poisson manifolds.
Proposition 4.1. Let (P, {·,·} ,I, H)be a dynamical I-Poisson manifold. Then the
Hamiltonian flow of Hlocally preserves I, i.e. for every q0∈Pthere exists an open
neighborhood U⊂Paround q0such that Φt
H|Uis defined for t∈(−ǫ, ǫ)and
(9) (Φt
H)∗I(Φt
H(U)) = I(U).
In the case that the Hamiltonian vector field XHis complete, this implies that, for all
t∈R, one has Φt
H
∗I ◦ Φt
H=Iand, in particular, that the ideal I(P)is preserved,
(Φt
H)∗I(P) = I(P).
Proof. Choose an open neighborhood W⊂Paround q0where I(W) is generated by
finitely many functions g1,...,gNfor some positive integer N. Then by the existence and
uniqueness theorem for ODEs there exist an open subset U⊂Wcontaining q0and an
interval (−ǫ, ǫ), ǫ > 0, such that Φt
H|Uis defined for t∈(−ǫ, ǫ). By Definition 4.3, there
exist functions λb
a∈C∞(U), a, b = 1,...,N, such that:
(10) {H, ga}=
N
X
b=1
λb
agb.
Using Equation (10), we obtain:
d
dt (Φt
H
∗ga) = Φt
H
∗{H, ga}=
N
X
b=1
(Φt
H
∗λb
a)(Φt
H
∗ga),
which defines a time-dependent linear ODE. By solving this equation, we obtain:
(11) Φt
H
∗ga(p) =
N
X
b=1 OE Zt
0Φt
H
∗λdtb
a
gb(p),
where λ◦Φt
His the matrix (Φt
H
∗λb
a)N
a,b=1 and OE stands for the time ordered exponential.
This proves the inclusion (Φt
H)∗I(Φt
H(U)) ⊂ I(U).
To prove equality, we first observe that the inclusion yields also (Φ−t
H)∗I(U)⊂ I(Φt
H(U))
for every t∈(−ǫ, ǫ). Thus, for every f∈ I(U), one has (Φ−t
H)∗f∈ I(Φt
H(U)). But on the
other hand, we have the obvious identity
f= (Φt
H)∗(Φ−t
H)∗f,
and therefore f∈(Φt
H)∗I(Φt
H(U)).
The following example shows that the condition of being locally finitely generated in the
definition of I-Poisson manifolds is crucial for Proposition 4.1 to hold true:
Example 12. Consider the Poisson manifold M=T∗R∼
=R2with coordinates (q, p)and
standard Poisson bracket
{f, g}=∂f
∂p
∂g
∂q −∂f
∂q
∂g
∂p .
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 18
Let Ibe the subsheaf of C∞vanishing on {q < 0} ⊂ M, which is not locally finitely
generated around every point on the p-axis, but still closed under the Poisson bracket.
Then the coordinate function pis an element of N(I)since Xp=∂
∂q preserves I. But
the Hamiltonian flow of Xpis given by Φt
Xp(q, p) = (q+t, p), which evidently does not
preserve Iif t > 0.
In order to define the category of I-Poisson manifolds, we introduce a notion of morphisms
and show that they can be composed:
Definition 4.6. Let ϕ: (P1,{·,·}1,I1)→(P2,{·,·}2,I2)be a smooth map between two
I-Poisson manifolds. We call it an I-Poisson map if the following three conditions are
satisfied:
ϕ∗(I2(P2)) ⊂ I1(P1),(12)
ϕ∗N(I2)⊂N(I1),(13)
{ϕ∗f, ϕ∗g}1−ϕ∗{f, g}2∈ I1(P1)∀f, g ∈N(I2).(14)
For dynamical I-Poisson manifolds we add the condition ϕ∗H2−H1∈ I1.
Proposition 4.2. The composition of two I-Poisson maps is an I-Poisson map.
Proof. Consider the following I-Poisson maps:
ϕ: (P1,{·,·}1,I1)→(P2,{·,·}2,I2)
ψ: (P2,{·,·}2,I2)→(P3,{·,·}3,I3).
Equations (12) and (13) for ψ◦ϕfollow directly from those equations for ψand ϕ. It is
thus enough to verify Equation (14) for the composition. For all f, g ∈N(I3) we have :
{f◦ψ◦ϕ, g ◦ψ◦ϕ}1− {f, g}3◦ψ◦ϕ
={(f◦ψ)◦ϕ, (g◦ψ)◦ϕ}1− {f◦ψ, g ◦ψ}2◦ϕ
+ ({f◦ψ, g ◦ψ}2− {f, g}3◦ψ)◦ϕ
∈ I1(P1) + ϕ∗I2(P2)⊂ I1(P1),
where we used Equations (12) and (14) for ϕand Equations (13) and (14) for ψin the last
line of the proof. A similar computation shows that morphisms of dynamical I-Poisson
manifolds can be composed as well.
Definition 4.7. The category of I-Poisson manifolds and dynamical I-Poisson man-
ifolds, IPois and dynIPois, respectively, consists of (dynamical) I-Poisson manifolds
together with I-Poisson maps.
Remark 4.4. Note that the three conditions in Definition 4.6 are the minimal condi-
tions for the map ϕ∗to induce a morphism of Poisson algebras ˜ϕ:R(I2)→ R(I1). In
particular, we obtain a functor: IPois op →PoisAlg, the category of Poisson algebras.
Remark 4.5. Viewing Poisson manifolds (P, {·,·})as I-Poisson manifolds (P, {·,·} ,0),
I-Poisson maps are precisely Poisson maps. This identifies the category of Poisson man-
ifolds Pois with a full subcategory of IPois.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 19
Remark 4.6. There is a functor from IPois to C3Alg, the category of coisotropic triples
of algebras as introduced in [DEW19]. On the level of objects, one assoicates the triple
(C∞(P), N(I),I(P)) to every I-Poisson manifold (P, {·,·} ,I), while a morphism ϕin
our sense gives rise to a morphism ϕ∗in C3Alg due to the first two defining conditions
(12) and (13).
5. Singular (Riemannian) foliations through I-Poisson manifolds
Let Mbe a smooth manifold. We denote by C∞
k(T∗M)⊂C∞(T∗M) the algebra of
homogeneous polynomials of degree kin the fiber coordinates of T∗Mwith coefficients
in C∞(M). Every vector field X∈X(M) defines an element X∈C∞
1(T∗M) on the
cotangent bundle of M, defined as
X(q, p) = hp, X |qi
for every (q, p)∈T∗M, where q∈Mand p∈T∗
qMand h·,·i denotes the canonical
pairing. This construction can be naturally extended to the sections of Γ(Sk(T M )) to
obtain elements in Ck(T∗M).
Lemma 5.1. Let X, Y ∈X(M)be two vector fields on M. Then
(15) X, Y T∗M=[X, Y ],
where {·,·}T∗Mis the canonical Poisson bracket on T∗Mand [·,·]is the Lie bracket of
vector fields on M.
Proof. Let (q1, ..., qn) be a local coordinate system on M, and (q1, ..., qn, p1, ..., pn) the
corresponding canonical local coordinates on T∗M. In this coordinate system Xand Y
can be written as X=Pn
i=1 Xipiand Y=Pn
i=1 Yipi, where Xi, Y iare the components
of V, W in the above coordinate system. The following calculation proves the lemma:
X, Y T∗M=
n
X
i=1
(
n
X
j=1
Xj∂Y i
∂qj−Yj∂Xi
∂qj)pi=
n
X
i=1
[X, Y ]ipi=[X, Y ].
Lemma 5.1 and the Leibniz rule for the Lie derivative of tensor fields imply:
Corollary 5.1. Let Sbe an element of Γ(Sk(T M )) for some k≥0, and Sbe its corre-
sponding element in C∞
k(T∗M). Then for every vector field X∈X(M)we have:
(16) X, ST∗M=LXS .
Now let (M, F) be a foliated manifold. Define a C∞(T∗M)-submodule JF⊂C∞
c(T∗M)
by
JF:= hX:X∈ FiC∞
c(T∗M).
Then we define the sub-presheaf IFof the sheaf of smooth functions on T∗Mby
IF(U) := {f∈C∞(U) : ρf ∈ JF∀ρ∈C∞
c(U)}(17)
for every open subset U⊂T∗M.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 20
Proposition 5.1. The presehaf IFdefined in Equation (17) is a subsheaf of the sheaf of
smooth functions on T∗M.
Proof. The locality of IFis evident, since IFis a sub-presheaf of the sheaf of smooth
functions on T∗M. To verify the gluing property, let {Ui}∞
i=1 be an arbitrary open cover
of T∗Mand let f∈C∞(T∗M) be such that f|Ui∈ IF(Ui) for every positive integer i. We
prove that f∈ IF(T∗M) as follows: it is enough to show that for every ρ∈C∞
c(T∗M),
we have ρf ∈ JF. Since supp(ρ) is compact, it can be covered by finitely many open
subsets Ui1,...,UiNin {Ui}∞
i=1. Choose a partition of unity σ0, σi1,...,σiNsubordinate
to the open cover {U0:= T∗M\supp(ρ), Ui1,...,UiN}of T∗Mand write
ρf =
N
X
k=1
ρσikf|Uik.
The latter implies that ρf ∈ JF, since by definition of IF(Uik), for each k= 1,...,N we
have ρσikf|Uik∈ JF.
We prove that the sheaf IFsatisfies the properties of Definition 4.2, in the following
lemmas:
Lemma 5.2. For every open subset U⊂T∗Mwe have
{IF(U),IF(U)}T∗M⊂ IF(U)
Proof. Let f, g ∈ IF(U). It is enough to show that for every ρ∈C∞
c(U) we have
ρ{f, g} ⊂ JF. Choose a compactly supported function σ∈C∞
c(U) such that σ|supp(ρ)≡1.
One obtains
ρ{f, g}T∗M={σf, ρg}T∗M− {σf, ρ}T∗Mg−ρf {σ, g}T∗M∈ JF,
since the first term belongs to JFby Lemma 5.1, the second term is inside JFby Definition
of IF(U), and the last term vanishes identically. Consequently {f, g}T∗M∈ IF(U).
Lemma 5.3. Let U⊂Mbe an open subset such that ι−1
UF=hX1,...,XNiC∞
c(U)for
finitely many vector fields X1,...,XN∈X(U). Then
IF(V) = hX1|V, . . . , XN|ViC∞(V),
for every open subset V⊂T∗U.
Proof. We first prove that hX1|V, . . . , XN|ViC∞(V)⊂ IF(V). Let PN
a=1 λaXa|Vbe an
element of hX1|V, . . . , XN|ViC∞(V)and take an arbitrary ρ∈C∞
c(V). By choosing a
compactly supported function h∈C∞
c(U) such that h|supp(ρ)≡1 (when viewing has an
element of C∞
0(T∗U)), we have
ρ
N
X
a=1
λaXa|V=
N
X
a=1
ρλahXa∈ JF,
since ρλa∈C∞
c(V) and hXa∈ F for all a= 1,...,N. To prove equality, let f∈ IF(V).
Choose a partition of unity {ρi}∞
i=1 subordinate to a locally finite cover {Vi}∞
i=1 of V. Since
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 21
for every iwe have ρif∈ JFand V⊂T∗U, there exist functions λ1
i,...,λN
i∈C∞
c(T∗U)
such that
ρif=
N
X
a=1
λa
i|VXa|V.
This implies
f=
∞
X
i=1
ρif=
∞
X
i=1
N
X
a=1
λa
i|VXa|V
=
N
X
a=1 ∞
X
i=1
λa
i|V!Xa|V,
which completes the proof.
Corollary 5.2. Let (M, F)be a foliated manifold. Then the triple (T∗M, {·,·}T∗M,IF)
is an I-Poisson manifold.
For every Riemannian manifold (M, g), its cotangent bundle T∗Mcarries a natural Hamil-
tonian function Hg:
Hg(q, p) = 1
2hp, g−1
♭(p)i
for every (q, p)∈T∗M, where g♭:TqM→T∗
qMis the musical isomorphism v7→ g(v, ·). In
local Darboux coordinates this becomes Hg(q1, ..., qn, p1, ..., pn) = 1
2Pn
i,j=1 gij pipjwhere
the matrix (gij ) is the inverse to the matrix of the Riemannian metric (gij ) in the coordi-
nates (q1, ..., qn). Equivalently, we can define also Hgusing the isomorphism Γ(S2(T M)) ∼
=
C∞
2(T∗M), under which it becomes identified with g−1:= Pn
i,j=1 gij ∂i⊙∂j, i.e. Hg=1
2g−1.
The following fact about Hgis standard knowledge, which we still prove for completeness.
Proposition 5.2. The Hamiltonian flow of Hgis the image of the geodesic flow under
the musical isomorphism, i.e. for every geodesic γ: (−ǫ, ǫ)→Mand every t∈(−ǫ, ǫ),
we have
Φt
Hg(γ(0), g( ˙γ(0),·)) = (γ(t), g( ˙γ(t),·)) .
Proof. Assume that (q1, . . . , qn) is a normal coordinate system centered at q∈M, i.e.
gij(q) = δij and ∂kgij (q) = 0 for i, j, k = 1,...,n. For every p∈T∗
qMwe have
XHg(q, p) =
n
X
i=1
pi
∂
∂qi|q.
Let γ: (−ǫ, ǫ)→Mbe a geodesic passing through qat t= 0; in particular, ¨qi(0) = 0.
Then ((γ(t), g( ˙γ(t),·)) is a curve on T∗Mpassing through (q, p) = ((γ(0), g( ˙γ(0)),·) at
t= 0; in local coordinates, ((γ(t), g( ˙γ(t)),·) = (q1(t),...,qn(t), p1(t),...,pn(t)) where
pi(t) = Pn
j=1 gij(q(t)) ˙qj(t). Then, since ˙pi(0) = 0, we have
d
dt|t=0((γ(t), g( ˙γ(t),·)) =
n
X
i=1
˙qi(0) ∂
∂qi|q.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 22
On the other hand, ˙qi(0) = g( ˙γ(0),∂
∂qi|q) = pi(0), which indeed gives
XHg(q, p) = d
dt|t=0((γ(t), g( ˙γ(t),·)) .
Lemma 5.4. Let (M, F)be a foliated manifold. We have
N(IF) = {f∈C∞(T∗M) : {f, JF}T∗M⊂ JF}.
Proof. The inclusion N(IF)⊂ {f∈C∞(T∗M) : {f, JF}T∗M⊂ JF}is satisfied by Def-
inition 4.3 and the fact that JFis equal to the set of compactly supported elements in
IF(T∗M). Now let f∈C∞(T∗M) be such that {f, JF}T∗M⊂ JF. Let U⊂T∗Mbe an
open subset and g∈ IF(U). For every ρ∈C∞
c(U) we have
ρ{f|U, g}T∗M={f, ρg}T∗M− {f, ρ}T∗Mg∈ JF,
since ρg ∈ JFand {f, ρ}T∗Mis compactly supported in U. The definition of IF(U) then
implies that {f|U,IF(U)}T∗M⊂ IF(U). Since Uis arbitrary, we obtain f∈N(IF).
Now we can state an equivalent definition of module SRFs through I-Poisson geometry.
Proposition 5.3. A singular foliation Fon a Riemannian manifold (M, g)defines a
module SRF, if and only if
(18) Hg∈N(IF).
Proof. Assume that (M, g, F) is a module SRF. By Lemma 3.1, for every X∈ F we have
LXg−1∈X(M)⊙ F .
Using the isomorphism Γ(S2(T M )) ∼
=C∞
2(T∗M) and Corollary 5.1, we obtain
X, HgT∗M=1
2LXg−1∈X(M)⊙ F ,
which together with the Leibniz rule imply {JF, Hg}T∗M⊂ JF. Lemma 5.4 then implies
that Hg∈N(IF). Conversely assume that Hg∈N(IF). After Proposition 3.1 we can
assume that F=hX1, . . . , XNiC∞
c(M). Using Lemma 5.3, Hg∈N(IF) implies that for
every a= 1,...,N there exist functions λ1
a,...,λN
a∈C∞
1(T∗M) such that
1
2LXag−1=Xa, Hg=
N
X
b=1
λb
aXb,
where we used Corollary 5.1 for the first equality. Lemma 3.2 then implies that LXg−1∈
X(M)⊙ F.
Now we are able to present the proof of Proposition 3.2.
Proof. [Proposition 3.2] Let (M, g, F) be a module SRF. As the statement is local, we
can assume that Fis finitely generated, i.e. there exist vector fields X1, ..., XN∈X(M)
for some positive integer N, such that F=hX1, ..., XNiC∞
c(M). By Lemma 5.3, IFis
generated by functions X1, ..., XN. By Proposition 5.3, for every a= 1,...,N there exist
functions λ1
a,...,λN
a∈C∞
1(T∗M) such that
Hg,Xa=
N
X
b=1
λb
aXb.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 23
Assume that γ: (−ǫ, ǫ)→Mis a geodesic such that ˙γ(0) ⊥Fγ(0), i.e. the geodesic
is orthogonal to the leaf at t= 0. Then the ideal IF(T∗M) vanishes at (q0, p0) =
(γ(0), g( ˙γ(0), .)) ∈T∗M. Since Φt
Hg(q0) is defined for −ǫ < t < ǫ, for every r < ǫ
there exists an open neighborhood U⊂Mof q0such that Φt
Hgis defined for t∈(−r, r)
on U. According to Proposition 5.2,
g( ˙γ(t), Xa(γ(t))) = Xa(γ(t), gγ(t)( ˙γ(t), .))
=Xa◦Φt
Hg|U(γ(0), gγ(0)( ˙γ(0), .))
= Φt
Hg|∗
UXa(q0, p0)
for a= 1,...,N. But now, according to Proposition 4.1, the function (Φt
Hg|U)∗Xa|Φt
Hg(U)
is an element in IF(U) for t∈(−r, r). This means that for tin this interval, ˙γ(t)⊥Fγ(t).
As r < ǫ is arbitrary, the proof is complete.
6. The functor Φand reduction
At the end of Section 3 we introduced the category SRF0and in Section 4 we introduced
the category of I-Poisson manifolds IPois. In this section we will provide a functor from
the first to the second category, by sending a module SRF (M, g, F) to the I-Poisson
manifold (T∗M, {·,·} ,IF) and every surjective Riemannian submersion π: (M1, g1)→
(M2, g2) to the bundle map ϕπ:= (g2)♭◦dπ◦(g2)−1
♭: (T∗M1,{·,·}1)→(T∗M2,{·,·}2), see
Theorem 6.1 below. ϕπis precisely the map making the following diagram commute:
T M1
T M2
T∗M1
T∗M2
dπ
(g2)♭
(g1)♭
ϕπ
The map ϕπis not a Poisson map in general:3
Example 13. Let π:R3→R2be the projection (x, y, z)7→ (x, y)with coordinates (x, y, z)
and (x, y)on R3and R2, respectively. The following metric tensors on R3and R2,
g3: = dx⊗dy+ (1 + x2)dy⊗dy−xdy⊗dz−xdz⊗dy+ dz⊗dzand
g2: = dx⊗dy+ dy⊗dy ,
respectively, turn πinto a Riemannian submersion. In the standard coordinates (x, y, z, px, py, pz)
and (x, y, px, py)on T∗R3and T∗R2, respectively, the map ϕπis given by
ϕπ(x, y, z, px, py, pz) = (x, y, px, py+xpz)
which is not a Poisson map, since {ϕ∗
πpx, ϕ∗
πpy}={px, py+xpz}=pz6= 0.
In the last example the obstruction for ϕπto be a Poisson map is that the horizontal
distribution of the Riemannian submersion π, which is generated by vector fields ∂
∂x and
∂
∂y +x∂
∂z , is not integrable; the corresponding connection has curvature.
3In contrast to what is claimed in [BWY21].
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 24
The map ϕπstill preserves the Poisson bracket up to some ideal of functions Iker dπ:
Definition 6.1. Let π:M1→M2be a submersion. The subsheaf of smooth functions
Iker dπon T∗M1is defined as the corresponding sheaf IFfor the regular foliation F:=
Γc(ker dπ).
When there is no ambiguity, for simplicity, we denote the ideal Iker dπ(T∗M1) by Iker dπ.
Remark 6.1. It is not difficult to see that for every open subset U⊂T∗M, The ideal
Iker dπ(U)is the vanishing ideal of the submanifold Ann(ker dπ)∩U⊂U. Here Ann(ker dπ)
stands for the annihilator of the subbundle ker dπ⊂T M1. Moreover, since Ann(ker dπ)
is an embedded submanifold, we have:
(19) C:= {(q, p)∈T∗M1:f(q, p) = 0 ∀f∈ Iker dπ} ≡ Ann(ker dπ).
Lemma 6.1. Let π: (M1, g1)→(M2, g2)be a Riemannian submersion. Then for every
f, g ∈C∞(T∗M2):
{f◦ϕπ, g ◦ϕπ}1− {f, g}2◦ϕπ∈ Iker dπ
(20)
{f◦ϕπ,Iker dπ}1⊂ Iker dπ.(21)
Proof. Choose local Darboux coordinates (qi
2, p2
i) on T∗M2and (qi
1, qα
1, p1
i, p1
α) on T∗M1,
such that qi
2◦π=qi
1(this is possible since πis assumed to be a submersion). In particular,
Ikerdπis generated by the momenta p1
α. Now note that at every point q∈M1,
dqπ(∂
∂qi
1|q) = ∂
∂qi
2|π(q),
since for every function f∈C∞(M2)
dqπ(∂
∂qi
1|q)·f=∂(f◦π)
∂qi
1(q) = ∂f
∂qj
2
(π(q))∂(qj
2◦π)
∂qi
1(q) = ∂f
∂qi
2(π(q)) .
In particular, since ϕπis a bundle map, we have
(22) qi
1=qi
2◦ϕπ.
Next we prove that upon restriction to the vanishing submanifold Cof Ikerdπ,
(23) C={(q, p)∈T∗M1:f(q, p) = 0 ∀f∈ Iker dπ} ≡ Ann(ker dπ),
one has p1
i=p2
i◦ϕπ. Indeed, let (q, p) be a point in T∗M1and X= (g1)−1
♭(p). Then
p1
i(q, p) = p(∂
∂qi
1|q) = g1(X, ∂
∂qi
1|q) = g1(X, (∂
∂qi
1|q)H) + g1(X, (∂
∂qi
1|q)V)
where ( ∂
∂qi
1|q)Hand ( ∂
∂qi
1|q)Vare the horizontal and vertical parts of the vector ∂
∂qi
1|qwith
respect to g1, respectively. Using that πis a Riemannian submersion and that there exist
functions Aαsuch that ( ∂
∂qi
1|q)V=PαAα(q)∂
∂qα
1|q, this implies:
p1
i(q, p) = g2(dqπ(X),∂
∂qi
2|π(q)) + X
α
g1(X, Aα(q)∂
∂qα
1|q).
Consequently, by definition of ϕπ,
p1
i(q, p) = p2
i◦ϕπ(q, p) + X
α
Aα(q)p1
α(q, p)
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 25
and thus
(24) p1
i|C=p2
i◦ϕπ|C.
Now for every f∈C∞(T∗M2), we have:
(25) ∂(f◦ϕπ)
∂qi
1(q, p) = ∂f
∂qj
2
(ϕπ(q, p)) ∂(qj
2◦ϕπ)
∂qi
1(q, p) + ∂f
∂p2
j(ϕπ(q, p)) ∂(p2
j◦ϕπ)
∂qi
1(q, p).
Since ∂
∂qi
1|Cis tangent to C, for every point (q, p)∈ C, we may use Equation (24) to
transform Equation (25) into:
∂(f◦ϕπ)
∂qi
1
(q, p) = ∂f
∂qj
2
(ϕπ(q, p)) ∂qj
1
∂qi
1
(q, p) + ∂f
∂p2
j(ϕπ(q, p)) ∂p1
j
∂qi
1
(q, p)(26)
=∂f
∂qi
2(ϕπ(q, p)) .
In a similar way, using the chain rule and that ∂
∂p1
i|Cand ∂
∂qα
1|Care tangent to C, for every
function f∈C∞(T∗M2) and every (q, p)∈ C, one finds
∂(f◦ϕπ)
∂p1
i(q, p) = ∂f
∂p2
i(ϕπ(q, p)) ,(27)
∂(f◦ϕπ)
∂qα
1(q, p) = 0 .(28)
For every two functions f, g ∈C∞(T∗M2), upon restriction to |Cwe have:
{f◦ϕπ), g ◦ϕπ}1|C=X
i∂(f◦ϕπ)
∂p1
i
∂(g◦ϕπ)
∂qi
1
−∂(g◦ϕπ)
∂p1
i
∂(f◦ϕπ)
∂qi
1|C
+X
α∂(f◦ϕπ)
∂p1
α
∂(g◦ϕπ)
∂qα
1−∂(g◦ϕπ)
∂p1
α
∂(f◦ϕπ)
∂qα
1|C
=X
i∂(f◦ϕπ)
∂p1
i
∂(g◦ϕπ)
∂qi
1−∂(g◦ϕπ)
∂p1
i
∂(f◦ϕπ)
∂qi
1|C
=X
i∂f
∂p2
i◦ϕπ|C∂g
∂qi
2◦ϕπ|C−∂g
∂p2
i◦ϕπ|C∂f
∂qi
2◦ϕπ|C
={f, g}2◦ϕπ|C
Here in the first equality we used just the definition of the Poisson bracket, in the second
one we used Equation (28), thereafter Equations (26) and (27), and finally again the
definition of the bracket. Note that every function on T∗M1vanishing on Cis an element
of Iker dπ, which proves Equation (20).
Equation (28) implies Equation (21) as well, since Iker dπis locally generated by coordinate
functions pαfor α= 1, . . . , k, and we have
{pα, f ◦ϕπ}1|C=∂(f◦ϕπ)
∂qα
1|C= 0 .
which gives {f◦ϕπ, pα}1∈ Iker dπ.
Corollary 6.1. The restriction ϕπ|C:C → T∗M2is a surjective submersion. It coincides
with the projection to the leaf space for the coisotropic reduction of C ⊂ T∗M1.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 26
Proof. Choosing the same local coordinates as in the proof of Lemma 6.1, (qi
1, qα
1, p1
i) give
local coordinates for Cand Equations (22) and (24) ensure that ϕπ(qi
1, qα
1, p1
i) = (qi
1, p1
i).
To study the obstruction for ϕπto be a Poisson map, we first prove the following lemma
which describes the horizontal distribution in terms of the map ϕπ:
Lemma 6.2. Let π: (N, h)→(M, g)be a Riemannian submersion and let Xbe a vector
field on M. Then the horizontal lift of Xis given by a vector field Von Nsatisfying
(29) V= (ϕπ)∗X ,
which is an element in C∞
1(T∗N).
Proof. Define V∈X(N) by V:= (ϕπ)∗(X)∈C∞
1(T∗M). Using Corollary 5.1 and
Lemma 6.1 for every function f∈C∞(M), we have
V·π∗f=V , π∗f=(ϕπ)∗X, π∗f
=(ϕπ)∗X, (ϕπ)∗f= (ϕπ)∗X, f
=π∗(X·f),
which means that Vis projectable and projects to X. In addition, for every vertical
vector v∈ker dqπand i= 1,...,n we have
h(v, V |q) = V(h♭(v)) = (ϕπ)∗X(h♭(v))
=X(ϕπ◦h♭(v)) = X(g♭◦dqπ(v))
= 0 ,
showing that Vis the horizontal lift of X.
The following identifies the obstruction for ϕπto be a Poisson map:
Proposition 6.1. Let π: (N, h)→(M, g)be a Riemannian submersion. Then the map
ϕπ:= g♭◦dπ◦h−1
♭is a Poisson map if and only if the horizontal distribution H ⊂ T M
of πis integrable.
Proof. Let {fi}n
i=1 be a local orthonormal frame around a point q∈Mand {ei}n
i=1
their horizontal lifts. By Lemma 6.2 we have ei= (ϕπ)∗fifor i= 1,...,n. If ϕπis a
Poisson map, the family of functions (ϕπ)∗(fi)∈C∞
1(T∗N) is closed under the Poisson
bracket, and consequently the horizontal distribution locally generated by vector fields ei
is integrable. This proves the if part of the proposition.
Conversely assume that His integrable. It is enough to check the condition of being
a Poisson map on smooth functions in C∞
0(T∗M)LC∞
1(T∗M) only. First, for every
f, g ∈C∞
0(T∗M) we have {f◦ϕπ, g ◦ϕπ}T∗N={f, g}T∗M= 0. Second, for every X∈
C∞
1(T∗M) and f∈C∞
0(T∗M) we have
X◦ϕπ, f ◦ϕπT∗N=XH·(f◦ϕπ)
= (X·f)◦ϕπ
=X, fT∗M◦ϕπ.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 27
Finally, by Lemma 6.2 and integrability of H, for every X , Y ∈C∞
1(T∗M) one obtains
X◦ϕπ, Y ◦ϕπT∗N=[XH, Y H]
=[X, Y ]H
=[X, Y ]◦ϕπ
=X, Y T∗M◦ϕπ.
Lemma 6.3. Let π: (N, h)→(M, g)be a Riemannian submersion. Then
Hh−Hg◦ϕπ∈ Iker dπ.
Proof. It is enough to show that the left-hand side vanishes on C, defined in Equation
(19). For every (q, p)∈ C, we have
Hh(q, p) = 1
2p, (h♭)−1(p)=1
2p, [(h♭)−1(p)]H
=1
2h((h♭)−1(p),[(h♭)−1(p)]H)
=1
2h([(h♭)−1(p)]H,[(h♭)−1(p)]H)
=1
2g(dqπ◦(h♭)−1(p),dqπ◦(h♭)−1(p))
=1
2ϕπ(p),(g♭)−1(ϕπ(p))
=Hg◦ϕπ(q, p).
Now we are able to prove the well-behavedness of module SRFs under Riemannian sub-
mersions.
Proof. [Proposition 3.3] Let (M, g, F) be a module SRF and π: (N, h)→(M, g) a
Riemannian submersion. By Lemma 3.3
(30) FN=hFH
M+ Γ (ker dπ)iC∞
c(N)
where FH
Mconsists of horizontal lifts of vector fields in FN. By Lemma 6.2 we have
JFN=(ϕπ)∗FM+Iker dπC∞
c(T∗N),
where FM:= X:X∈ FM. Finally, it remains to check Hg∈N(Iπ−1F). By Lemma
5.4 it is enough to verify the following:
Hg,(ϕπ)∗FM+Iker dπ=Hg−Hh◦ϕπ,(ϕπ)∗FM+Hh◦ϕπ,(ϕπ)∗FM
+{Hg−Hh◦ϕπ,Iker dπ}+{Hh◦ϕπ,Iker dπ}
⊂(ϕπ)∗FM+Iker dπ.
Here we used Lemmas 6.1 and 6.3 to prove the inclusion.
The following theorem is the main result of this section:
Theorem 6.1. The map sending every module SRF (M, g, F)to the corresponding dy-
namical I-Poisson manifold (T∗M, {·,·}T∗M,IF, Hg)and every morphism πof SRFs
within SRF0to the map ϕπdefines a functor Φ : SRF0→dynIPois.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 28
Proof. It is enough to show that Φ preserves the morphisms. A morphism πwithin SRF0
is a Riemannian submersion π: (N, h, FN)→(M, g, FM) such that FN=π−1(FM).
Similar to the previous proof we have
JFN=(ϕπ)∗FM+Iker dπC∞
c(T∗N)
and therefore the pullback (ϕπ)∗IFM(T∗M) and Iker dπlie inside IFN(T∗N). By Lemma
6.1, for every f∈N(IFM) we have
f◦ϕπ,(ϕπ)∗FM+Iker dπT∗N⊂f, FMT∗M◦ϕπ+{f◦ϕπ,Iker dπ}T∗N+Iker dπ
⊂(ϕπ)∗FM+Iker dπ
which implies that {f◦ϕπ,JFN}T∗N⊂ JFN, and consequently (ϕπ)∗N(IFM) lies inside
N(IFN). Using Lemma 6.1 again, for every f, g ∈N(IFM)
{f◦ϕπ, g ◦ϕπ}T∗N− {f, g}T∗M◦ϕπ∈ Iker dπ⊂ IFN(T∗N).
These together with Lemma 6.3 complete the proof.
Theorem 6.2. Let (M1,F1)and (M1,F1)be Hausdorff Morita equivalent singular folia-
tions. Then the Poisson algebras R(IF1)and R(IF2)are isomorphic.
Here R(IFi)≡N(IFi)/IFi(T∗Mi), i= 1,2, see Definition 4.4.
The proof of this theorem will be a consequence of the following two lemmas.
Lemma 6.4. Let π: (N, h)→(M, g)be a surjective Riemannian submersion with con-
nected fibers and Fbe a finitely generated SF on M. If f◦ϕπ∈ Iπ−1F(T∗N)for some
f∈C∞(T∗M), then f∈ IF(T∗M).
Proof. Let F=hX1, . . . , XNiC∞
c(M)and let Y1,...,YK∈X(N) be generators of the
regular foliation Γc(ker dπ) for some positive integers Nand K. Lemma 3.3 implies that
π−1F=hXH
1,...,XH
N, Y1, . . . , YKiC∞
c(N). Consequently, for every open subset V⊂T∗N,
we obtain
(31) Iπ−1F(V) = hX1◦ϕπ|V, . . . , XN◦ϕπ|V, Y1|V, . . . , YK|ViC∞(V),
where we used Lemmas 6.2 and 5.3.
Let us assume for a moment that there exists a global section s:T∗M→ C for the surjec-
tive submersion ϕπ|C:C → T∗M(see Corollary 6.1). Since f◦ϕπ∈ Iπ−1F(T∗N), Equa-
tion (31) implies that there exist smooth functions λ1,...,λN, η1,...,ηK∈C∞(T∗N)
such that
(32) f◦ϕπ=
N
X
a=1
λa·Xa◦ϕπ+
K
X
b=1
ηb·Yb.
Since ϕπ◦s◦ϕπ=ϕπand Yb|C= 0, composing both sides of Equation (32) by s◦ϕπgives
f◦ϕπ=f◦ϕπ◦s◦ϕπ=
N
X
a=1
(λa◦s◦ϕπ)·Xa◦ϕπ◦s◦ϕπ
= N
X
a=1
(λa◦s)·Xa!◦ϕπ.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 29
This implies that f=PN
a=1 (λa◦s)·Xa∈ IF(T∗M), since ϕπis surjective.
If a global section does not exist, we can choose an open covering {Ui}∞
i=1 of T∗Msuch
that for every positive integer ithere exists a local section si:Ui→ C. Using the same
argument as for the global case, we may show that f|Ui∈ IF(Ui) for each i. Since IFis
a sheaf on T∗M, we have f∈ IF(T∗M).
Lemma 6.5. Let π: (N, h)→(M, g)be a surjective Riemannian submersion with con-
nected fibers and Fbe a finitely generated SF on M. Then for every F∈N(Iπ−1F), there
exists some f∈C∞(T∗M)such that F−f◦ϕπ∈ Iπ−1F(T∗N).
Proof. We proceed as in the beginning of the proof of Lemma 6.4, establishing Equation
(31) and assuming first again that there is a global section s:T∗M→ C = Ann(ker dπ)
for the surjection ϕπ|C:C → T∗M. Define, in addition, f:= F◦s∈C∞(T∗M).
We now will prove that for every x∈T∗M, there exists an open neighborhood Vx⊂T∗N
such that (F−f◦ϕπ)|Vx∈ Iπ−1F(Vx). Since Iπ−1Fis a sheaf, this implies the desired
F−f◦ϕπ∈ Iπ−1F(T∗N). The proof is divided into the following three cases:
Case 1. [x6∈ C]: We choose an open subset Vx⊂T∗Nsuch that Vx∩ C =φ. Let
ρ∈C∞(T∗N) with supp(ρ)⊂T∗N\ C and ρ|Vx≡1. Then since ρ·(F−f◦ϕπ) vanishes
on C, we obtain
(F−f◦ϕπ)|Vx=ρ·(F−f◦ϕπ)|Vx∈ Iker dπ(Vx)⊂ Iπ−1F(Vx).
Case 2. [x∈s(T∗M)⊂ C]: Choose local coordinates (qi, qα) centered at the base-
point of xand (qi
M) on Nand M, respectively, were i∈ {1,...,m := dim M}and α∈
{m+1,...,n := dim N}, which are compatible with the submersion π, i.e. π(qi, qα) = (qi).
Let (qi, qα, pi, pα) be the corresponding Darboux coordinates on some open neighborhood
Vx⊂T∗Ncentered at x. As a consequence, in particular ϕπ(qi, qα, pi,0) = (qi, pi) (see
Corollary 6.1) and Iker dπ(Vx) = hpαiC∞(Vx). For simplicity also assume that, in these local
coordinates, s◦ϕπ(qi, qα, pi,0) = (qi,0, pi,0). Then, for every arbitrary point (qi
0, qα
0, p0
i,0)
in Vx∩ C, we have
(F−f◦ϕπ)(qi
0, qα
0, p0
i,0) = F(qi
0, qα
0, p0
i,0) −F(qi
0,0, p0
i,0)
=Z1
0
d
dtF(qi
0, tqα
0, p0
i,0)dt
=Z1
0 X
βnqβ
0pβ, F oT∗N!(qi
0, tqα
0, p0
i,0)dt
=Z1
0 X
β
qβ
0{pβ, F }T∗N!(qi
0, tqα
0, p0
i,0)dt .(33)
Since F∈N(Iπ−1F), for every βthere exist smooth functions λ1
β,...,λN
β, η1
β,...,ηK
β∈
C∞(Vx) such that {pβ, F }T∗N=Paλa
β·(Xa◦ϕπ|Vx) + Pbηb
β·(Yb|Vx). Implementing this
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 30
into Equation (33) gives
(F−f◦ϕπ)(qi
0, qα
0, p0
i,0)
=Z1
0 X
β
qβ
0 X
a
λa
β·(Xa◦ϕπ|Vx)!!(qi
0, tqα
0, p0
i,0)dt
=X
aΛa·(Xa◦ϕπ|Vx)(qi
0, qα
0, p0
i,0)(34)
where Λa∈C∞(Vx) is defined as
Λa(qi, qα, pi, pα) := Z1
0 X
β
qβλa
β!(qi, tqα, pi, pα)dt .
Equation (34) implies that (F−f◦ϕπ)−PaΛa·(Xa◦ϕπ|Vx) vanishes on Vx∩ C and
consequently this difference is an element of Iker dπ(Vx). Since PaΛa·(Xa◦ϕπ|Vx)∈
Iπ−1F(Vx) we obtain
(F−f◦ϕπ)|Vx∈ Iπ−1F(Vx).
Case 3. [x∈ C \ s(T∗M)]: Define x0=s◦ϕπ(x). Since xand x0belong to the same fiber
of ϕπ|C, there exist compactly supported functions h1,...,hl∈ Iker dπfor some positive
integer lsuch that their Hamiltonian flows connect x0to x, i.e.
x= Φ1
hl◦...◦Φ1
h1(x0).
Then the global section s′:= Φ1
hl◦...◦Φ1
h1◦spasses through the point x. After Case
2for the function f′:= F◦s′, there exists an open neighborhood Vxaround xsuch that
(F−f′◦ϕπ)|Vx∈ Iπ−1F(Vx). It remains to show that (f′−f)◦ϕπ∈ Iπ−1F(T∗N). For
arbitrary y∈ C, defining Φt
h0:= IdT∗Nand y0=s◦ϕπ(y) gives
(f′−f)◦ϕπ(y) = (F◦s′−F◦s)◦ϕπ(y)
=
l
X
i=1
(F◦Φ1
hi◦...◦Φ1
h0−F◦Φ1
hi−1◦...◦Φ1
h0)(y0)
=
l
X
i=1 Z1
0
d
dtF◦Φt
hi◦Φ1
hi−1◦...◦Φ1
h0(y0)dt
=
l
X
i=1 Z1
0
{hi, F }T∗N◦Φt
hi◦Φ1
hi−1◦...◦Φ1
h0(y0)dt .(35)
Since F∈N(Iπ−1F), for every ithere exist smooth functions λ1
i,...,λN
i, η1
i,...,ηK
i∈
C∞(T∗N) such that {hi, F }T∗N=Paλa
i·(Xa◦ϕπ) + Pbηb
i·(Yb). Implementing this into
Equation (35), making use of the fact that the flows of the his preserve C, and noting that
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 31
the Ybs vanish on C, this gives
(f′−f)◦ϕπ(y)
=
l
X
i=1 Z1
0 X
a
λa
i·(Xa◦ϕπ)!◦Φt
hi◦Φ1
hi−1◦...◦Φ1
h1(y0)dt
=X
a
Λa(y)·(Xa◦ϕπ(y)) .(36)
Here we defined Λa∈C∞(T∗N) by
Λa(z) :=
l
X
i=1 Z1
0
λa
i◦Φt
hi◦Φ1
hi−1◦...◦Φ1
h1◦s◦ϕπ(z)dt∀z∈T∗N .
Equation (36) implies that (f′−f)◦ϕπ−PaΛa(y)·(Xa◦ϕπ(y)) vanishes on Cand,
equivalently, it thus belongs to Iker dπand since PaΛa·(Xa◦ϕπ)∈ Iπ−1F(T∗N). This
gives (f′−f)◦ϕπ∈ Iπ−1F(T∗N), which completes the proof in Case 3.
If a global section does not exist, we can choose a locally finite open covering {Ui}∞
i=1
of T∗Mwith a partition of unity {ρi}∞
i=1 subordinate to it, such that for every positive
integer ithere exists a local section si:Ui→ C. Similar to the global case, we can show
that for fi:= F◦si∈C∞(Ui), we have F|ϕ−1
π(Ui)−f◦ϕπ|ϕ−1
π(Ui)∈ Iπ−1F(ϕ−1
π(Ui)).
Defining f:= P∞
i=1 ρifi, we claim that F−f◦ϕπ∈ Iπ−1F(T∗N). This is equivalent to
showing that for every σ∈C∞
c(T∗N) we have σ·(F−f◦ϕπ)∈ Jπ−1F. Since supp(σ)
is compact, it can be covered by finitely many open subsets ϕ−1
π(Ui1),...,ϕ−1
π(Uin) of the
covering {ϕ−1
π(Ui)}∞
i=1. This gives
σ·(F−f◦ϕπ) =
∞
X
i=1
σ·(ρi◦ϕπ)·(F−f◦ϕπ)(37)
=
n
X
a=1
σ·(ρia◦ϕπ)·(F|ϕ−1
π(Uia)−fia◦ϕπ|ϕ−1
π(Uia))
∈ Jπ−1F,
since σ·(ρia◦ϕπ)∈C∞
c(ϕ−1
π(Uia)). This completes the proof.
Proof. [Theorem 6.2] It is enough to show that for every surjective submersion π:N→
Mwith connected fibers over a foliated manifold (M, F), the Poisson algebras R(IF) and
R(Iπ−1F) are isomorphic.
To do so, we first choose Riemannian metrics gMand gNsuch that πbecomes a Rie-
mannian submersion. This can be done as follows: choose a Riemannian metric gM
on M, a fiber metric g⊥on ker dπ⊂T N , and a subbundle H ⊂ T N complementary
to ker dπ; one then declares these two subbundles to be orthogonal to one another and
defines gN= (π∗gM)|H+g⊥.
We now prove that the map ˜ϕπ:R(IF)→ R (Iπ−1F) is injective as follows: Let f∈
N(IF) be such that f◦ϕπ∈ Iπ−1F(T∗N), we prove that f∈ IF(T∗M). Choose an open
covering {Ui}∞
i=1 of Msuch that for every positive integer ithe pullback ι−1
UiFis finitely
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 32
generated. Lemma 6.4 then implies that f|T∗Ui∈ IF(T∗Ui) and, since IFis a sheaf, we
obtain f∈ IF(T∗M).
It remains to prove that ˜ϕπis surjective. It follows from showing that, for every F∈
N(Iπ−1F), there exists f∈N(IF) such that F−f◦ϕπ∈ Iπ−1F(T∗N). To do so we choose
an open covering {Ui}∞
i=1 of Msuch that, for every positive integer i, the pullback ι−1
UiF
is finitely generated. Let {Va}∞
a=1 be a locally finite refinement of the covering {T∗Ui)}∞
i=1
of T∗Mand let {ρa}∞
a=1 be a partition of unity subordinate to {Va}∞
a=1. Lemma 6.5 then
implies that for every athere exists fa∈C∞(Va) such that F|ϕ−1
π(Va)−fa◦ϕπ|ϕ−1
π(Va)∈
Iπ−1F(ϕ−1
π(Va)). Using the same argument as in the proof of Lemma 6.5 (see Equation
(37)), for f:= P∞
a=1 ρafawe have F−f◦ϕπ∈ Iπ−1F(T∗N). To complete the proof, we
show that f∈N(IF) as follows: Since f◦ϕπ∈N(Iπ−1F), Equation (20) of Lemma 6.1
implies that {f, JF}T∗M◦ϕπ⊂ϕ∗
πJF⊂ Iπ−1F(T∗N). As a consequence of Lemma 6.4
we have {f, JF}T∗M⊂ JF, which together with Lemma 5.4 gives f∈N(IF).
Remark 6.2. The isomorphism found in Theorem 6.2 is not canonical and depends on the
choice of compatible metrics, making the submersions in the Hausdorff Morita equivalence
Riemannian.However, for Morita equivalent module SRFs (M1, g1,F1)and (M2, g2,F2),
there is a canonical isomorphism of the pairs (R(IFi),[Hgi]) for i= 1,2, where the equality
of [Hg1]and [Hg2]under the isomorphism follows from Lemma 6.3.
Appendix A. Almost Killing Lie algebroids
In the appendix we recall the notion of almost Killing Lie algebroids as defined previ-
ously in [KS19] and provide their relation to module SRFs defined in this paper. (See, in
particular, Proposition A.1 below, but also Theorem 3.1 in the main text).
Definition A.1. Let A→Mbe a vector bundle, ρ:A→T M a vector bundle mor-
phism called the anchor map, and a skew-symmetric bracket [·,·]Aon Γ(A). The triple
(A, ρ, [·,·]A)is called an almost Lie algebroid if the induced map ρ: Γ(A)→X(M)pre-
serves the brackets, and the Leibniz rule is satisfied:
[s, f s′]A= (ρ(s)·f)s′+f[s, s′]A.
Definition A.2. Let (A, ρ, [·,·]A)be an almost Lie algebroid over Mand E→Ma vector
bundle over M. An A-connection on Eis a C∞(M)-linear map A∇on Γ(A)with values
in derivations of the C∞(M)-module Γ(E), i.e.
A∇s(fe) = (ρ(s).f )e+fA∇se ,
for every f∈C∞(M),e∈Γ(E)and s∈Γ(A).
An almost Lie algebroid (A, ρ, [·,·]A) together with an ordinary connection on A,∇: Γ(A)→
Γ(T∗M⊗A), defines an A-connection A∇on T M by:
A∇sX:= Lρ(s)X+ρ(∇Xs),(38)
for every s∈Γ(A) and X∈X(M). Note that by assuming the Leibniz rule and the
commutativity of A∇swith contractions, these derivations can be extended to the the
tensor powers of T M and T∗M.
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 33
Definition A.3. Let (A, ρ, [·,·]A)be an almost Lie algebroid over a Riemannian manifold
(M, g)and ∇: Γ(A)→Γ(T∗M⊗A)a connection on A. Then (A, ∇)and (M, g)are
called compatible if
A∇g= 0 ,
where the A-connection A∇is defined by Equation (38). The triple (A, ∇, g)is called a
Killing almost Lie algebroid over M.
Lemma A.1. Let (A, ρ, [·,·]A)be an almost Lie algebroid over a Riemannian manifold
(M, g). The triple (A, ∇, g)is a Killing almost Lie algebroid if and only if for every
X, Y ∈X(M)and s∈Γ(A)we have
(Lρ(s)g)(X, Y ) = g(ρ(∇Xs), Y ) + g(X, ρ(∇Ys)) .
Proof. By Equation (38), for every vector field X∈X(M)
(A∇sg)(X, X) = A∇s(g(X, X )) −2g(A∇sX, X)
= (Lρ(s)g)(X, X)−2g(ρ(∇Xs), X).
Consequently, A∇g= 0 if and only if
(Lρ(s)g)(X, X) = 2g(ρ(∇Xs), X ).
Proposition A.1. Let (M, F)be an SF on a Riemannian manifold (M, g). Then the
triple (M, g, F)is a module SRF if and only if it is locally generated by Killing almost
Lie algebroids, i.e. ∀q∈M, there exist an open neighborhood U∈Mcontaining qand a
Killing almost Lie algebroid (AU,∇, gU)over (U, gU)such that ρ(Γc(AU)) = ι−1
UF.
Proof. Assume that (M, g, F) is a module SRF and q∈M. Then there exists an open
neighborhood U∈Mcontaining qsuch that ι−1
UFis generated by finitely many vector
fields V1,...,VN∈X(U) for some positive integer U. By involutivity of ι−1
UF, the trivial
vector bundle AUof rank Nwith a frame e1,...,eN∈Γ(AU) together with the anchor
map ρ:AU→T M ,ea7→ Vafor a= 1,...,N, can be equipped with an almost Lie
algebroid structure. By Lemma 3.2 there exist 1-forms ωb
a∈Ω1(U) such that
LVag=
N
X
b=1
ωb
a⊙ιVbg∀a, b = 1,...,N .
Now if we define ∇ea=PN
b=1 ωb
a⊗eb, for every X, Y ∈X(U), we have
(Lρ(ea)g)(X, Y ) = (LVag)(X, Y )
=
N
X
b=1 (ιXωb
a)g(Vb, Y ) + (ιYωb
a)g(X, Vb)
=g ρ N
X
b=1
ιXωb
aeb!, Y !+g X, ρ N
X
b=1
ιYωb
aeb!!
=g(ρ(∇Xea), Y ) + g(X, ρ(∇Yea)) .
SINGULER RIEMANNIAN FOLIATIONS AND I-POISSON MANIFOLDS 34
Consequently, by Lemma A.1, (AU,∇U, gU) is a Killing almost Lie algebroid and we
have ρ(Γc(AU)) = ι−1
UF. Conversely, Assume that (M, F) is locally generated by Killing
almost Lie algebroids. Let q∈M, and take a neighborhood U∈Mcontaining qwith
a Killing almost Lie algebroid (AU,∇, gU) over (U, gU) such that ρ(Γc(AU)) = ι−1
UF. By
choosing Usmall enough, we can assume that AUis trivial and there is a global frame
e1, ..., eN∈Γ(AU). Then there exist 1-forms ωb
a∈Ω1(U) such that
∇ea=
N
X
b=1
ωb
a⊗eb∀a, b = 1,...,N .
If we define Va=ρ(ea) for a, b = 1,...,N, by Lemma A.1, for every X, Y ∈X(U) we
have
(LVag)(X, Y ) =
N
X
b=1 (ιXωb
a)g(Vb, Y ) + (ιYωb
a)g(X, Vb)
= N
X
b=1
ωb
a⊙ιVbg!(X, Y ).
So by Lemma 3.2, Equation (4) is locally satisfied and, by Proposition 3.1, (M, g, F) is a
module SRF.
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Email address:nahariATmath.univ-lyon1.fr, stroblATmath.univ-lyon1.fr
Institut Camille Jordan, Universit´
e Claude Bernard Lyon 1, Universit´
e de Lyon, 43
boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
