Integrable systems on singular symplectic manifolds: From local to global
IINTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS:FROM LOCAL TO GLOBAL
ROBERT CARDONA AND EVA MIRANDAA
BSTRACT . In this article we consider integrable systems on manifolds endowed with singular sym-plectic structures of order one. By singular symplectic structures of order one we mean structureswhich are symplectic away from an hypersurface along which the symplectic volume either goes toinfinity or to zero in a transversal way (singularity of order one) resulting either in a b -symplecticform or a folded symplectic forms. The hypersurface where the form degenerates is called criticalset. In this article we give a new impulse to the investigation of action-angle coordinates for thisstructures initiated in [30] and [29] by proving an action-angle theorem for folded symplectic inte-grable systems, establishing new cotangent models for these systems and investigating duality with b -integrable systems via desingularization. We provide global constructions of integrable systemsand investigate obstructions for global existence of action-angle coordinates in both scenarios. Thenew topological obstructions found emanate from the topology of the critical set of the singular sym-plectic manifold Z . The existence of these obstructions in turn implies the existence of singularitiesfor the integrable system on Z .
1. I
NTRODUCTION
In this article we investigate the integrability of Hamiltonian systems on manifolds endowedwith a smooth -form which is symplectic away from an hypersurface Z (called the critical set )and which degenerates in a controlled way (of order one) along it. Either this form lowers itsrank at Z and it induces a form on Z with maximal rank or its associated symplectic volumeblows-up with a singularity of order one. The manifolds endowed with the first type of singu-lar structure are called folded symplectic manifolds and the ones endowed with the second one arecalled b -symplectic forms. Folded symplectic manifolds can be thought as symplectic manifolds with a fold, Z that ”mirrors” the symplectic structure on both sides. The study of folded symplecticmanifolds complements that of their ”duals” to b -symplectic manifolds which have been largely in-vestigated since [15] and [13] and are better described as Poisson manifolds whose Poisson bracketlooses rank along an hypersurface keeping some transversality properties. This article is also aninvitation to consider more degenerate cases (higher order singularities) which will be studiedelsewhere and the models provided here can be considered as a toy model for more complicatedsingularities.The research of integrability of Hamiltonian systems on these manifolds is of interest both froma Poisson and symplectic point of view. The existence of action-angle coordinates on symplecticmanifolds has been of major importance as, other than integrating the system itself, it provides Robert Cardona is supported by an FPI grant under the Maria de Maeztu-BGSMath excellence programme. EvaMiranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize2016. Both authors are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) and referencenumber 2017SGR932 (AGAUR) . a r X i v : . [ m a t h . S G ] J u l ROBERT CARDONA AND EVA MIRANDA semilocal normal forms for integrable Hamiltonian systems which allows, for instance, to under-stand perturbation theory of these systems (KAM theory). The existence of action-angle coordi-nates of integrable systems is also useful for quantization as already observed by Einstein whenreformulating the Bohr-Sommerfeld quantization conditions [10].Integrable systems on these singular symplectic manifolds define natural Lagrangian foliationson them and thus naturally yield real polarizations on these manifolds. In particular they areof interest to study geometric quantization of symplectic manifolds with boundary as one of thesources of examples for these singular structures. On symplectic manifolds with boundary defor-mation quantization is already well-understood [36] and formal geometric quantization has beenobject of recent study in [41] for non-compact manifolds and in [20, 21] and [4] for b m -symplecticmanifolds. More specifically, the existence of action-angle coordinates for these structures pro-vides a primitive first model for geometric quantization by counting the integer fibers of the in-tegrable system. As proved in [23, 22] this model has been tested to be successful in geometricquantization of toric symplectic manifolds and refines the idea of Bohr-Sommerfeld quantization.Understanding action-angle coordinates for integrable systems on singular symplectic manifoldscan be a first good step in the study of geometric quantization of singular symplectic manifolds.The study of folded symplectic manifolds comprises the case of origami manifolds [6] where addi-tional conditions are imposed on the critical set and a natural global toric action exists. Origamimanifolds inherit their denomination from origami paper templates where a superposition ofDelzant polytopes [7] gives rise to a toric action on a class of folded symplectic manifolds. Sym-plectic origami provides an example of integrable system on folded symplectic manifolds butthere are other examples motivated by physical systems such as the folded spherical pendulumor the Toda systems where the interacting particles are far-away.In this article we show the existence of b -integrable systems on b -symplectic manifold of di-mension having a critical set a Seifert manifold, and via the desingularization technique we ob-tain folded integrable systems on the associated desingularized folded symplectic manifold. Weprove existence of action-angle coordinates `a la Liouville-Mineur-Arnold exploring the Hamilton-ian actions by tori on folded symplectic manifolds. The action-angle theorem provides modelsof cotangent type by replacing the cotangent bundle of a Liouville torus by another algebroidover the Liouville torus. We use the desingularization techniques in [18] to relate this cotangentmodel to cotangent models in the b -case obtained in [30]. The cotangent models provide semilocaltoric actions in a neighbourhood of the Liouville tori which do not always extend to the wholemanifold.We end up this article investigating the obstruction theory of global existence of action-anglecoordinates exhibiting a new topological obstruction for the singular symplectic manifolds thatlives on the critical set of the singular symplectic form. This yields examples of integrable systemson b -symplectic manifolds and folded symplectic manifolds with critical set non diffeomorphic toa product of a symplectic leaf with a circle. For those systems the toric action does not even extendto a neighborhood of the critical set. We end up this article observing that the existence of finiteisotropy for the transverse S -action given by the modular vector field obstructs the uniformiza-tion of periods of the associated torus action on the b -symplectic manifold and automatically yieldsthe existence of singularities of the integrable system on the critical locus of the b -symplectic struc-ture. Organization of this article:
In Section 2 we introduce the basic tools in b -symplectic and foldedsymplectic geometry. In Section 3 we investigate Hamiltonian dynamics on folded symplecticmanifolds and introduce folded integrable systems. In Section 4 we provide a list of motivating NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 3 examples for integrable systems on folded symplectic manifolds. In Section 5 we prove an action-angle theorem (Theorem 5) for folded symplectic manifolds which we also rephrase as a cotangentlift 16. We end up this section investigating the duality of folded and b -integrable systems. Section6 contains constructions of integrable systems on 4-dimensional b -symplectic manifolds having ascritical set a Seifert manifold (Theorem 21) and on any folded symplectic manifold which desingu-larizes it. Section 7 investigates the existence of global action-angle coordinates and highlights thenon-triviality of the mapping torus as topological obstruction to global existence of action-anglecoordinates (Theorems 26 and 27). 2. P RELIMINARIES
In this article we consider forms ω on even dimensional manifolds M n which are symplecticaway from a hypersurface Z and such that ω n either cuts the zero section of the bundle Λ n ( T ∗ M ) transversally or goes to infinity in a controlled way along Z . So, in particular ω n defines a volumeform away from Z .For the class of forms for which ω n either cuts the zero section of the bundle Λ n ( T ∗ M ) transver-sally we require an extra condition to guarantee maximal rank (see below). These forms are called folded symplectic forms as they can be visualized as symplectic manifolds which are folded alongthe folding hypersurface.2.1. Basics on folded symplectic manifolds.
We recall here some basic facts on folded symplecticmanifolds.
Definition 1.
Let M be a n -dimensional manifold. We say that ω ∈ Ω ( M ) is folded-symplectic if (1) dω = 0 , (2) ω n (cid:116) O , where O is the zero section of (cid:86) n ( T ∗ M ) , hence Z = ( ω n ) − ( O ) is a codimension submanifold, (3) i : Z → M is the inclusion map, i ∗ ω has maximal rank n − .We say that ( M, ω ) is a folded-symplectic manifold and we call Z ⊂ M the folding hypersurface. The following theorem is an analog of Darboux theorem for folded symplectic forms [33]:
Theorem 1 (Martinet) . For any point p on the folding hypersurface Z of a folded symplectic manifold ( M n , ω ) there is a local system of coordinates ( x , y , . . . , x n , y n ) centered at p such that Z is locally givenby x = 0 and ω = x dx ∧ dy + dx ∧ dy + . . . + dx n ∧ dy n . Let ( M, ω ) be a n -dimensional folded symplectic manifold. Let i : Z (cid:44) → M be the inclusionof the folding hypersurface Z . The induced restriction i ∗ ω has a one-dimensional kernel at eachpoint. We denote by V the bundle ker i ∗ ω defined at Z , and L = V ∩ T Z the null line bundle.The following theorem [5, Theorem 1] is a Moser type theorem for folded symplectic manifoldswhich extends the local normal form above to a neighborhood of Z . For the null line bundle weconsider α a one-form such that α ( v ) = 1 for a non-vanishing section v of L . Theorem 2.
Suppose that Z is compact. Then there exists a neighborhood U of Z and an orientationpreserving diffeomorphism, ϕ : Z × ( − ε, ε ) → U ROBERT CARDONA AND EVA MIRANDA where ε > and U such that ϕ ( x,
0) = x for all x ∈ Z and ϕ ∗ ω = p ∗ i ∗ ω + d (cid:0) t p ∗ α (cid:1) , with p : Z × ( − ε, ε ) → Z the projection onto the first factor, the map i : Z (cid:44) → M the inclusion and t thereal coordinate on the interval ( − ε, ε ) . A special class of folded-symplectic manifolds are origami symplectic manifolds for which thenull foliation L defines a fibration. Definition 2. An origami manifold is a folded symplectic manifold ( M, ω ) whose null foliation is fibrat-ing with oriented circle fibers, π , over a compact base , B . S ZB π The form ω is called an origami form and the null foliation is called the null fibration . Remark 1.
On an origami manifold, the base B is naturally symplectic with symplectic form ω B on B satisfying i ∗ ω = π ∗ ω B . Example.
Consider the unit sphere S n ⊂ R n +1 given by equation (cid:80) ni ( x i + y i ) + z = 1 with globalcoordinates x , y , . . . , x n , y n , z on R n +1 . Let ω be the restriction to S n of the form dx ∧ dy + . . . + dx n ∧ dy n which in polar coordinates reads r dr ∧ dθ + . . . + r n dr n ∧ dθ n . Then ω is a folded symplecticform. The folding hypersurface is the sphere given by the intersection with the plane z = 0 . It is easy tocheck that the null foliation is the Hopf foliation (see for instance [6] ). Basics on b -symplectic manifolds. In this section we give a crash course on b -symplectic/Poissonmanifolds .The study of b -symplectic manifolds starts with a similar definition to that of folded symplecticmanifolds but in the context of Poisson geometry. Given a symplectic form ω we can naturallyassociate a Poisson bracket to any pair of smooth functions f, g ∈ C ∞ ( M n ) from the symplecticstructure as follows { f, g } = ω ( X f , X g ) where the vector fields X f and X g stand for the Hamiltonian vector fields with respect to ω .From the equation above it is simple to check that { f, g } = X f ( g ) so the bracket defines abiderivation (Leibnitz rule), it is antisymmetric and because X { f,g } = [ X f , X g ] , it also satisfies theJacobi identity (i.e., {{ f, g } , h } + {{ g, h } , f } + {{ h, f } , g } = 0 for any triple of smooth functions f, g and h .A general Poisson structure is defined as a general antisymmetric bracket on any manifold (notnecessarily even dimensional) {· , ·} : C ∞ ( M ) × C ∞ ( M ) −→ C ∞ ( M ) satisfying Leibnitz rules andJacobi identity.Because a Poisson bracket defines a biderivation, it is possible to work with it like a bivectorfield Π ∈ Γ(Λ ( T M )) .The correspondence between Poisson brackets and Poisson bivector fields is clarified by theequation Π( df, dg ) = { f, g } . NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 5
As the Jacobi identify defines an additional constraint not every bivector field defines a Poissonstructure. Bivector fields which are Poisson satisfy the integrability equation [Π , Π] = 0 where thebracket is the Schouten bracket, the natural extension of Lie brackets to bivector fields.In total analogy with the symplectic case given a function we may define the Hamiltonian vectorfield via the equation: X f := Π( df, · ) . Observe that, in particular the equation { f, g } = X f ( g ) alsoholds in the general Poisson context.Let us now consider a Poisson bivector field on an even dimensional manifold which is sym-plectic away from an hypersurface. When these Poisson bivector fields fulfill transversality con-ditions along the hypersurface many tricks from symplectic geometry can be applied. These arecalled b -Poisson manifolds and have been studied and analyzed in detail starting in [15]. Definition 3.
Let ( M n , Π) be a Poisson manifold. If the map p ∈ M (cid:55)→ (Π( p )) n ∈ n (cid:94) ( T ( M n )) is transverse to the zero section, then Π is called a b -Poisson structure on M . The pair ( M n , Π) is calleda b -Poisson manifold . The vanishing set of Π n is a hypersurface denoted by Z and called the criticalhypersurface of ( M n , Π) . A list of examples can be found and analyzed in detail in [35]. In the particular case of surfaces,these structures coincide with the stable Poisson structures classified by [37]. The next example isthe typical
Radko sphere . Example.
We endow the -sphere S with the coordinates ( h, θ ) , where h denotes the height function and θ is the angle. The Poisson structure written as Π = h ∂∂h ∧ ∂∂θ vanishes transversally along the equator Z = { h = 0 } and thus it defines a b -Poisson structure on the pair ( S , Z ) . The product of two b -Poisson manifolds is not a b -Poisson manifold but the product of a b -Poisson surface with a symplectic manifold is a b -Poisson manifold as described in the examplebelow. Example.
For higher dimensions we may consider the following product structures: let ( S , Z ) be thesphere in the example above and ( S n , π S ) be a symplectic manifold, then ( S × S, π S + π S ) is a b -Poissonmanifold of dimension n + 2 . We may replace ( S , Z ) by any Radko compact surface ( R, π R ) (see forinstance [15] ). Other examples come from foliation theory and from the theory of cosymplectic manifolds:
Example.
Let ( N n +1 , π ) be a Poisson manifold of constant corank . Let us assume that there exists avector field X which is a Poisson vector field and f : S → R a smooth function. The bivector field given by Π = f ( θ ) ∂∂θ ∧ X + π defines a b -Poisson structure on the product S × N whenever the function f vanishes linearly and thevector field X is transverse to the symplectic leaves of N . In this case, the critical hypersurface is formed bythe union of as many copies of N as zeros of f . The example above is generic in the sense that any b -Poisson structure can be described in thisway in a neighborhood of a critical hypersurface N . The critical hypersurface N has a naturalcosymplectic structure associated to it. In particular, this example realizes a given cosymplecticmanifold as a connected component of a cosymplectic manifold which is a critical set of a b -Poissonmanifold. This is the content of example 19 in [15]. ROBERT CARDONA AND EVA MIRANDA
Around any point in Z , the b -Darboux theorem (see [15] and [36]) guarantees that it is alwayspossible to find local coordinates with respect to which the b -Poisson structure as stated below: Theorem 3 ( b -Darboux) . For any point p ∈ Z on the critical hypersurface of a b -Poisson manifold wemay find local coordinates centered at p for which the b -Poisson structure Π can be written as: Π = n − (cid:88) i =1 ∂∂x i ∧ ∂∂y i + t ∂∂t ∧ ∂∂z . Thus b -Poisson manifolds and symplectic manifolds have many things in common. Indeed it ispossible to work with the language of forms by admitting dff where f is the defining function of Z as a legal form of an extended complex. This is the complex of b -forms originally introduced byRichard Melrose [34] to study the index theorem on manifolds with boundary.In order to introduce this language properly we briefly recall the construction of b -forms: Givena pair ( M, Z ) where Z is an hypersurface a b -vector field is a vector field on M tangent to Z . Thespace of b -vector fields can be naturally identified as sections of a vector bundle on M called the b -tangent bundle b T M . When we refer to b -forms we consider sections of the exterior algebra ofthe its dual, the b -cotangent bundle b T ∗ M := ( b T M ) ∗ .Any b -form of degree k can be written as ω = dff ∧ α + β where α and β are k − and k smoothDe Rham forms respectively and f is a defining function for Z . Definition 4. ( b -functions ) A b -function on a b -manifold ( M, Z ) is a function which is smooth away fromthe critical set Z , and near Z has the form c log | t | + g, where c ∈ R , g ∈ C ∞ , and t is a local defining function. The sheaf of b -functions is denoted b C ∞ . A closed b -form of degree which is nondegenerate as a section of the bundle Λ ( b T ∗ M ) iscalled a b -symplectic form . As proved in [15] there is a one-to-one correspondence between b -symplectic forms and b -Poisson forms. In particular we may re-state the Darboux normal form inthe language of b -forms as done below Theorem 4. [ b -Darboux theorem] Let ω be a b -symplectic form on ( M, Z ) and p ∈ Z . Then we can finda coordinate chart ( U, x , y , . . . , x n , y n ) centered at p such that on U the hypersurface Z is locally definedby y = 0 and ω = dx ∧ dy y + n (cid:88) i =2 dx i ∧ dy i . For any b -function on a b -symplectic manifold ( M, ω ) the b -Hamiltonian vector field is the one X f n defined by ι X fn ω = − df n .A T k action on a b -symplectic manifold ( M n , ω ) is called b -Hamiltonian if the fundamentalvector fields are the b -Hamiltonian vector fields of functions which Poisson commute. Such anaction is called toric if k = n .The critical hypersurface Z of a b -symplectic structure has an induced regular Poisson structurewhich can also be visualized as a cosymplectic manifold (see [15, 17]).In [17] it was shown that if Z is compact and connected, then the critical set Z is the mappingtorus of any of its symplectic leaves L by the flow of the any choice of modular vector field u : Z = ( L × [0 , k ]) / ( x, ∼ ( φ ( x ) ,k ) , NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 7 where k is a certain positive real number and φ is the time- k flow of u . In particular, all thesymplectic leaves inside Z are symplectomorphic.This yields the following definition: Definition 5 (Modular period) . Taking any modular vector field u Ω mod , the modular period of Z is thenumber k such that Z is the mapping torus Z = ( L × [0 , k ]) / ( x, ∼ ( φ ( x ) ,k ) , and the time- t flow of u Ω mod is translation by t in the [0 , k ] factor above. The (twisted) b -cotangent lift. The cotangent lift can also be defined on the b -cotangent bundleof a smooth manifold. In this case there are two different -forms that provide the same geometri-cal structure on the b -cotangent bundle (a b -symplectic form). These are the canonical (Liouville) b -form and the twisted b -form. Both forms of degree have the same differential ( a smooth b -symplectic form) but are indeed non-smooth forms. The b -cotangent lift in each of the cases isdefined in a different manner. These were studied in detail in [30]. In this article we focus on thetwisted b -cotangent lift as it gives the right model for the structure of a b -integrable system. Definition 6.
Let T ∗ T n be endowed with the standard coordinates ( θ, a ) , θ ∈ T n , a ∈ R n and consideragain the action on T ∗ T n induced by lifting translations of the torus T n . Define the following non-smoothone-form away from the hypersurface Z = { a = 0 } : λ tw,c log | a | dθ + n (cid:88) i =2 a i dθ i . Then, the form ω := − dλ tw,c is a b -symplectic form on T ∗ T n , called the twisted b -symplectic form on T ∗ T n .In coordinates: ω tw,c := ca dθ ∧ da + n (cid:88) i =2 dθ i ∧ da i . (1)Observe that this twisted forms comes endowed with a local invariant: The constant c . Theinterpretation of this invariant is that this gives the period of the modular vector field.We call the lift together with the b -symplectic form (1) the twisted b -cotangent lift with modularperiod c on the cotangent space of a torus.As it was deeply studied in [30] the lifted action can be extended to groups of type S × H whichturns out to be b -Hamiltonian in general.2.2.2. b -Integrable systems. A b -symplectic manifold/ b -Poisson manifold can be seen as a standardPoisson manifold. Along the critical set Z the Liouville tori determined by a standard integrablesystem have dimension n − which is not convenient to model integrable systems on b -symplecticmanifolds where the critical set Z represent the direction of infinity in celestial mechanics and wewould expect to have n -dimensional tori.This is why in this context it is more natural to talk abou b -integrable system as follows: Definition 7. A b -integrable system on a n -dimensional b -symplectic manifold ( M n , ω ) is a set of nb -functions which are pairwise Poisson commuting F = ( f , . . . , f n − , f n ) with df ∧ · · · ∧ df n (cid:54) = 0 as asection of ∧ n ( b T ∗ ( M )) on a dense subset of M and on a dense subset of Z . A point in M is regular if thevector fields X f , . . . , X f n are linearly independent (as smooth vector fields) at it. ROBERT CARDONA AND EVA MIRANDA
For these systems an action-angle coordinate, proved in [29], shows the existence of a semilocalinvariant in the neighbourhood of Z (the modular period): Theorem 5.
Let ( M, ω, F = ( f , . . . , f n − , f n = log | t | )) be a b -integrable system, and let m ∈ Z be aregular point for which the integral manifold containing m is compact, i.e. a Liouville torus F m . Then thereexists an open neighborhood U of the torus F m and coordinates ( θ , . . . , θ n , σ , . . . , σ n ) : U → T n × B n such that ω | U = n − (cid:88) i =1 dσ i ∧ dθ i + cσ n dσ n ∧ dθ n , (2) where the coordinates σ , . . . , σ n depend only on F and the number c is the modular period of the componentof Z containing m . In [30] this normal form was identified as a cotangent model:
Theorem 6.
Let F = ( f , . . . , f n ) be a b -integrable system on the b -symplectic manifold ( M, ω ) . Thensemilocally around a regular Liouville torus T , which lies inside the exceptional hypersurface Z of M , thesystem is equivalent to the cotangent model ( T ∗ T n ) tw,c restricted to a neighbourhood of ( T ∗ T n ) . Here c isthe modular period of the connected component of Z containing T . b -symplectic manifolds and folded symplectic manifolds as duals. In [18] the followingtheorem is proved in the more general setting of b m -symplectic structures with singularities ofhigher order: Theorem 7.
Let ω be a b -symplectic structure on a compact manifold M and let Z be its critical hyper-surface. There exists a family of folded symplectic forms ω (cid:15) which coincide with the b -symplectic form ω outside an (cid:15) -neighborhood of Z . As a consequence of this result any b -symplectic manifold admits a folded symplectic struc-ture. However, it is well-known that the converse statement does not hold as not every foldedsymplectic form can be presented as a desingularization of a b -symplectic structures. In particu-lar, any compact orientable -dimensional manifold admits a folded symplectic form [2] but notevery -dimensional compact manifold admits a b -symplectic manifold. For instance the 4-sphere S does not admit a b -symplectic structure as it was proven in [15] that the class determined bythe b -symplectic form is non-vanishing.As we will see in the next section not every folded integrable system on a folded symplecticmanifold can be obtained via desingularization from a b -integrable system.3. H AMILTONIAN DYNAMICS ON FOLDED SYMPLECTIC MANIFOLDS
Let ( M, ω ) be a folded symplectic manifold, with folding hypersurface Z . Consider p a point in Z , applying Theorem 1 the folded-symplectic form ω can be written as following in a neighbor-hood U of a point on the folding hypersurface as: ω = tdt ∧ dq + n (cid:88) i =2 dx i ∧ dy i . with t = 0 as folding hypersurface. The singularity in ω prevents the Hamiltonian equation ι X ω = − df from having a solution for every possible function f . So not every function f ∈ C ∞ ( U ) defineslocally a Hamiltonian vector field. NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 9
Example.
Let ( U ; t, q, ..., x n , y n ) be a chart where ω takes the folded-Darboux form mentionedabove. Take for example the function f = t . By imposing the Hamiltonian equation we get that X = t ∂∂q , which is not a well defined smooth vector field.Fortunately, we can characterize the set of functions which define smooth Hamiltonian vectorfields. Lemma 8.
A function f : M → R in a folded symplectic manifold ( M, ω ) has an associated smoothHamiltonian vector field X f if and only if df | Z ( v ) = 0 for every v ∈ V . Furthermore X f is tangent to Z .Proof. Assume that f has a well-defined smooth Hamiltonian vector field at a point p in Z . TakeDarboux coordinates ( t, q, ..., x n , y n ) at a neighborhood U of p . In these coordinates, the form canbe written as: ω = tdt ∧ dq + n (cid:88) i =2 dx i ∧ dy i . Any vector field can be written as X = a ∂∂t + b ∂∂q + ... + a n ∂∂x n + b n ∂∂y n . Imposing the Hamiltonianequation ι X ω = − df we obtain that a = − ∂f∂q t b = ∂f∂t t a i = − ∂f∂q , i = 2 , ..., nb i = ∂f∂t , i = 2 , ..., n. In order to have the coefficients a and b well defined, the following equation needs to hold ∂f∂q = tH and ∂f∂t = tF where H and G are functions in M . The second equation implies that f = t f + f ( q, x , ..., y n ) and the first equation implies that ∂f ∂q = 0 . Hence f has the form f = t f + f ( x , ..., y n ) , (3)which implies that df ( ∂∂t ) | Z = df ( ∂∂q ) | Z = 0 since Z = { t = 0 } . The converse is obviously true.It follows from equation (3) that ∂f∂q = t ∂f ∂q which implies that a = 0 . We deduce that X f , theHamiltonian vector field of f , is tangent to Z . (cid:3) We denote these functions as folded functions.
Definition 8.
A function f : M → R in a folded symplectic manifold ( M, ω ) is a folded function if df | Z ( v ) = 0 for every v ∈ V = ker ω | Z . Note that even if a Hamiltonian vector field X f is always tangent to Z , one can obtain nonvanishing components of X f in the null line bundle L . If one takes n folded functions, we willalways have df ∧ ... ∧ df n | Z = 0 when we look it as a section of Λ n T ∗ M . However, the n functionscan define n linearly independent Hamiltonian vector fields. This yields the following definition: Definition 9.
An integrable system on a folded symplectic manifold ( M, ω ) with critical surface Z is a setof functions F = ( f , ..., f n ) such that they define Hamiltonian vector fields which are independent on adense set of Z and M , and commute with respect to ω . Around the regular points of the integrable system, the expression of the functions can be sim-plified and as a consequence the Poisson bracket of the functions is well-defined:
Lemma 9.
Near a regular value of an integrable system, there exist coordinates ( t, q, x , ..., y n ) such that ω = tdt ∧ dq + (cid:80) ni =2 dx i ∧ dy i , and the integrable system has the form f = t / f = g ( t, q, x , ..., y n ) t k + h ( x , y , ..., x n , y n ) ... f n = g n ( t, q, x , ..., y n ) t k n + h n ( x , y , ..., x n , y n ) , for k , . . . , k n ∈ N all of them ≥ and t is a defining function of Z .Proof. Denote the inclusion of Z in M by i : Z (cid:44) → M . Since the pullback to Z of the foldedsymplectic form i ∗ ω has rank n − , there are at most n − independent Hamiltonian vectorfields tangent to Z such that (cid:104) X , ..., X n − (cid:105) has no component in ker i ∗ ω . This implies that at anyregular point p ∈ Z of an integrable system, one of the n independent Hamiltonian vector fields X , ..., X n has a component in ker i ∗ ω . We might assume it is the first one X .Let us show that in the points close to p in Z , this vector field X can be written as X = v + X (cid:48) ,where v ∈ ker i ∗ w and X (cid:48) ∈ (cid:104) X , ..., X n (cid:105) .Indeed if X had a component in the complement of ker i ∗ ω ∪ (cid:104) X , ..., X n (cid:105) , it would have acomponent either in the symplectic orthogonal space to (cid:104) X , ..., X n (cid:105) with respect to i ∗ ω or in T Z ⊥ .However, we know that the Hamiltonian vector fields with respect to ω are tangent to Z , and so X would have a component in the symplectic orthogonal to (cid:104) X , ..., X n (cid:105) and would not Poissoncommute with them. In particular, we can take a new basis of Hamiltonian vector fields generating (cid:104) X , ..., X n (cid:105) in a neighborhood of p such that X lies exactly in the kernel of i ∗ ω in U ∩ Z .Take local coordinates in a neighborhood U of p such that X = ∂∂q . Consider the one form α = dq and some symplectic coordinates ( x , y , ..., x n , y n ) of i ∗ ω . We can now use Theorem 2 toconclude that ω = tdt ∧ dq + n (cid:88) i =2 dx i ∧ dy i . In these coordinates the vector field X is the Hamiltonian vector field of t / , and the rest ofHamiltonian functions will be of the form (3). This concludes the proof. (cid:3) Folded cotangent bundle.
In this subsection we recall the construction of [25], a dual of the b -cotangent bundle for folded symplectic manifolds. Definition 10.
Let M a manifold and Z a closed hypersurface. Let V a rank subbundle of i ∗ Z T M so thatfor all p ∈ Z the fiber V p is transverse to T p Z . We define for each open subset U ⊂ M Ω V ( U ) := { α ∈ Ω ( U ) | α | V = 0 } , the space of -forms on U vanishing on V . If U ∩ Z = ∅ then it is just Ω ( U ) . Following [25] there exists a vector bundle T ∗ V M called the folded cotangent bundle , of rank n whose global sections are isomorphic to Ω V ( M ) . This vector bundle is unique up to isomorphism,independently of the chosen V . For a small open neighborhood U of a point in Z , there exists suit-able coordinates ( x , ..., x n − ) in U ∩ Z and a coordinate t such that ( x , ..., x n − , t ) are coordinatesin U and T ∗ V ( U ) is generated by dx , ..., dx n − , tdt . The dual bundle to T ∗ V M is denoted by T V M and called the folded tangent bundle. NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 11
In this bundle there is a canonical folded symplectic form which is obtained by taking a Liou-ville form λ f which is canonical as it satisfies the Liouville-type equation (cid:104) λ f | p , v (cid:105) = (cid:104) p, ( π p ) ∗ ( v ) (cid:105) for every v ∈ T V ( T ∗ V M ) and p ∈ T ∗ V M . In coordinates ( x , ..., x n , p , ..., p n ) we have λ = p x dx + n (cid:88) i =2 p i dx i . Its derivative gives rise to a folded symplectic structure ω f = dλ = x dp ∧ dx + n (cid:88) i =2 dp i ∧ dx i which looks like the Darboux-type folded symplectic structure. The introduction of this bundleallows to restate the definition of a folded integrable system in terms of the folded cotangentbundle. Definition 11.
An integrable system on a folded symplectic manifold ( M, ω ) is a set of folded functions F = ( f , ..., f n ) for which df ∧ ... ∧ df n (cid:54) = 0 as sections of Λ n T ∗ V M on a dense set of M and Z , and whoseHamiltonian vector fields commute with respect to ω . Even if ω does not define a Poisson bracket in Z because the Hamiltonian vector fields arenot defined for non-folded functions, the bracket is well defined for folded functions and thecommutation condition ω ( X f i , X f j ) = 0 for two Hamiltonian vector fields is still well-defined.4. M OTIVATING EXAMPLES
In this section we present a series of examples of folded integrable systems. In particular, weexhibit examples of folded integrable systems whose dynamics cannot possibly be modeled by b -integrable systems. This will motivate the development of the theory of folded integrable systems,and in particular of the existence of action-angle coordinates.4.1. Double collision in two particles system.
In the literature of celestial mechanics like the re-stricted 3-body problem or the n-body problem several regularization transformation associated toad-hoc changes (like time reparametrization) bring singularities into the symplectic structure. Be-low we describe a model of double collision in two particle systems where McGehee type changesare implemented. We model a system of two particles under the influence of a potential energyfunction of the form U ( x ) = −| x | − α , with α > . In the phase space ( x, y ) ∈ R × R it is a Hamil-tonian system with Hamiltonian function F = | y | − | x | − α . Let us introduce a notation for twoconstants: denote β = α/ and γ = β +1 . By implementing the change of coordinates: (cid:40) x = r γ e iθ y = r − βγ ( v + iw ) e iθ and scaling with a new time parameter τ such that dt = rdτ we obtain the equations of motion r (cid:48) = ( β + 1) rvv (cid:48) = w + β ( v − θ (cid:48) = ww (cid:48) = ( β − ωv . We will model the collision set { r = 0 } in the case β = 1 as the folding hypersurface of afolded symplectic manifold endowed with a folded integrable system. Let us consider the folded symplectic form ω = rdr ∧ dv + dθ ∧ dw in the manifold T ∗ ( R × S ) ∼ = R × S × R with coordinates ( r, v, θ, w ) . We take the folded Hamiltonian function H = − r w + ( v − w . Observe that dH = − r vdv + ( w + v − rdr + ( w − r w ) dw , and the Hamiltonian vector field is X H = − rv ∂∂r + ( w + v − ∂∂v + ( w + r w ) ∂∂θ The equations of motion in the critical hypersurface { r = 0 } coincide with the equations of motionin the collision manifold of the original problem, hence providing a folded Hamiltonian model forit. In fact, even the linear asymptotic behavior close to collision is captured by the model. Observethat X commutes with ∂∂θ , which is a Hamiltonian vector field for the function f = w . Hence thedynamics are modelled by a folded integrable system given by F = ( f = H, f ) in T ∗ ( R × S ) with folded symplectic structure rdr ∧ dv + dθ ∧ dw .4.2. Folded integrable systems in toric origami manifolds.
Not all integrable systems on foldedsymplectic manifolds come from standard systems on symplectic manifolds after singularizationtransformations or regularization techniques as in the example above. Take for instance R withthe standard symplectic structure ω = dx ∧ dy + dx ∧ dy . The function f = x + y + x + y and f = x y − x y commute with respect to ω . There is a natural folding map from the sphere S to ¯ D , that we denote π . It is a standard fact that π ∗ ω is a folded symplectic structure in S ,and in fact an origami symplectic structure. Taking F = ( π ∗ f , π ∗ f ) yields an example of a foldedintegrable system in S with its induced folded symplectic structure. Note that this is an exampleof an integrable system on a singular symplectic manifold which is not b -symplectic, as shown bythe obstructions in [15] and [32].4.3. Symplectic manifolds with fibrating boundary.
Consider a symplectic manifold with bound-ary such that close to the boundary the symplectic form tends to degenerate and admits adaptedMartinet-Darboux charts such that the boundary has local equation x = 0 and the symplecticform degenerates on the boundary with the following local normal form: ω = x dx ∧ dy + dx ∧ dy + . . . + dx n ∧ dy n . Let us take as starting point some integrable system naturally defined on a manifold with bound-ary. Assume the folding hypersurface fibrates by circles over a compact symplectic base (origamitype). It would be enough to consider an integrable system on the (2 n − -symplectic base f , . . . f n and add t as f . The set ( t , f , . . . , f n ) defines a folded integrable system. Observethat complete integrability comes as a consequence of Theorem 2.4.4. Product of folded surfaces with symplectic manifolds endowed with integrable systems.
Take an orientable surface Σ , and ω a non-vanishing two form. Denote t any function in Σ whichis transverse to the zero section. The critical set is a finite number of closed curves γ j , j = 1 , ..., k .Then the function t defines a folded integrable system in (Σ , tω ) , where tω is a folded symplecticstructure. Let F = ( f , ..., f n ) be an integrable system in a symplectic manifold ( M n , ω ) . Then ( t , f , ..., f n ) defines a folded integrable system in the manifold M n × Σ endowed with the foldedsymplectic form ω f = tω ⊕ ω . In fact, taking any ( n + 1) -tuple of the form ( t (cid:80) ni =1 λ i f i , f , ..., f n ) for some non trivial n -tuple of constants λ i yields a folded integrable system. The critical set is ofthe form Z = (cid:116) kj =1 γ j × M n . NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 13
Origami templates.
The study of toric folded symplectic manifolds was initiated in [6] in theorigami case and by Hockensmith in the general case.Toric actions and integrable systems have always been hand-in-hand. In particular the action-angle coordinate theorem proof that we will provide in this article uses intensively this corre-spondence. So in particular a toric manifold provides examples of integrable systems which aredescribed by a global Hamiltonian action of a torus. Indeed any integrable system can be semilo-cally described in these terms (as we will see in the next section).The classical theory of toric symplectic manifolds is closely related to a theorem by Delzant [7]which gives a one-to-one correspondence between toric symplectic manifolds and a special typeof convex polytopes (called Delzant polytopes) up to equivalence. Grosso modo, toric symplecticmanifolds can be classified by their moment polytope, and their topology can be read directlyfrom the polytope in terms of equivariant cohomology. In [26, 27] the authors examine the toricorigami case and describe how toric origami manifolds can also be classified by their combinato-rial moment data.Origami templates form a visual way to describe toric origami manifolds and thus in particularintegrable systems on folded symplectic manifolds.F
IGURE
1. Origami template of Example 4.2Toric origami manifolds are classified by combinatorial origami templates which overlap Delzant’spolytopes in an special way providing pictorially beautiful examples of folded integrable systems.F
IGURE
2. Origami template corresponding to the radial blow-up of two Hirze-bruch surfaces.
The folded spherical pendulum.
Consider the spherical pendulum on S defined as follows:Take spherical coordinates ( θ, φ ) with θ ∈ (0 , π ) and φ ∈ (0 , π ) if we denote each momentum as P θ and P φ respectively, the Hamiltonian function is H = 12 ( P θ + 1sin θ P φ ) + cos θ. Instead of taking the standard symplectic form in T ∗ S we consider the folded symplectic form ω = P φ dP φ ∧ dφ + dP θ ∧ dθ. Computing the Hamiltonian vector field associated to H we get X H = 1sin θ ∂∂φ + P θ ∂∂θ + (sin θ + cos θ sin θ P φ ) ∂∂P θ . This vector field clearly commutes with ∂∂φ , which is the Hamiltonian vector field of f = P φ .Observe furthermore that dH ∧ dP φ = − (sin θ + cos θ sin θ P φ )2 P φ dθ ∧ dP φ , which is nondegenerate on a dense set of M and on a dense set of Z when seen as a section ofthe second exterior product of the folded cotangent bundle. The manifold is in fact M = T ∗ ( S \{ N, S } ) , i.e. we are taking out the poles of the sphere. In this sense M is equipped with an origamisymplectic form: the critical set is T ∗ ( S \ { N, S } ) and the null line bundle is an S fibrationgenerated by ∂∂ϕ .Observe that dynamically this system is different from the standard spherical pendulum. When P φ = 0 , the vector field can have a non vanishing ∂∂φ component.4.7. A folded integrable system which cannot be modelled as a b -integrable system. Consider S with the folded symplectic form ω = hdh ∧ dθ . A folded function whose exterior derivative is anon-vanishing one-form (when considered as section of the folded cotangent bundle) on a denseset of Z . This function defines a folded integrable system. Take for instance f = cos θh , whichsatisfies this condition. Computing its Hamiltonian vector field we obtain X f = h sin θ ∂∂h + 2 cos θ ∂∂θ . This vector field vanishes at some points in the critical locus Z = { h = 0 } . A b -integrable system ona surface Σ is defined by a function f = c log h + g with g ∈ C ∞ (Σ) . In particular its Hamiltonianvector field cannot vanish at any point on the critical hypersurface, as it is happening in thisexample of folded integrable system. Thus, even if the structure hdh ∧ dθ can be seen as thedesingularization of h dh ∧ dθ , the dynamics of this folded integrable system cannot be modeledusing the b -symplectic structure.4.8. Cotangent lifts for folded symplectic manifolds.
In this section we describe the cotangentlift in the set-up of folded symplectic manifolds.When the group acting on the base is a torus this procedure provides examples of folded inte-grable systems.Consider a Lie group G acting on M by an action φ : G × M −→ M . Definition 12.
The cotangent lift of φ is the action on T ∗ M given by ˆ φ g := φ ∗ g − , where g ∈ G . NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 15
We have the following commuting diagram: T ∗ M T ∗ MM M (cid:47) (cid:47) ˆ φ g (cid:15) (cid:15) π (cid:15) (cid:15) π (cid:47) (cid:47) φ g where π is the projection from T ∗ M to M . The cotangent bundle has the symplectic form ω = − dλ where λ is the Liouville form. This form is defined by the property (cid:104) λ p , v (cid:105) = (cid:104) p, ( π p ) ∗ ( v ) (cid:105) , where v ∈ T ( T ∗ M ) and p ∈ T ∗ M . It can be shown easily that the cotangent lift is a Hamiltonian actionwith momentum map µ : T ∗ M → g ∗ given by (cid:104) µ ( p ) , X (cid:105) := (cid:104) λ p , X | p (cid:105) = (cid:104) p, X | π ( p ) (cid:105) . Here X denotes the fundamental vector field of X associated to the action. The Liouville form isinvariant by the action which implies the invariance of the moment map. In particular, the map isPoisson.The construction called b -symplectic cotangent lift for b -symplectic manifolds done in [30] canbe similarly done in the folded symplectic case which we will do below.For the standard Liouville form in the folded cotangent bundle, the singularity is in the basespace, and we would like to have it on the fiber. A different form, that we call twisted foldedLiouville form can be defined on T ∗ V S with coordinates ( θ, p ) : λ tw = p dθ . this way the singularity is in the fiber, and we can apply it to define a folded cotangent lift onthe torus. Let T n be the manifold and the group acting by translations, and take the coordinates ( θ , ..., θ n , a , ..., a n ) on T ∗ M . The standard symplectic Liouville form in these coordinates is λ = n (cid:88) i =1 p i dθ i . The moment map µ can : T ∗ T n → t ∗ of the lifted action with respect to the canonical symplecticform is µ can ( θ, p ) = (cid:88) i p i dθ i where the θ i are seen as elements of t ∗ . In fact, one can identify the moment map as just theprojection of T ∗ T n into the second component since T ∗ T n ∼ = T n × R n .This torus action in the cotangent bundle of the torus can be seen as a folded-Hamiltonian ac-tion with respect to a folded symplectic form. Similarly to the Liouville one-form we define thefollowing singular form away from the hypersurface Z = { p = 0 } : p dθ + n (cid:88) i =2 p i dθ i . The negative differential gives rise to a folded symplectic form called twisted folded symplecticform on T ∗ T n : ω tw,f := p dθ ∧ dp + n (cid:88) i =2 dθ i ∧ dp i . The moment map is then µ tw,f = ( p , p , . . . , p n ) , where we identify t ∗ with R n as before.We call this lift the folded cotangent lift . Note that, in analogy to the symplectic case, the com-ponents of the moment map define a folded integrable system on ( T ∗ T n , ω tw,f ) . Remark 2.
As we will see in the next chapter the example of folded cotangent lifts is the stan-dard example as the action-angle theorem that we prove identifies the neighbourhood model of aLiouville torus with the folded cotangent lift.5. C
ONSTRUCTION OF ACTION - ANGLE COORDINATES
In this section we prove existence of action-angle coordinates for singular symplectic manifoldsof order one. One may have the temptation to use the desingularization and the action-anglecoordinate theorem proved in [29] to conclude. However, as we saw in previous sections, notevery folded integrable system can be seen as a desingularized b -integrable system and thus acomplete proof is needed.5.1. Topology of the integrable system.
We first show that for a folded integrable system thereis a foliation by Liouville tori in the neighborhood of a regular fiber of the integrable system. Forthis it is important to observe the following:The foliation given by the Hamiltonian vector fields of F coincides with the foliations describedby the level sets of ¯ F = ( t, f , ..., f n ) because by definition the Hamiltonian vector field of t / istangent to the level set of this function, and hence also to the level sets of t . The same argument in[31] gives a commutative diagram U T n × B n B nϕ ¯ F π and proves:
Proposition 10.
Let p ∈ Z be a regular point of a folded-integrable system ( M, ω, F ) . Assume that theintegral manifold F p is compact. Then there is neighborhood U of F p and a diffeomorphism ϕ : U ∼ = T n × B n which takes the foliation F to the trivial foliation { T n × { b }} b ∈ B n . In what follows we prove a Darboux-Carath´eodory theorem for folded symplectic manifolds tocomplete (locally) a set of Poisson commuting functions to a maximal set. We do this adapting thearguments of the proof of Darboux theorem provided in [1] . The Darboux-Carath´eodory theoremwill be a key point in the proof of existence of action-angle coordinates.
Theorem 11 (Folded Darboux-Carath´eodory theorem) . Let m ∈ Z be a point of the folding hypersur-face of a folded symplectic manifold ( M, ω ) and let t be the function defining Z . Consider f = t / , ..., f n , n functions whose Hamiltonian vector fields are smooth, independent at m and commute pairwise withrespect to ω . Then in a neighborhood U of m there exists n functions q , ..., q n such that (1) the n functions ( t, f , ..., f n , q , ..., q n ) form a system of coordinates on U , centered at m ; NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 17 F IGURE
3. Fibration by Liouville tori: In the middle fiber (in blue) of the point F ( p ) , the neighbouring Liouville tori in red.(2) in these coordinates the folded symplectic form is written as ω = tdq ∧ dt + n (cid:88) i =2 dq i ∧ df i Remark 3.
Comparing this theorem with the analog in [29] observe that unlike the b -symplecticcase where the modular period turns out to be a semilocal invariant for the action-angle theorem,there are no semilocal invariants for folded symplectic manifolds. Proof.
The first function is f = t / and the rest can be written as f i = t k i g i ( t, q, x , ..., y n ) + h i ( x , ..., y n ) , with k i ≥ . Because the Hamiltonian vector fields are independent we obtain dh i (cid:54) =0 for all i = 2 , ..., n . Observe that the system of functions t, f , ..., f n are functionally independent.The Hamiltonian vector fields X f i commute with respect to ω and on M \ Z , the Poisson bracket { f i , f j } = 0 and thus [ X f i , X f j ] = 0 along M \ Z . By continuity the bracket [ X f i , X f j ] vanisheseverywhere on M and the distribution (cid:104) X f , ..., X f n (cid:105) is involutive.For any i , the distribution given by all but one Hamiltonian vector field D i = (cid:104) X f j (cid:105) j (cid:54) = i is alsoinvolutive and integrates to a submanifold L i , which lies inside the Lagrangian submanifold L generated by X , ..., X n at p . For each i there is an (cid:15) i small enough such that the flow φ ti of X f i isdefined in a neighborhood U for | t | < (cid:15) i and such that it does not leave the neighbourhood of thetorus. Then for any x ∈ U ∩ L there is a point ξ i = ξ i ( x ) ∈ L i ∩ U and one time t i ( x ) such that x = ϕ t i i ( ξ i ) . Define now q i : U → R x (cid:55)→ t i ( x ) . Observe that since [ X f i , X f j ] = 0 the leaves are sent to leaves. Since q i ( φ ti ( L i ∩ U )) = t is con-stant for each t we deduce dq i ( X j ) = δ ij everywhere on U . In particular in U \ Z the functions ( f , ..., f n , q , ..., q n ) constitute a coordinate system and the form is completely determined as ω = n (cid:88) i =1 df i ∧ dq i . Since ω, f i and q i are smooth in U , this is the form everywhere in U , including the points in Z . Itis clear that the functions ( t, f , ..., f n ) define a coordinate system. (cid:3) Remark 4.
The same theorem applies if we take a set of k < n commuting functions which areindependent f , ..., f k . We can find then a set of coordinates such that ω = (cid:80) li =1 dq i ∧ df i + (cid:80) ni = k +1 dx i ∧ dy i with f = t . Equivariant relative Poincar´e’s lemma for folded symplectic forms.
We start this sectionwith some lemmas which we will need for the proof of the action-angle theorem. They concernRelative Poincar´e’s lemma for folded symplectic forms and their equivariant versions. Recall from[40, page 25].
Theorem 12 (Relative Poincar´e lemma) . Let N ⊂ M a closed submanifold and i : N (cid:44) → M the inclusionmap. Let ω a closed k -form on M such that i ∗ ω = 0 . Then there is a ( k − -form α on a neighborhood of N in M such that ω = dα . This Relative Poincar´e lemma can be used in the particular case in which the form is folded andthe submanifold is a Liouville torus.
Proposition 13.
In a neighborhood U ( L ) of a Liouville torus the folded symplectic form can be written ω = dα. If ω is invariant by a compact group action, α can be assumed to be invariant by the same compact groupaction.Proof. Let i : L (cid:44) → M be the natural inclusion of the Liouville torus on the folded symplecticmanifold, since i ∗ ω = 0 we may apply the following relative Poincar´e theorem.Let us check that the hypotheses of the Relative Poincar´e lemma are met. The form is closedand we only need to check i ∗ ω = 0 . Since every Y i is Hamiltonian with Hamiltonian function σ i , we obtain that ι Y i ω = dσ i . And therefore the tangent space to L is generated by Y , ..., Y n .However, we know that i ∗ dσ i = 0 , since L is the level set of the integrable system. This impliesthat ι Y i i ∗ ω = 0 and hence i ∗ ω = 0 .Now define a new ¯ α as ¯ α = (cid:90) G ρ ∗ g αdµ, where µ is a Haar measure and ρ g is the group action. This 1-form is G -invariant, and as ρ preserves ω we obtain, ω = (cid:90) G ρ ∗ g ωdµ = (cid:90) G dρ ∗ g αdµ Thus ω = d ( (cid:82) G ρ ∗ g ( α ) dµ ) . In particular this proves that the primitive ¯ α is invariant by the action.i.e, for any Y i fundamental vector field of the torus action one obtains, L Y i ¯ α = 0 . Thus finishingthe proof of the proposition. (cid:3)
Statement and proof of the action-angle coordinate theorem.
We proceed now with thestatement and the proof of the action-angle theorem.
Theorem 14.
Let F = ( f , ..., f n ) be a folded integrable system in ( M, ω ) and p ∈ Z a regular point inthe folding hypersurface. We assume the integral manifold F p containing p is compact. Then there exist anopen neighborhood U of the torus F p and a diffeomorphism ( θ , ..., θ n , τ, σ , ..., σ n ) : U → T n × B n , NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 19 where τ is a defining function of Z and such that ω U = τ dθ ∧ dτ + n (cid:88) i =2 dθ i ∧ dp i . Moreover, functions τ, p , ..., p n depend only on F .The S -valued functions θ , ..., θ n are called angle coordinates and the R -valued functions τ , p , ..., p n are called action coordinates. Besides the lemmas in the former subsection we will need the following technical lemma.In [31] (see Claim 2 in page 1856) it is shown that given a complete vector field Y of period and a bivector field P such that L Y L Y P = 0 then L Y P = 0 . If instead of a bivector field we take a -form, the proof can be easily adapted as follows. Lemma 15. If Y is a complete vector field of period and ω is a -form such that L Y L Y ω = 0 then L Y ω = 0 .Proof. Denote v = L Y ω . Denote φ t the flow of Y . For any point p we have ddt (cid:0) φ ∗ t ω φ − t ( p ) (cid:1) = ( φ t ) ∗ ( L Y ω φ − t ( p ) )= φ t ∗ v φ − t ( p ) = v p In the last equality we used that L Y v = 0 . Integrating we obtain ( φ t ) ∗ ω φ − t ( p ) = ω p + tv p . At time t = 1 the flow is the identity because Y has period 1 and hence v p = 0 . (cid:3) We now proceed to the action-angle theorem proof.
Proof.
We may assume that the integrable system is of the form f = t / , f , ..., f n by Lemma 9.The vector fields X f , ..., X f n define a torus action on each Liouville torus T n × { b } b ∈ B n . We wouldlike an action defined in a neighborhood of the type T n × B n . For the first part of the proof wefollow the proofs in [31] and [29] and construct a toric action. For this we consider the classicalaction of the joint-flow (which is an R n -action) and prove uniformization of periods to induce a T (cid:110) -action.We denote by ϕ ti the time- t -flow of the Hamiltonian vector fields X f i . Consider the joint flow ofthese Hamiltonian vector fields. ϕ : R n × ( T n × B n ) −→ T n × B n (cid:0) ( t , . . . , t n ) , ( x, y ) (cid:1) (cid:55)−→ ϕ t ◦ · · · ◦ ϕ t n n ( x, y ) . The vector fields X f i are complete and commute with one another so this defines an R n -actionon T n × B n . When restricted to a single orbit T n × { b } for some b ∈ B n , the kernel of this action isa discrete subgroup of R n , a lattice Λ b . We call Λ b the period lattice of the orbit. The rank of Λ b is n because the orbit is assumed to be compact.The lattice Λ b will in general depend on b . The idea of uniformization of periods is to modifythe action to get constant isotropy groups such that Λ b = Z n for all b . For any b ∈ B n − × { } andany a i ∈ R the vector field (cid:80) a i X f i on T n × { b } is the Hamiltonian vector field of the function a i t / n (cid:88) i =2 a i f i . To perform the uniformization we pick smooth functions ( λ , λ , . . . , λ n ) : B n → R n such that(1) ( λ ( b ) , λ ( b ) , . . . , λ n ( b )) is a basis for the period lattice Λ b for all b ∈ B n (2) λ i vanishes along { } × B n − for i > . Here, λ ji denotes the j th component of λ i .Such functions λ i exist such that they satisfy the first condition (perhaps after shrinking B n ) by theimplicit function theorem, using the fact that the Jacobian of the equation Φ( λ, m ) = m is regularwith respect to the s variables. We will now show why they can be chosen to satisfy the secondcondition.We define a uniformized flow using the functions λ i as ˜Φ : R n × ( T n × B n ) → T n × B n (cid:0) ( s , . . . , s n ) , ( x, b ) (cid:1) (cid:55)→ Φ (cid:32) n (cid:88) i =1 s i λ i ( c ) , ( x, b ) (cid:33) . The period lattice of this R n action is Z n , and therefore constant hence the initial action clearlydescends to the quotient to define a new action of the group T n .We want to find now functions σ , ..., σ n such that their Hamiltonian vector fields are preciselythe ones constructed above Y i = (cid:80) nj =1 λ ji X f j . Denote λ as c . We compute the Lie derivative ofthe vector fields Y i using Cartan’s formula: L Y i ω = dι Y i ω + ι Y i dω = d ( − n (cid:88) j =1 λ ji df j )= − n (cid:88) j =1 dλ ji ∧ df j We deduce that L Y i L Y i ω = L Y i ( − n (cid:88) j =1 dλ ji ∧ df j ) . In the last equality we have used the fact that λ ji are constant on the level sets of F . Lemma 15applied to the vector fields Y i yields L Y i ω = 0 and the folded-symplectic structure is preserved.The next step is to prove that the collection of -forms ι Y i ω are exact in the neighbourhoodof a Liouville torus. So the new action is indeed Hamiltonian. We apply proposition 13 in aneighbourhood of a Liouville torus and the symplectic form ω can be written as ω = dα . Now since NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 21 L Y i ω = 0 , consider the toric action generated by the vector fields Y i . Applying the equivariantversion of Proposition 13 with the group G = T n the form ω is G -invariant and we can find a new ¯ α which is at the same time a primitive for the folded symplectic structure ω and T n -invariant.Cartan’s formula yields: ι Y i ω = − ι Y i d ¯ α = − dι Y i ¯ α. Thus we deduce that the fundamental vector fields Y i are indeed Hamiltonian with Hamiltonianfunctions ι Y i ¯ α . Denoting by σ , ..., σ n these Hamiltonian functions, they are now the natural can-didates for ”action” coordinates. Observe that σ = ct since λ i = 0 for all i < n , for some constant c > .By Theorem 11 (Darboux-Carath´eodory theorem) there exists a coordinate system ( σ , ..., σ n , q , ..., q n ) such that ω = dσ ∧ dq + n (cid:88) i =2 dσ i ∧ dq i . Since the vector fields X σ i are Hamiltonian fundamental vector fields of the T n -action, in the localchart the flow of the vector fields gives a linear action on the q i coordinates and the functions σ = ct / , ..., σ n that were defined in an open set U of the point can be extended to the whole set U (cid:48) = σ − ( σ ( U )) (a neighbourhood of the Liouville torus). For the sake of simplicity we denotethese extensions using the same notation. The Hamiltonian vector fields of σ i have period one,so the functions q i can be viewed as angle variables θ i . It only remains to check if the extendedfunctions define a system of coordinates in the neighbourhood of the torus and that ω has thedesired Darboux-type form.Observe that ω ( ∂∂σ i , ∂∂θ i ) = δ ij by the own definition of θ i . By abuse of notation we denote by X θ i the vector fields which solve the equation: ι X θi ω = − dθ i .In the original neighborhood U we had that ω ( ∂∂σ i , ∂∂σ j ) = ω ( X θ i , X θ j ) = 0 . Applying thedefinition of exterior derivative, using that ω is closed and that the vector fields commute weobtain: dω ( X θ i , X θ j , X σ k ) = X θ i ( ω ( X θ j , X σ k )) − X θ j ( ω ( X θ i , X σ k ))+ X σ k ( ω ( X θ i , X θ j ))= 0 Using that ω ( X σ i , X θ j ) = δ ij for all i and j , we obtain X σ k ( ω ( X θ i , X θ j )) = 0 . In particular, by following the flow of the vector fields X σ k we prove that the relation ω ( ∂∂σ i , ∂∂σ j ) =0 holds in the whole neighborhood U (cid:48) . We conclude that ω has the desired form ω = ctdt ∧ dθ + n (cid:88) i =2 dσ i ∧ dθ i . In particular the functions t, σ ..., σ n , θ , ..., θ n are independent on U and hence define a coordinatesystem. Taking p i := − σ i and τ = √ ct then the form in the neighborhood of the torus is written ω = τ dθ ∧ dτ + n (cid:88) i =2 dθ i ∧ dp i . This concludes the proof. (cid:3)
Action-angle coordinates and cotangent lifts.
Observe that using the language of cotangentlifts introduced in Section 4.8, this theorem can be stated as follows:
Theorem 16.
Let F = ( f , ..., f n ) be a folded integrable system on a folded symplectic manifold ( M, ω ) .Around a regular Liouville torus inside Z , the system is equivalent to the system ( p , ..., p n ) in T ∗ T n , withcoordinates ( θ , ..., θ n , p , ..., p n ) and twisted folded symplectic form ω = p dp ∧ dθ + (cid:80) ni =2 dp i ∧ dθ i ,restricted to a neighborhood of the zero section. Remark 5.
By this local model, one can also think of a neighborhood of a regular point of a foldedintegrable system as the ”folding” of a regular symplectic neighborhood of an integrable system.Concretely, consider the standard action-angle coordinates in ( T n × R n , ω = (cid:80) ni =1 dθ i ∧ dp i ) withthe set of functions F = ( p , . . . , p n ) . Then the local model for folded integrable systems is just thepullback of the symplectic form ω and the set of functions F by the folding map F : T n × B n −→ T n × B n ( θ i , p , . . . , p n ) (cid:55)−→ ( θ i , p , p , . . . , p n ) . Singular cotangent models and desingularization.
In [30] several examples of b -integrablesystems are provided using the b -cotangent lift. In fact, the construction can be generalized to thecontext of b m -symplectic manifolds. From the definition of cotangent lift and the results in [30]which we recalled in subsection 2.2.1 we obtain: Proposition 17.
The twisted b -cotangent lift of the action of an abelian group G of rank n on M n yieldsa b -Hamiltonian action in T ∗ M . If the action is free or locally free, the twisted cotangent lift yields a b -integrable system. In [19] the desingularization of torus actions was explored in detail. As a consequence of theo-rem 6.1 in [19] where an equivariant desingularization procedure is established for effective torusactions, we obtain the following desingularized models . Proposition 18.
The equivariant desingularization takes the twisted b -cotangent lift of an action of a torus T n on M to a twisted folded cotangent lift model.Proof. Denote t the defining function of the critical hypersurface Z . The moment map of the actionin the S coordinate is a function of the form f = c log( | p | ) for some constant c , where p denotesthe momentum coordinate in T ∗ S . Its Hamiltonian vector field is X f = c ∂∂θ . Take f (cid:48) = cp as newmomentum map component for the folded symplectic structure in T ∗ M . (cid:3) This construction provides a machinery to produce examples of folded integrable systems viadesingularization of b -integrable systems. Even though, we know that not all integrable systemon a folded symplectic manifold comes from desingularization. NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 23
6. C
ONSTRUCTIONS OF INTEGRABLE SYSTEMS
In this section, we study the existence of integrable systems on b -symplectic manifolds andtheir possible desingularization into folded integrable systems. We construct ad-hoc integrablesystems on any -dimensional b -symplectic manifold from integrable systems defined on the leafof a cosymplectic manifolds. In what follows we will always assume that the symplectic foliationon the critical set Z contains a compact leaf, and thus Z is a symplectic mapping torus by [17,Theorem 19].6.1. Structure of a b -integrable system in Z . We start analyzing how a b -integrable system be-haves on Z , the critical hypersurface of a b -symplectic manifold ( M, ω ) . Claim 1.
Let F stand for a b -integrable system on a b -symplectic manifold ( M, ω ) . Then for a fixedsymplectic leaf L of Z there is a dense set of points in L that are regular points of F .Proof. Assume that the set of regular points in a fixed leaf L is not dense. Then we can find anopen neighbourhood U in L which does not contain any regular point, i.e. df ∧ · · · ∧ df n = 0 (when seen as a section of Λ n ( b T ∗ M ) ). However, in order for F to define a b -integrable system,one of the functions has to be a genuine (i.e., non-smooth) b -function in a neighborhood of Z . Inother words, f = c log | t | + g with c (cid:54) = 0 and g a smooth function. We can assume that f = f is agenuine b -function in a neighborhood U (cid:48) in Z containing U . Since c (cid:54) = 0 , it defines a Hamiltonianvector field whose flow is transverse to the symplectic leaf L . The function f Poisson commuteswith all the other integrals, and so the the flow of f preserves df ∧ · · · ∧ df n .Denote by ϕ t the flow of X f . Then the set V = { ϕ t ( U (cid:48) ) | t ∈ (0 , ε ) } is an open subset of Z where df ∧ · · · ∧ df n = 0 . This is a contradiction with the fact that F = ( f , . . . , f n ) defines a b -integrable system. (cid:3) Using this claim, we deduce the structure of the b -integrable system in Z . Proposition 19.
Let ( M, ω ) be a b -symplectic manifold admitting a b -integrable system, then in a neighbor-hood of Z where f = c log t , its Hamiltonian vector field defines an S -action along Z which is transverseto the symplectic foliation on Z . Furthermore ( f , ..., f n ) induces an integrable system on each symplecticleaf L on Z which is invariant by the monodromy of the action.Proof. Using Proposition 3.5.3 in [28] (or remark 16 in [29]) we may assume that near a connectedcomponent of Z the first function is f = c log t , where c is the modular period of that connectedcomponent, and f , ..., f n are smooth. The action-angle theorem shows that in a neighborhood ofa regular torus level set, the Hamiltonian vector field of c log t is periodic of period .By Claim 1, there is a dense set of regular points in a fixed leaf L of Z . If we denote ϕ t theflow associated to X , then ϕ is a symplectomorphism of L with a dense set of fixed points: allthe regular points of the system in that leaf. We deduce that this symplectomorphism has tobe the identity, and the Hamiltonian vector field of c log t defines an S -action transverse to thesymplectic leaves of Z .By hypothesis, the Hamiltonian vector field X c log t commutes with the Hamiltonian vectorfields X f , ..., X f n which implies that the flow ϕ t of the S -action preserves each function f , ..., f n .The flow also preserves the symplectic foliation in Z . Thus, fixing a symplectic leaf L , the flow ϕ t satisfies ϕ t ( L ) ∼ = L and ϕ ∗ t ( f , ..., f n ) = ( f , ..., f n ) . This shows that on each leaf the functions f , ..., f n induce the same integrable system. In particular this integrable system in L is preserved by the first return map of the monodromy in that fixed leaf, implying that the system is invariantby that finite group action. (cid:3) Corollary 20.
Let Z be the critical set of of a b -symplectic manifold endowed with a b -integrable system.Then each connected component Z i of Z is a mapping torus with gluing mapping a periodic symplectomor-phism of a symplectic leaf L of Z i . Remark 6.
In the jargon of three-dimensional geometry, the connected components of the criticalset Z of a 4-dimensional manifolds are then Seifert manifolds with orientable base and vanishingEuler number.6.2. Construction of b -integrable systems. As a consequence of Corollary 20, in order to construct b -integrable systems we will assume that Z is the mapping torus of a periodic symplectomorphismof a compact leaf L on Z . This symplectomorphism defines a finite group action on L . This iswhy in order to construct b -integrable systems on dimensional b -symplectic manifolds, we startby proving that we can always find a non-constant function which is invariant under a symplecticfinite group action on a surface. Claim 2.
Let Z k be a finite group acting of a symplectic surface Σ . Then there exists a non-constant analyticfunction F invariant by the group action.Proof. Take f a generic analytic function in Σ . Consider the averaged function given by the aver-aging trick : F ( x ) := k − (cid:88) i =1 f ( i.x ) . By construction this analytic function is invariant by the action of Z k . Given a point p in Σ , thedifferential of F vanishes at p if and only if df p + df .p + ...df ( k − .p = 0 . Observe that for a generic f , there exists a point where this condition is not fulfilled. In particular, we deduce that dF p (cid:54) = 0 at some point p , and hence F is not a constant function. (cid:3) In the claim above we might replace the analytic condition by a Morse function F . See forinstance [39] for a proof of the existence and density of invariant Morse functions by the action ofa compact Lie group. Theorem 21.
Let ( M, ω ) be a b -symplectic manifold of dimension with critical set Z which is a mappingtorus associated to a periodic symplectomorphism. Then ( M, ω ) admits a b -integrable system.Proof. In this case, a leaf of the critical set is a surface L . Take a neighborhood of Z of the form U = Z × ( − ε, ε ) . Denote by X the Hamiltonian vector field of the function log t for some definingfunction of Z . By hypothesis, X defines a Poisson S -action in Z transverse to the leaves as studiedin [3]. This S -action can have some monodromy. Denote by α and β the defining one and twoforms of ω at Z . That is, in U we can assume that ω has the form ω = α ∧ dtt + β with α ∈ Ω ( Z ) , β ∈ Ω ( Z ) . Recall that both forms are closed and i ∗ L β is a symplectic form in a leaf L of Z .We have the following description of the critical set. There is an equivariant cover L × S × ( − ε, ε ) of U , and we denote by p the projection to U . This equivariant cover can be equipped with the b -Poisson structure ω = π ∗ Z ˜ α ∧ dtt + π ∗ Z ˜ β, where ˜ α = p ∗ α and ˜ β = p ∗ β . Then U is Poisson isomorphic by [3, Corollary 17] to the quotient ofthe equivariant by the action of a finite group Z k in the leaf given by the return time flow of the S -action and extended trivially to the neighborhood L × S × ( − ε, ε ) . NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 25
The action of Z k acts by symplectomorphisms on L . By Lemma 2, there is an analytic function F in L which is invariant by the action. In particular, F can be extended to ˜ F in all Z by the S -action. If π is the projection in U = Z × ( − ε, ε ) to the first component, then we extend ˜ F to U byconsidering π ∗ ˜ F and denote it again ˜ F .We construct in the neighborhood U the pair of functions ( f , f ) = ( ϕ ( t ) c log t, ϕ ( t ) ˜ F ) in U . Thefunction ϕ ( t ) denotes a bump function which is constantly equal to for t ∈ ( − δ, δ ) and constantlyequal to for | t | > δ (cid:48) , with δ < δ (cid:48) < ε . Observe the functions f and f are linearly independent in b T ∗ M in a dense set of Z × ( − δ (cid:48) , δ (cid:48) ) . The Hamiltonian vector field of ϕ ( t ) f generates the transverse S action extended to U , and the Hamiltonian vector field of ˜ F is tangent to the symplectic leavesin each Z × { t } . Hence { f , ˜ F } = 0 . Now using the properties of the Poisson bracket we have { f , ϕ ( t ) ˜ F } = −{ ϕ ( t ) ˜ F , f } = { ϕ ( t ) , f } ˜ F + { ˜ F , f } = 0 + 0 , where we used that f only depends on the coordinate t . We obtain an integrable system in theneighborhood of the critical locus U . To obtain an integrable system in all M , we do it as in theproof of existence of integrable systems in symplectic manifolds as shown by Brailov (cf. [12]).That is, cover M \ U by Darboux balls, each of them equipped with a local integrable system ofthe form f (cid:48) i = x i + y i . By cutting off this system using a function ϕ ( (cid:80) i =1 ( x i + y i )) , we can obtainfor each Darboux ball a globally defined pair of functions f i = ϕ.f (cid:48) i . We can now cover M \ U by afinite amount of balls B i whose intersection is only the union of their boundaries. We choose ϕ ineach ball such that the locally defined integrable systems vanish in all derivatives exactly at theseboundaries. The closed set of zero measure where the globally constructed n -tuple of functionsare not independent is composed of the boundaries of the balls, and includes Z × {− ε, ε } . Thisis illustrated in Figure 4, where only some balls are depicted close to the boundary of Z × [ − ε, ε ] .The closed set where the functions vanish are represented by the black-colored boundaries. U ∼ = Z × ( − ε, ε ) M \ UB , B , . . .Z F IGURE
4. Some Darboux balls filling M \ U .This allows to glue the system in each ball and with the system we constructed in U , yielding apair of commuting functions F , F such that dF ∧ dF (cid:54) = 0 in a dense set of M and Z . (cid:3) Remark 7.
The proof generalizes to higher dimensions as long as one can construct an integrablesystem in the symplectic leaf invariant by the finite group action. This is the content of Claim 6.2for the case of a symplectic surface.
Remark 8.
The original construction of integrable systems in regular symplectic manifolds via thecovering of Darboux balls yields an integrable system without any interesting property. However,the construction in Theorem 21 gives rise to a lot of examples of b -integrable systems that near thesingular set Z can be very rich from a semi-global point of view. The following theorem is Theorem B in [11]:
Theorem 22.
Any cosymplectic manifold of dimension 3 is the singular locus of orientable, closed, b-symplectic manifolds.
In particular, whenever the cosymplectic manifold has periodic monodromy, it can be realizedas the critical locus of a b -symplectic manifold with a b -integrable system. Theorem 22 requiresspecifically that Z is connected. If we drop that requirement, there is a direct construction (Exam-ple 19 in [15]) to realize any cosymplectic manifold as the singular locus of a b -symplectic manifoldthat we will introduce later.The proof of Theorem 21 can be adapted to obtain folded integrable systems in the desingular-ized folded symplectic manifold resulting from applying Theorem 7. Corollary 23.
Let ( M, ω ) be a b -symplectic manifold in the hypotheses of Theorem 21. Then the desingu-larized folded symplectic manifold ( M, ω ε ) admits a folded integrable system.Proof. The desingularization given by Theorem 7 sends ω to ω ε , which is a folded symplectic struc-ture in M with critical hypersurface Z . The induced structure on Z remains unchanged: it is acosymplectic manifold with compact leaves whose monodromy is periodic. The S -action gener-ated by the modular vector field becomes the null line bundle of ω ε . Such line bundle is generatedin a neighborhood of Z by the Hamiltonian vector field of t , where t is defining function of Z . ByClaim 6.2, there is an analytic function invariant by the first return map X t . One can do exactlythe same construction as in the proof of Theorem , taking as first function f = ϕ ( t ) t instead of ϕ ( t ) log t in the neighborhood U of Z . (cid:3)
7. G
LOBAL ACTION - ANGLE COORDINATES : C
ONSTRUCTIONS AND EXISTENCE
In this section we extend toric actions on the symplectic leaf on the critical set of a b -symplecticmanifold and folded-symplectic manifold to a toric action in the neighbourhood of the criticalset Z . Thus obtaining global action-angle coordinates. For certain compact extensions of thisneighbourhood we obtain global action-angle coordinates on the compact manifold. In doingso, we explore obstructions for the existence of global-action angle coordinates which lie on thecritical set Z .For a global toric action which we combine with the finite group transversal action given by thecosymplectic structure on Z to produce an example of integrable system on any b -symplectic/foldedsymplectic manifold with toric symplectic leaves on the critical set.By doing so, we explore the limitations that this construction has to admit an extension to aglobal toric action and thus admit global action-angle coordinates. This limitation lays on thetopology of the critical set Z which can be an obstruction for the global existence of action-anglecoordinates. In other words, this construction admits global action-angle coordinates if and onlyif the toric structure of the symplectic leaf of the critical set Z extends to a toric action of the b -symplectic/folded symplectic manifolds. Toric symplectic manifolds are well-understood thanksto [16] and [14].In this section we will need to following lemma (which is Corollary 16 in [16]): Lemma 24.
Let ( M n , Z, ω ) be a b -symplectic manifold with a toric action and L a symplectic leaf of Z .Then Z ∼ = L × S . NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 27
Let L be a toric symplectic manifold of dimension n − and let F = ( f , . . . , f n ) be its momentmap.We know from Delzant’s theorem [7] that the image of F is a Delzant polytope. From the defi-nition of moment map the components of F Poisson commute and are functionally independentso they form an integrable system on L . Consider now φ be a symplectomorphism of L which isequivariant with respect to the toric action and let Z = L × [0 , / ∼ be the cosymplectic manifoldassociated to it. Extend the integrable system on Z to an integrable system on Z just by observingthat by hypothesis the toric action commutes with the symplectomorphism defining the cosym-plectic manifold. Observe that the integrable system F on the leaf extends to Z only if Z is aproduct or F is invariant by the monodromy. Denote by ( α, ω ) the pair of 1 and 2-forms associ-ated to the cosymplectic structure i.e, ω restricted to the symplectic leaves defines the symplecticstructure on Z and α is a closed form defining the codimension one symplectic foliation.Following the extension theorem (Theorem 50 in [15]) we consider now the open b -symplecticmanifold U = Z × ( − (cid:15), (cid:15) ) with b -symplectic form, ω = dff ∧ π ∗ ( α ) + π ∗ ( ω ) where π : U → Z stands for the projection in the first component of U , Z and f is the definingfunction for the critical set Z .Consider the map ˆ F = ( c log( t ) , π ∗ ( f ) , . . . , π ∗ ( f n )) on with c the modular period of Z wherewe abuse notation and we write the components on the covering L × [0 , of the mapping torus Z . In this section we prove, Theorem 25.
The mapping ˆ F = ( c log( t ) , π ∗ ( f ) , . . . , π ∗ ( f n )) defines a b -integrable system on the open b -symplectic manifold Z × ( − (cid:15), (cid:15) ) thus extending the integrable system defined by the toric structure of L .The toric structure of L extends to a toric structure on the b -symplectic manifold Z × ( − (cid:15), (cid:15) ) if and only ifthe cosymplectic structure of Z is trivial, i.e., Z = L × [0 , .Proof. Observe that the functions f , . . . , f n define an integrable system on the cosymplectic mani-fold Z as the gluing symplectomorphism that defines the mapping torus commutes with the torusaction defined by F . So this torus action descends to the quotient Z and the functions f i are well-defined on the mapping torus Z . From the definition of b -symplectic form the projection π is aPoisson map and thus { π ∗ ( f i ) , π ∗ ( f j ) } = { f i , f j } = 0 for all i, j ≥ . Observe also that functionalindependence on a dense set W of L , of the functions f , . . . , f n on L (a factor of U ) together withthe functional independence of the pure b -function c log( t ) from the functions π ∗ ( f ) , . . . , π ∗ ( f n ) implies the functional independence on the dense open set W × I with the product topology.Furthermore, the Poisson bracket { c log( t ) , π ∗ ( f j ) } = 0 from the expression of b -symplecticstructure. Thus the system ˆ F defines an integrable system on Z × ( − (cid:15), (cid:15) ) .To conclude observe that the action-angle coordinates associated to the global toric action on L extends to Z (and thus to a neighborhood Z × ( − (cid:15), (cid:15) ) if and only if the action extends to a toricaction. We now use Lemma 24 above to conclude that the toric structure extends to Z if and onlyif the mapping torus is trivial, i.e., Z = L × [0 , . This ends the proof of the theorem. (cid:3) Observe that given any cosymplectic compact manifold Z , then following the construction fromExample 19 in [15], Z × S admits a b -symplectic structure simply by considering the dual b -Poisson structure (where π is the corank regular Poisson structure associated to the cosymplecticstructure and X is a Poisson vector field transverse to the symplectic foliation in Z as it was provedin [17]). The function f is a function vanishing linearly. The critical locus of this b -Poisson structurehas as many copies of the original Z as zeros of the function f . Π = f ( θ ) X ∧ ∂∂θ + π The theorem above admits its compact version:
Theorem 26.
The mapping ˆ F = ( c log( f ( θ )) , π ∗ ( f ) , . . . , π ∗ ( f n )) defines a b -integrable system on the b -symplectic manifold Z × S thus extending the integrable system defined by the toric structure of L . Thetoric structure of L extends to a toric structure on the b -symplectic manifold Z × S if and only if thecosymplectic structure of Z is trivial, i.e., Z = L × [0 , . As a corollary we can detect situations in which the topological obstruction to global existenceof action-angle coordinates lies in the non-triviality of the mapping torus defined by the criticalset Z . Theorem 27.
Any b -integrable system on b -symplectic manifold extending a toric system on a symplecticleaf of Z does not admit global action-angle coordinates whenever the critical set Z is not a trivial mappingtorus Z = L × [0 , . Below we show an example of a b -symplectic manifold M of dimension an admits some b -integrable system which is not toric even though the leaves on the critical hypersurface are toric.Observe that the b -integrable system cannot define a toric action (and thus admit global action-angle coordinates) because of the topological structure of Z . Example (Topological obstruction to semi-local action-angle coordinates) . Consider a product ofspheres S × S with coordinates ( h , θ , h , θ ) and standard product symplectic form ω = dh ∧ dθ + dh ∧ dθ . The map ϕ : S × S −→ S × S ( p, q ) (cid:55)−→ ( q, p ) is a symplectomorphism satisfying that ϕ = Id . The induced map in homology swaps the generators of H ( S × S ) ∼ = H ( S ) ⊕ H ( S ) . This shows that ϕ is not in the connected component of the identity,since this would imply that induced map in homology would act trivially [24, Theorem 2.10] . Thus, themapping torus with gluing diffeomorphism ϕ cannot be a trivial product S × S × S .The pair of functions F = ( f , f ) = ( h + h , h h ) are invariant with respect to ϕ and hence descendto the mapping torus. Furthermore, they define an integrable system in S × S , since they clearly Poissoncommute and satisfy that df ∧ df = ( h − h ) dh ∧ dh (cid:54) = 0 almost everywhere. In particular, by Remark7, any b -symplectic manifold with critical set Z admits a b -integrable system. However since the criticalhypersurface is not a trivial product, any b -integrable system will not be toric in a neighborhood of Z .By the discussion before Theorem 26, we know that we can realize this cosymplectic manifold N as aconnected component of a critical hypersurface of a compact b -symplectic manifold diffeomorphic to M = N × S . Any b -integrable system in M will not be toric even in a neighborhood of Z . Observe that with the magic of the desingularization trick we obtain examples of folded-integrablesystems without global action-angle coordinates just by applying Theorem 27 and the behaviourof torus actions under desingularization studied in [19].
NTEGRABLE SYSTEMS ON SINGULAR SYMPLECTIC MANIFOLDS: FROM LOCAL TO GLOBAL 29
Theorem 28.
Let F be a folded integrable system obtained by desingularization of a b -integrable system,if the critical set Z of the original b -symplectic structure is not a trivial mapping torus, then the foldedintegrable system F does not admit global action-angle coordinates. Let us finish this article with a couple of concluding remarks: • For symplectic manifolds the obstructions to global action-angle coordinates started withDuistermaat in his seminal paper [9] where Duistermaat related the existence of obstruc-tions to the existence of monodromy which in its turn was naturally associated to the exis-tence of singularities.In this article we have concluded that for a singular symplectic manifold there are topo-logical obstructions for existence of global action-angle coordinates that are detectable atfirst sight: The critical set Z has to be a trivial mapping torus Z = L × [0 , thus theexistence of monodromy associated to this mapping torus is also an obstruction. • Furthermore, the existence of action-angle coordinates yields a free action of a torus inthe neighbourhood of a regular torus action thus the existence of isotropy groups for thecandidate of torus action defining the system, automatically implies that the locus withnon-trivial isotropy groups is singular for the integrable system. The same holds for asub-circle. In particular:
Corollary 29.
Let F be an b -integrable system on a b -symplectic manifold and denote by T the union ofthe exceptional orbits of the S -action described in Proposition 19. Then the system has singularities at theset T . Thus not only the topology of the critical set Z yields an obstruction to existence of globalaction-angle coordinates but it also detects singularities of integrable systems. In particular alongthe exceptional orbits for the transverse S -action given by Proposition 19. This motivates us tostudy singularities of integrable systems on singular symplectic manifolds, study which we willpursue in a different article. R EFERENCES [1] V. I. Arnold.
Mathematical Methods of Classical Mechanics . Grad. Texts in Math. 60, Springer-Verlag, Berlin (1978r).[2] A. Cannas da Silva.
Fold-forms for four-folds.
J. Symplectic Geom. 8 (2010), no. 2, 189-203.[3] R. Braddell, A. Kiesenhofer, E. Miranda.
Cotangent models for group actions in b -Poisson manifolds . Preprint (2018)arXiv:1811.11894.[4] M.Braverman, Y. Loizides, Y. Song, Geometric quantization of b-symplectic manifolds , arXiv:1910.10016.[5] A. Cannas da Silva, V. Guillemin, C. Woodward.
On the unfolding of folded symplectic structures . Math. Res. Lett.,7(1):35-53, 2000.[6] A. Cannas da Silva, V. Guillemin, A. R. Pires.
Symplectic origami . Int. Math. Res. Not. IMRN 2011, no. 18, 4252-4293.[7] T. Delzant.
Hamiltoniens p´eriodiques et images convexes de l’application moment . Bulletin de la SMF tome 116 num. 3 p.315-339, 1988.[8] A. Delshams, A. Kiesenhofer, E. Miranda,
Examples of integrable and non-integrable systems on singular symplecticmanifolds , J. Geom. Phys. 115 (2017), 89–97. .[9] J.J Duistermaat,
On global action-angle coordinates , Communications on pure and applied mathematics, , (1980),687–706.[10] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein , Verhandlungen der Deutschen PhysikalischenGesellschaft. 19: 82-92 (1917).[11] P. Frejlich, D. Mart´ınez, E. Miranda.
A note on symplectic topology of b -manifolds . J. Symplectic Geom. 15 (2017), no.3, 719-739.[12] A.T. Fomenko, V.V. Trofimov. Integrable Systems on Lie Algebras and Symmetric Spaces . Gordon and Breach, Amster-dam (1988).[13] M. Gualtieri, S. Li.
Symplectic groupoids of log symplectic manifolds . International Mathematics Research Notices,2014(11), 3022-3074. [14] M. Gualtieri, S. Li, A. Pelayo, T. Ratiu.
The tropical momentum map: a classification of toric log symplectic manifolds.
Math. Ann. 367 (2017), no. 3–4, 1217–1258.[15] V. Guillemin, E. Miranda, A.R. Pires.
Symplectic and Poisson geometry on b -manifolds . Adv. Math. 264 (2014), 864-896.[16] V. Guillemin, E. Miranda, A.R. Pires, G. Scott. Toric actions on b-symplectic manifolds . Int. Math. Res. Not. IMRN2015, no. 14, 5818–5848.[17] V. Guillemin, E. Miranda, A.R. Pires.
Codimension one symplectic foliations and regular Poisson structures . Bull. Braz.Math. Soc. (N.S.) 42 (2011), no. 4, 607–623.[18] V. Guillemin, E. Miranda, J. Weitsman.
Desingularizing b m -symplectic structures . Int. Math. Res. Not. IMRN 2019, no.10, 2981–2998.[19] V. Guillemin, E. Miranda, J. Weitsman. Convexity of the moment map image for torus actions on b m -symplectic manifolds .Phil. Trans. Roy. Soc. A, 2018 376 20170420; DOI: 10.1098/rsta.2017.0420.[20] V. Guillemin, E. Miranda and J. Weitsman, On geometric quantization of b -symplectic manifolds , Adv. Math. , 941-951 (2018)[21] V. Guillemin, E. Miranda and J. Weitsman, On geometric quantization of b m -symplectic manifolds , Math. Z., to appear(2020)[22] V. Guillemin and S. Sternberg, The Gelfand-Cetlin system and quantization of the complex flag manifolds.
J. Funct. Anal.52 (1983), no. 1, 106–128.[23] M. Hamilton, Locally toric manifolds and singular Bohr-Sommerfeld leaves. Mem. Amer. Math. Soc. 207 (2010),no. 971, vi+60 pp. ISBN: 978-0-8218-4714-5[24] A. Hatcher.
Algebraic topology . Cambridge Univ. Press, Cambridge, 2002.[25] D. Hockensmith.
A classification of toric, folded-symplectic manifolds . PhD Thesis arXiv:1511.08108, 2015.[26] T.S. Holm, A.R. Pires.
The fundamental group and Betti numbers of toric origami manifolds.
Algebr. Geom. Topol. 15(2015), no. 4, 2393–2425.[27] T.S. Holm, A.R. Pires.
The topology of toric origami manifolds . Math. Res. Lett. 20 (2013), no. 5, 885–906.[28] A. Kiesenhofer.
Integrable systems on b-symplectic manifolds . PhD Thesis, October 2016.[29] A. Kiesenhofer, E. Miranda, G. Scott.
Action-angle variables and a KAM theorem for b -Poisson manifolds . Journal desMath´ematiques Pures et Appliqu´ees, J. Math. Pures Appl. (9) 105 (2016), no. 1, 66-85.[30] A. Kiesenhofer, E. Miranda. Cotangent models for integrable systems . Comm. Math. Phys. 350 (2017), no. 3,1123–1145.[31] C. Laurent-Gengoux, E. Miranda, P. Vanhaecke.
Action-angle Coordinates for Integrable Systems on Poisson Manifolds .International Mathematics Research Notices, Volume 2011, Issue 8, 1 January 2011, pages 1839–1869.[32] I. Marcut, B. Osorno.
On cohomological obstructions for the existence of log-symplectic structures . J. Symplectic Geom.12 (2014), no. 4, 863-866[33] J. Martinet.
Sur les singularit´es des formes diff´erentielles . Ann. Inst. Fourier (Grenoble) 20 1970 fasc. 1, 95-178.[34] R. Melrose.
Atiyah-Patodi-Singer Index Theorem (book). Research Notices in Mathematics, A.K. Peters, Wellesley,1993.[35] E. Miranda and A. Planas,
Equivariant classification of b m -symplectic surfaces , Regul. Chaotic Dyn. 23 (2018), no. 4,355–371.[36] R. Nest, B. Tsygan. Formal deformations of symplectic manifolds with boundary . J. Reine Angew. Math. 481, 1996, pp.27–54.[37] O. Radko.
A classification of topologically stable Poisson structures on a compact oriented surface . Journal of SymplecticGeometry, 1, no. 3, 2002, pp. 523–542.[38] G. Scott.
The Geometry of b k Manifolds . J. Symplectic Geom. 14 (2016), no. 1, 7195.[39] A. Wasserman.
Equivariant differential topology . Topology 8 (1969), 127-150.[40] A. Weinstein.
Lectures on symplectic manifolds . Published for the Conference Board of the Mathematical Sciences bythe American Mathematical Society (1977), Providence. R.I.[41] J. Weitsman,
Non-abelian symplectic cuts and the geometric quantization of non-compact manifolds , Letters in Mathemat-ical Physics 56 (2001) 31-40.L
ABORATORY OF G EOMETRY AND D YNAMICAL S YSTEMS , D
EPARTMENT OF M ATHEMATICS , U
NIVERSITAT P OLIT ` ECNICADE C ATALUNYA AND
BGSM
ATH , B
ARCELONA , S
PAIN
E-mail address : [email protected] L ABORATORY OF G EOMETRY AND D YNAMICAL S YSTEMS D EPARTMENT OF M ATHEMATICS , U
NIVERSITAT P OLIT ` ECNICADE C ATALUNYA AND
BGSM
ATH , B
ARCELONA , S
PAIN , AND , IMCCE, CNRS-UMR8028, O
BSERVATOIRE DE P ARIS ,PSL U
NIVERSITY , S
ORBONNE U NIVERSIT ´ E , 77 A VENUE D ENFERT -R OCHEREAU , 75014 P
ARIS , F
RANCE
E-mail address ::