SOME PROPERTIES OF LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS

SOME PROPERTIES OF LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS STEFAN HALLER Abstrat. The aim of this note is to give an overview of what loally onformal sympleti manifolds are and to present some results, well known from sympleti geometry, whih an be generalized to this slightly larger ategory. 1. Loally onformal sympleti manifolds A loally onformal sympleti manifold, l..s. for short, is a triple (M, Ω, ω), where M is a finite dimensional smooth manifold, Ω is a non-degenerate 2-form on M and ω is a losed 1-form on M, suh that d ω Ω = 0, where d ω : Ω (M) Ω +1 (M), d ω α := dα + ω α is the Witten deformed differential. We will always assume, that M is onneted and M =. Remark 1. The dimension of an l..s. manifold has to be even. Sine Ω n is nowhere vanishing an l..s. manifold possesses a anoni orientation. Remark 2. If dim(m) > 2 then ω is uniquely determined by Ω. Indeed if d ω Ω = d ω Ω = 0 and ω x ω x at a point x M, then (ω x ω x ) Ω x = 0 and hene Ω x = (ω x ω x ) α for some α T x M, but this ontradits the non-degeneray of Ω at the point x. Remark 3. If (Ω, ω) is an l..s. struture on M and a C (M, R) then (e a Ω, ω da) is again an l..s. struture on M. Two l..s. strutures (Ω, ω) and (Ω, ω ) on M are alled onformally equivalent if (Ω, ω ) = (e a Ω, ω da) for some a C (M, R). We will write (Ω, ω) (Ω, ω ) in this ase. Note that the anoni orientation does only depend on the onformal equivalene lass of (M, Ω, ω). Remark 4. Suppose ω is a 1-form on M and onsider the trivial line bundle E := M R M with the onnetion ω Xf := X f + ω(x)f. For its urvature one easily heks R ω = dω. Via the obvious identifiation Ω (M; E) = Ω (M) the ovariant exterior derivative of E-valued differential forms is d ω = d ω. For losed ω one has d ω d ω = 0, orresponding to the fat that the onnetion ω is flat in this ase. Moreover, for two 1-forms ω 1 and ω 2 one has d ω1+ω2 (α β) = (d ω1 α) β + ( 1) α α (d ω2 β), (1) whih orresponds to the fat ω1 ω2 = ω1+ω2. 1991 Mathematis Subjet Classifiation. 53D99. Key words and phrases. loally onformal sympleti manifold, simple group, Erlanger Programm, sympleti redution. The author is supported by the Fonds zur Förderung der wissenshaftlihen Forshung (Austrian Siene Fund), projet number P14195-MAT.. 1

2 STEFAN HALLER Remark 5. Let ω be a losed 1-form on M. Sine d ω d ω = 0 one obtains a twisted derham ohomology Hd ω(m). It is easy to see, that this is the ohomology of the loally onstant sheave of d ω -onstant funtions, i.e. loally defined funtions f, satisfying d ω f = 0. Note, that sine d ω Ω = 0, Ω defines a d ω -ohomology lass [Ω] Hd 2 ω(m), but even if M is losed this lass might vanish, f. Example 3 below. From the derivation like formula (1) one sees, that the wedge produt indues : H k1 d ω 1 (M) H k2 d ω 2 (M) H k1+k2 d (M). ω 1 +ω 2 If ω = ω da for a C (M, R) then e a : Hd ω(m) = H dω (M). So for [ω] = 0 H 1 (M) the twisted derham ohomology is isomorphi to the ordinary derham ohomology. For [ω] 0 H 1 (M) one easily shows Hd 0 ω(m) = 0 and H0 d (M) = 0, ω where the latter denotes ohomology with ompat supports. Together with the Poinaré duality PD : Hd k ω(m) ( H dim(m) k (M) ) d, ω PD(α)(β) = M α β, one obtains H dim(m) d (M) = 0 and H dim(m) ω d (M) = 0 for all [ω] 0 H 1 (M), see ω [9] and [10]. Example 1. Sympleti manifolds are l..s. manifolds with vanishing ω. Moreover (M, Ω, ω) is onformally equivalent to a sympleti manifold iff [ω] = 0 H 1 (M). Example 2. If ω is a losed 1-form on a manifold N then (T N, d π ω θ, π ω) is an l..s. manifold, where θ is the anonial 1-form on T N and π : T N N denotes the projetion. Example 3 (Banyaga). If (N, α) is a ontat manifold then (N S 1, d ν α, ν) is an l..s. manifold, where ν is the standard volume form on S 1. For example S 3 S 1 is an l..s. manifold, although it does not admit a sympleti struture, for H 2 (S 3 S 1 ) = 0. Note, that H d ν (N S1 ) = 0, sine one has a Künneth theorem and H d ν (S1 ) = 0. Example 4 (Guedira and Lihnerowiz). There is a natural orrespondene between even dimensional transitive Jaobi manifolds and l..s. manifolds. In partiular, every even dimensional leave of a Jaobi manifold posses a natural l..s. struture, f. [9]. The odd dimensional transitive Jaobi manifolds, are in natural one-to-one orrespondene with ontat strutures. Partiularly every odd dimensional leave of a Jaobi manifold is a ontat manifold. Example 5 (Lee). Suppose U is an open overing of M and for every U U, we have given an l..s. struture (Ω U, ω U ) on U, suh that (Ω U, ω U ) U V (Ω V, ω V ) U V, for all U, V U. One easily shows, f. [16], that in this situation there exists an l..s. struture (Ω, ω) on M, suh that (Ω, ω) U (Ω U, ω U ) for all U U. Obviously (Ω, ω) is unique up to onformal equivalene. For an l..s. manifold (M, Ω, ω) let us denote by Diff (M, Ω, ω) := { g Diff (M) (g Ω, g ω) (Ω, ω) } the group of ompatly supported diffeomorphisms preserving the onformal equivalene lass of (Ω, ω). The orresponding Lie algebra of vetor fields is X (M, Ω, ω) := { X X (M) X R : L ω XΩ = X Ω },

SOME PROPERTIES OF LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS 3 where L ω X : Ω (M) Ω (M), L ω Xα := L X α + ω(x)α, for a vetor field X X(M) and a losed 1-form ω. Note that Diff (M, Ω, ω) and X (M, Ω, ω) do only depend on the onformal equivalene lass of (M, Ω, ω). Remark 6. Note, that for two losed 1-forms ω 1 and ω 2 one has L ω1+ω2 X (α β) = (L ω1 X α) β + α (Lω2 X β). Moreover L ω X Lω Y Lω Y Lω X = Lω [X,Y ], Lω X dω d ω L ω X = 0, Lω X i Y i Y L ω X = i [X,Y ] and d ω i X + i X d ω = L ω X, for a losed 1-form ω and vetor fields X, Y X(M). Theorem 1 (Weinstein hart). Suppose (M, Ω, ω) is an l..s. manifold. Then Diff (M, Ω, ω) is a regular Lie group with Lie algebra X (M, Ω, ω) in the sense of [15]. Partiularly, if M is ompat then Diff (M, Ω, ω) is a regular Fréhet-Lie group with Lie algebra X(M, Ω, ω). The proof, see [10] and [12], is a generalization of a well known onstrution due to A. Weinstein, f. [24]. 2. Infinitesimal invariants Reall that X X (M, Ω, ω) iff L ω X Ω = XΩ for some onstant X R. The following mapping, alled extended Lee homomorphism f. [9] or [23], ϕ : X (M, Ω, ω) R, ϕ(x) := X is a Lie algebra homomorphism, where R is onsidered as Abelian Lie algebra. Indeed, using the formulas in Remark 6 we get L ω [X,Y ] Ω = Lω XL ω Y Ω L ω Y L ω XΩ = L ω X( Y Ω) L ω Y ( X Ω) = (L X Y )Ω + Y L ω XΩ (L Y X )Ω X L ω Y Ω = ( Y X X Y )Ω = 0. Note that X (M, Ω, ω) and ϕ do only depend on the onformal equivalene lass of (M, Ω, ω). Indeed, if L ω X Ω = XΩ and a C (M, R), we get L ω da X (e a Ω) = (L da ea )Ω + e a L ω XΩ = X (e a Ω), X sine obviously L da X ea = 0. Remark, that ϕ has to vanish if M is non-ompat or [ω] = 0 H 1 (M). So it does not appear in the sympleti ase, at least as long as one onsiders ompatly supported vetor fields. The next invariant generalizes the Flux homomorphism to the l..s. ase. The onept is essentially due to E. Calabi, f. [7]. The mapping ψ : ker ϕ H 1 d ω (M), ψ(x) := [i XΩ] is a surjetive Lie algebra homomorphism, where Hd 1 (M) is onsidered as Abelian ω Lie algebra. It is well defined and onto, sine X X (M, Ω, ω) iff d ω i X Ω = L ω X Ω = 0 and sine Ω is non-degenerate. To see that ψ vanishes on brakets one uses the formulas in Remark 6 to get i [X,Y ] Ω = L ω Xi Y Ω i Y L ω XΩ = d ω i X i Y Ω + i X d ω i Y Ω = d ω i X i Y Ω + i X L ω Y Ω i X i Y d ω Ω = d ω i X i Y Ω.

4 STEFAN HALLER If ω = ω da then e a : Hd (M) = H (M), see Remark 5, and ψ (X) = ω d ω e a ψ(x). In partiular, ker ψ does only depend on the onformal equivalene lass of (M, Ω, ω). Suppose M is of dimension 2n. Then the following is a surjetive Lie algebra homomorphism ρ : ker ψ H 2n d (n+1)ω (M) /( H 0 d ω (M) [Ωn ] ), ρ(x) := [h X Ω n ], where h X is a funtion with ompat support, suh that d ω h X = i X Ω. Again ker ρ does only depend on the onformal equivalene lass of (M, Ω, ω). But note, that the odomain of ρ is non-trivial only if M is non-ompat and [ω] = 0 H 1 (M), i.e. (M, Ω, ω) is onformally equivalent to a sympleti manifold. The invariant ρ is sometimes alled Calabi invariant, f. [7], [2], [10] and [11]. Remark 7. Note, that X ker ρ iff there exist h Ω 0 (M) and α Ω 2n 1 (M), suh that i X Ω = d ω h and hω n = d (n+1)ω α. The following is due to E. Calabi in the sympleti ase. Theorem 2. Let (M, Ω, ω) be an l..s. manifold. Then ker ρ is a perfet Lie algebra, i.e. [ker ρ, ker ρ] = ker ρ. Remark 8. Theorem 2 permits to ompute the derived series 1 of the Lie algebra X (M, Ω, ω), see [10]. The only algebras in the derived series of X (M, Ω, ω) are ker ϕ, ker ψ and ker ρ, but there is a ase 2, where [X (M, Ω, ω), X (M, Ω, ω)] = ker ρ and ker ψ ker ρ. However, in any ase one has D 2 X (M, Ω, ω) = ker ρ. If (M, Ω, ω) is an l..s. manifold and U M open, we will denote by ϕ U, ψ U and ρ U the orresponding invariants of the l..s. manifold (U, Ω U, ω U ). Note that we have X (U, Ω U, ω U ) X (M, Ω, ω), ker ϕ U ker ϕ, ker ψ U ker ψ and ker ρ U ker ρ, given by extending a vetor field by zero. Lemma 1 (Fragmentation lemma). Let (M, Ω, ω) be an l..s. manifold, U be an open overing of M and suppose X X (M, Ω, ω). Then for every X ker ρ there exist N N, U 1,..., U N U and X i ker ρ Ui, suh that X = N i=1 X i. Proof. By Remark 7 we find h Ω 0 (M) and α Ω 2n 1 (M), suh that i X Ω = d ω h and hω n = d (n+1)ω α. Choose N N and U 1,..., U N U whih over supp α. Choose a partition of unity {λ 0, λ 1,..., λ N } subordinated to the open overing {M \supp α, U 1,..., U N }, i.e. supp λ 0 M \supp α and supp λ i U i for 1 i N. For 0 i N define h i Ω 0 (M), by h i Ω n := d (n+1)ω (λ i α) and X i X (M, Ω, ω) by i Xi Ω := d ω h i. Sine we have N N h i Ω n = d (n+1)ω λ i α = d (n+1)ω α = hω n, i=0 i=0 we get N i=0 h i = h and hene N i=0 X i = X. Moreover X 0 = 0 and X i ker ρ Ui for 1 i N. 1 The derived series of a Lie algebra g is defined by D 0 g := g and D i+1 g := [D i g, D i g]. 2 M non-ompat, [ω] = 0 H 1 (M) and vanishing sympleti pairing {, } : Hd 1 ω (M) H1 d ω (M) H2n d (n+1)ω (M), {α 1, α 2 } := α 1 α 2 [Ω n 1 ].

SOME PROPERTIES OF LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS 5 The following an be found in [1], where it is used to show Theorem 2 in the sympleti ase. Lemma 2. Let U V be open subsets of R 2n equipped with the standard sympleti form Ω = dx 1 dx 2 +, suh that Ū V. For all X ker ρ U there exist Y i, Z i ker ρ V, suh that X = 2n i=1 [Y i, Z i ]. Partiularly ker ρ U [ker ρ V, ker ρ V ]. Proof of Theorem 2. We have to show ker ρ [ker ρ, ker ρ]. In view of Lemma 1 we may assume, that (M, Ω, ω) is an open ball in R 2n equipped with the standard sympleti struture, but this ase follows from Lemma 2. A well known theorem of L.E. Pursell and M.E. Shanks see [19] states, roughly speaking, that a smooth manifold is ompletely determined by its Lie algebra of vetor fields. More preisely, if there exists an isomorphism of the Lie algebras of vetor fields then there exists a unique diffeomorphism between the manifolds, induing the given Lie algebra isomorphism. H. Omori proved several generalizations, namely the Lie algebra of vetor fields preserving a sympleti form resp. a volume form uniquely determines the manifold together with the sympleti resp. volume struture up to multipliation with a onstant, see [18]. We will show an analogous statement for l..s. strutures, i.e. any of the Lie algebras X (M, Ω, ω), ker ϕ, ker ψ, ker ρ uniquely determines the l..s. manifold (M, Ω, ω) up to onformal equivalene. More preisely one has the following Theorem 3. Let (M i, Ω i, ω i ), i = 1, 2, be two l..s. manifolds and assume that κ is a Lie algebra isomorphism from one of the Lie algebras X (M 1, Ω 1, ω 1 ), ker ϕ 1, ker ψ 1, ker ρ 1 onto one of the Lie algebras X (M 2, Ω 2, ω 2 ), ker ϕ 2, ker ψ 2, ker ρ 2. Then there exists a unique diffeomorphism g : M 1 M 2, suh that κ = g. Moreover we have (M 1, Ω 1, ω 1 ) (M 1, g Ω 2, g ω 2 ). For the proof we first state a slightly weaker statement and a simple lemma. Proposition 1. Let (M i, Ω i, ω i ), i = 1, 2, be two l..s. manifolds and let κ : ker ρ 1 ker ρ 2 a Lie algebra isomorphism. Then there exists a unique diffeomorphism g : M 1 M 2 suh that κ = g. Moreover (M 1, Ω 1, ω 1 ) (M 1, g Ω 2, g ω 2 ). Lemma 3. Let g be a Lie algebra suh that ad : g L([g, g]) is injetive and let λ : g g be a Lie algebra homomorphism, suh that λ [g,g] = id. Then λ = id. Proof. For X g and Y [g, g] we have [X, Y ] = λ([x, Y ]) = [λ(x), λ(y )] = [λ(x), Y ], hene ad(x λ(x)) = 0 L([g, g]) and so λ(x) = X. Proof of Theorem 3. We have a Lie algebra isomorphism κ : g 1 g 2. Sine D 2 g i = ker ρ i, see Remark 8, κ restrits to an isomorphism κ ker ρ1 : ker ρ 1 ker ρ 2. From Proposition 1 we obtain a unique diffeomorphism g : M 1 M 2 suh that κ ker ρ1 = g ker ρ1. Moreover (M 1, Ω 1, ω 1 ) (M 1, g Ω 2, g ω 2 ). So g g 2 is one of the Lie algebras X (M, Ω 1, ω 1 ), ker ϕ 1, ker ψ 1, ker ρ 1 and we either have g g 2 g 1 or g g 2 g 1. Assume we are in the first ase (for the seond onsider g 1 ). Then λ := g 1 κ = g κ : g 1 g 1 is a Lie algebra homomorphism and we know that λ D2 g 1 = id. Moreover we obviously have for every vetor field Z X(M 1 ) the following property: [Z, X] = 0 for all X ker ρ 1 Z = 0.

6 STEFAN HALLER Using ker ρ 1 D i+1 g 1 we obtain ad : D i g 1 L([D i g 1, D i g 1 ]) = L(D i+1 g 1 ) is injetive for all i. So we an apply Lemma 3 twie to obtain λ D 1 g 1 = id and λ = λ D0 g 1 = id, i.e. g = κ. Proof of Proposition 1. We will only sketh the proof, sine it is very similar to the sympleti ase, as soon as one has the fragmentation Lemma 1. For more details see [10] or [18] in the sympleti ase. Suppose (M, Ω, ω) is an l..s. manifold. Then for every x M, I x := {X ker ρ X is flat at x} is a maximal ideal in ker ρ, and every maximal ideal of ker ρ is of this form. This permits to define a bijetion g : M 1 M 2 by I g(x) = κ(i x ). For every A M one has Ā = { x M } I y I x, and hene g is a homeomorphism. Next one shows, that for X ker ρ and x M y A X(x) 0 [X, ker ρ] + I x = ker ρ. So X i ker ρ 1 are linearly independent at x iff κ(x i ) are linearly independent at g(x). Using a Darboux hart and vetor fields in ker ρ 1 involving the oordinate funtions, one shows, that g is a diffeomorphism and that (g Ω 2, g ω 2 ) (Ω 1, ω 1 ). 3. Integrated invariants A well known theorem of W.P. Thurston states that Diff (M) o is a simple group, see [22]. His proof used a theorem due to M.R. Herman see [14], whih solves the problem for the torus. The group of volume preserving diffeomorphisms is not simple in general, but there exists a homomorphism and its kernel is simple. This was shown by W.P. Thurston, see [6]. A. Banyaga showed an analogous statement in the sympleti ase, see [2]. In [11] this was generalized to l..s. manifolds. The infinitesimal invariants ϕ, ψ and ρ an by integrated to group homomorphisms Φ, Ψ and R, see [11]. The kernels of these homomorphisms do only depend on the onformal equivalene lass of (M, Ω, ω). Theorem 4. Let (M, Ω, ω) be an l..s. manifold. Then ker R is a simple group. The main ingredients in the proof are Banyagas theorem for sympleti open balls and the following fragmentation lemma, whih an be found in [10] or [11]. Lemma 4 (Fragmentation lemma). Let (M, Ω, ω) be an l..s. manifold and let U be an open overing of M. Then for any g C ( (I, 0), (ker R, id) ) there exist N N, U i U and g i C ( (I, 0), (ker R Ui, id) ), suh that g t = gt 1 gt N, for all t I = [ 1, 1]. 3 Remark 9. Theorem 4 permits to ompute the derived series 4 of the onneted omponent Diff (M, Ω, ω) o. All groups in this derived series are ker Φ, ker Ψ or ker R and it is preisely the integral ounterpart of the derived series of X (M, Ω, ω), 3 For an l..s. manifold (M, Ω, ω) and U M open, RU denotes the orresponding invariant for the l..s. manifold (U, Ω U, ω U ). In partiular every g i in the proposition is supported in U i. 4 Reall, that for a group G, the derived series is defined by D 0 G := G and D i+1 G := [D i G, D i G].

SOME PROPERTIES OF LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS 7 f. Remark 8. Partiularly one obtains D 2 Diff (M, Ω, ω) o = ker R for every l..s. manifold (M, Ω, ω). Filipkiewiz showed that a smooth manifold is uniquely determined by its group of diffeomorphisms. That is, if two manifold have isomorphi diffeomorphism groups then the underlying manifolds are diffeomorphi, see [8]. He used tehniques developed in [25] and [21] who proved analogous statements in the topologial setting. There are many generalizations to other geometri strutures, see [3], [4], [5] and [20]. A similar result is true for l..s. manifolds. More preisely we have Theorem 5. Let (M i, Ω i, ω i ) be two l..s. manifolds, i = 1, 2, and suppose κ : G 1 G 2 is an isomorphism from one of the groups Diff (M 1, Ω 1, ω 1 ), ker Φ 1, ker Ψ 1, ker R 1 onto one of the groups Diff (M 2, Ω 2, ω 2 ), ker Φ 2, ker Ψ 2, ker R 2. Then there exists a unique homeomorphism g : M 1 M 2 suh that κ(h) = g h g 1 for all h G 1. Moreover g is a diffeomorphism and (M 1, Ω 1, ω 1 ) (M 1, g Ω 2, g ω 2 ). To prove Theorem 5 one first observes, that sine one has the fragmentation Lemma 4 and Proposition 1, a result due to T. Rybiki, see [20], an be applied, whih yields the following Proposition 2. Let (M i, Ω i, ω i ), i = 1, 2, be two l..s. manifolds and suppose κ : ker R 1 ker R 2 is an isomorphism of groups. Then there exists a unique homeomorphism g : M 1 M 2 suh that κ(h) = g h g 1 for all h ker R 1. Moreover g is a diffeomorphism and (M 1, Ω 1, ω 1 ) (M 1, g Ω 2, g ω 2 ). Moreover, we have a group analogue to Lemma 3: Lemma 5. Let G be a group suh that onj : G Aut([G, G]) is injetive and let λ : G G be a homomorphism, suh that λ [G,G] = id. Then λ = id. Proof. For g G and h [G, G] we have [g, h] = λ([g, h]) = [λ(g), λ(h)] = [λ(g), h], hene onj g 1 λ(g) = id Aut([G, G]), and by injetivity λ(g) = g. Proof of Theorem 5. The restrition of κ is an isomorphism κ D 2 G 1 : D 2 G 1 D 2 G 2. In any ase D 2 G i = ker R i for i = 1, 2, by Remark 9. So we may apply Proposition 2 and obtain a unique homeomorphism g : M 1 M 2 suh that κ(h) = ghg 1 for all h ker R 1 = D 2 G 1. Moreover g is a diffeomorphism and (M 1, Ω 1, ω 2 ) (M 1, g Ω 2, g ω 2 ). So it remains to show that κ(h) = onj g (h) := ghg 1 for all h G 1. From (M 1, Ω 1, ω 2 ) (M 1, g Ω 2, g ω 2 ) we see that onj g 1(G 2 ) G 1 or onj g 1(G 2 ) G 1. Assume we are in the first ase (otherwise onsider g 1 ). Then λ := onj g 1 κ : G 1 G 1 is a homomorphism and λ D2 G 1 = id. Moreover, for h Diff (M 1 ) we have: [h, k] = id k ker R 1 h = id, sine ker R 1 ats 2-transitive on M 1. Using ker R 1 D i+1 G 1 we obtain onj : D i G 1 Aut([D i G 1, D i G 1 ]) = Aut(D i+1 G 1 ) is injetive for all i. So we an apply Lemma 5 twie and obtain first λ D1 G 1 = id and finally λ = λ D0 G 1 = id, i.e. κ = (onj g ) G1. 4. Sympleti redution The aim of this last setion is to show that, in the regular ase, the sympleti redution is possible for l..s. manifolds. In the paper of Marsden and Weinstein [17] it has been formalized the fat that if an n-dimensional symmetry group ats

8 STEFAN HALLER on a Hamiltonian system then the number of degrees of freedom an be redued by n, and the dimension of the phase spae is redued by 2n. This is still true for l..s. manifolds. For proofs see [13]. Every l..s. manifold is a Jaobi manifold, so C (M, R) has a Lie algebra struture. More expliitly, the braket is given by {f 1, f 2 } := Ω(X f1, X f2 ) = L ω X f2 f 1 = L ω X f1 f 2. Here X f denotes the Hamiltonian vetor field to f C (M, R) given by i Xf Ω = d ω f. One has X {f1,f 2} = [X f1, X f2 ] and 0 H 0 d ω(m) C (M; R) Ham(M, Ω, ω) 0 is, up to a sign, a entral extension of Lie algebras, where Ham(M, Ω, ω) denotes the Lie algebra of Hamiltonian vetor fields. For ompat M one has Ham(M, Ω, ω) = ker ψ, but note that in this setion we onsider vetor fields with arbitrary support. Let G be a finite dimensional Lie group with Lie algebra g, ating from the left on M. We will write l g : M M for the ation of the element g G. We will always assume that G ats sympletially, i.e. l g Diff (M, Ω, ω) for all g G. For Y g let ζ Y X(M, Ω, ω) denote the fundamental vetor field to Y. If there exists a Lie algebra homomorphism ˆµ : g C (M, R), suh that Xˆµ(Y ) = ζ Y for all Y g, we define µ : M g by µ(x), Y = ˆµ(Y )(x), Y g, x M, and all µ an equivariant moment mapping for the ation. Note, that in this ase the ation of G o is Hamiltonian. One easily shows Proposition 3. If H 2 (g; Hd 0 ω(m)) = 0, where H0 dω(m) is onsidered as trivial g-module, then there exists an equivariant moment mapping. Moreover, if an equivariant moment mapping exists, then the set of all equivariant moment mappings is naturally parametrized by H 1 (g; Hd 0 ω(m)). Remark 10. If g is semi simple then there always exists a unique equivariant moment mapping, sine H 2 (g; Hd 0 ω(m)) = 0 and H1 (g; Hd 0 ω(m)) = 0 by the seond and the first Whitehead lemma, respetively. If the l..s. manifold is not onformally equivalent to a sympleti manifold, i.e. [ω] 0 H 1 (M), then there always exists a unique equivariant moment mapping, for Hd 0 ω(m) = 0 in this ase. Suppose that the l..s. struture is exat, i.e. Ω = d ω θ, and that the ation preserves θ, i.e. L ω ζ Y θ = 0. Then ˆµ(Y ) := i ζy θ defines an equivariant moment mapping. If the G-ation admits an equivariant moment mapping µ, then the vetor fields whih are Ω-orthogonal to the orbits span an involutive distribution. Suppose L is a maximal onneted submanifold of M, tangential to this distribution. Then ˆµ(L) R + µ(x 0 ) for every x 0 L. However ˆµ need not be onstant along L. Let g L := {Y g : ad Y ˆµ(x 0 ) = 0}, whih does not depend on x 0 g. Then g L are preisely those Y g, for whih ζ Y is tangential to L. Sine L is maximal, the onneted subgroup of G orresponding to g L g leaves L invariant. In [13] the following generalization of sympleti redution, see [17], is proved: Theorem 6. Let G be a finite dimensional Lie group ating sympletially on an l..s. manifold (M, Ω, ω) and assume that the ation admits an equivariant moment mapping µ. Suppose L is a maximal onneted submanifold of M with T x L = ζ g (x). Let G L be a subgroup of G whih preserves L and has g L as Lie

SOME PROPERTIES OF LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS 9 algebra, and assume that G L ats freely and properly on L. Then P L := L/G L admits a unique (up to onformal equivalene) l..s. struture ( Ω, ω), suh that (L, i Ω, i ω) (L, π Ω, π ω), where i : L M denotes the inlusion and π : L P L denotes the projetion. Remark 11. Suppose (M, Ω, ω) is an l..s. manifold with sympleti G-ation and equivariant moment mapping µ. If a C (M, R) and (Ω, ω ) = (e a Ω, ω da), then e a µ is an equivariant moment mapping for the same ation on (M, Ω, ω ). Moreover L is Ω-orthogonal to the orbits iff it is Ω -orthogonal to them, and in this ase g L = g L. So the redued spae does only depend on the onformal equivalene lass of (M, Ω, ω). Example 6. Let G be a disrete group ating freely, properly and sympleti on an l..s. manifold (M, Ω, ω). Then g = 0, µ = 0 is an equivariant moment mapping, and the only possible hoie for L is L = M. If we hoose G L = G then P L = M/G arries an l..s. struture. Notie that even if we start with a sympleti manifold, M/G need not be onformally equivalent to a sympleti manifold. Example 7. Suppose (N, α) is a ontat manifold, suh that the Reeb vetor field E generates a free S 1 -ation. Then N/S 1 inherits a sympleti struture. On the other hand the l..s. manifold (N S 1, d ν α, ν), see Example 3, has an S 1 -ation generated by E 0. This ation is Hamiltonian with equivariant moment mapping µ = onst. Then L = N {point}, g L = R and G L = S 1 satisfy the assumptions of Theorem 6, and L/G L = N/S 1 as l..s. manifolds. Remark 12. Moreover one an show, that a Hamiltonian system whih is invariant under the ation, desends to a Hamiltonian system on the redued spae, see [13]. Referenes [1] A. Avez, A. Lihnerowiz and A. Diaz-Miranda, Sur l algèbre des automorphismes infinitésimaux d une variété sympletique, J. Diff. Geom. 9 (1974), 1 40. [2] A. Banyaga, Sur la struture du groupe des difféomorphismes qui préservent une forme sympletique, Comm. Math. Helv. 53 (1978), 174 227. [3] A. Banyaga, On isomorphi lassial diffeomorphism groups. I, Pro. Amer. Math. So. 98 (1986), 113 118. [4] A. Banyaga, On isomorphi lassial diffeomorphism groups. II, J. Diff. Geom. 28 (1988), 23 35. [5] A. Banyaga and A. MInerney, On isomorphi lassial diffeomorphism groups. III, Ann. Global Anal. Geom. 13 (1995), 117 127. [6] A. Banyaga, The Struture of Classial Diffeomorphism Groups, Mathematis and its Appliations, Kluwer Aad. Publ., 1997. [7] E. Calabi, On the group of automorphisms of a sympleti manifold, in Problems in Analysis, Symposium in Honor of S. Bohner, 1969, Prineton University Press, Prineton, NJ, 1970, 1 26. [8] R.P. Filipkiewiz, Isomorphisms between diffeomorphism groups, Ergodi Theory Dyn. Sys. 2 (1982), 159 171. [9] F. Guedira and A. Lihnerowiz, Géometrie des algèbres de Lie loales de Kirillov, J. Math. Pures Appl. 63 (1984), 407 484. [10] S. Haller, Perfetness and Simpliity of Certain Groups of Diffeomorphisms, Thesis, University of Vienna, 1998. [11] S. Haller and T. Rybiki, On the group of diffeomorphisms preserving a loally onformal sympleti struture, Ann. Global Anal. Geom. 17 (1999), 475 502. [12] S. Haller and T. Rybiki, Integrability of the Poisson algebra on a loally onformal sympleti manifold, The proeedings of the 19th winter shool Geometry and Physis (Srní, 1999) Rend. Cir. Mat. Palermo (II) Suppl. 63 (2000), 89 96.

10 STEFAN HALLER [13] S. Haller and T. Rybiki, Redution for loally onformal sympleti manifolds, J. Geom. Phys. 37 (2001), 262 271. [14] M.R. Herman, Sur le groupe des difféomorphismes du tore, Ann. Inst. Fourier 23 (1973), 75 86. [15] A. Kriegl and P.W. Mihor, The Convenient Setting of Global Analysis, Mathematial Surveys and Monographs 53, Amer. Math. So., 1997. [16] H.-C. Lee, A kind of even-dimensional differential geometry and its appliation to exterior alulus, Amer. J. Math. 65 (1943), 433 438. [17] J. Marsden and A. Weinstein, Redution of sympleti manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121 130. [18] H. Omori, Infinite Dimensional Lie Transformation Groups, Leture Notes in Mathematis, Springer Verlag 427, 1974. [19] L.E. Pursell and M.E. Shanks, The Lie algebra of smooth manifolds, Pro. Amer. Math. So. 5 (1954), 468 472. [20] T. Rybiki, Isomorphisms between groups of diffeomorphisms, Pro. Amer. Math. So. 123 (1995), 303 310. [21] F. Takens, Charaterization of a differentiable struture by its group of diffeomorphisms, Bol. So. Bras. Mat. 10 (1979), 17 25. [22] W.P. Thursten, Foliations and groups of diffeomorphisms, Bull. Amer. Math. So. 80 (1974), 304 307. [23] I. Vaisman, Loally onformal sympleti manifolds, Int. J. Math. Math. Si. 8 (1985), 521 536. [24] A. Weinstein, Sympleti manifolds and their Lagrangian submanifolds, Adv. Math. 6 (1971), 329 346. [25] J.V. Whittaker, On isomorphi groups and homeomorphi spaes, Ann. Math. 78 (1963), 74 91. Institute of Mathematis, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria, E-mail: stefan@mat.univie.a.at