Yunhyung CHOMin Kyu KIMDong Youp SUH

1 Introduction

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces.We first consider the following simple situation which provides the motivation of our work.

Let Σgbe a two-dimensional smooth closed oriented manifold with genus g,and let Diff(Σg)denote the diffeomorphism group of Σg.Suppose that G is a compact connected Lie group acting on Σgeffectively(A G-action on a manifold M is called effective if the identity element 1 ∈ G is the unique element which fi xes whole M.),i.e.,there is an injective Lie group homomorphism

Since G is compact and connected,by averaging any given Riemannian metric over the Haar measure of G,we can get a G-invariant metric Ω on Σgso that we may regard G as a closed Lie subgroup of the identity component Iso(Σg,Ω)0of the full isometry group Iso(Σg,Ω).In particular,we have the following inequalities

where the last inequality comes from the classical fact that

for any n-dimensional complete Riemannian manifold(Mn,h),see[6]for more details.Note that the isometry groups of a 2-sphere S2and a 2-torus T2are well-known such that

(1)Iso(S2,h)0is a subgroup of SO(3)for any metric h on S2,and

(2)Iso(T2,h)0is a subgroup of SO(2)×SO(2)for any metric h on T2.

In the case when the genus g ≥ 2,then the isometry group of Σgis finite with respect to any metric on Σg,see[4,Chapter 7]for more details.In fact,the latter statement can be deduced from the following well-known formula

wheredenotes the set of fixed points of the S1-action on Σg.

Thus the condition g≥2 is an obstruction to the existence of an action of a compact Lie group of positive dimension on compact Riemann surfaces.Together with the fact that χ(Σg)=2− 2g,one might expect that the existence of a non-trivial compact connected Lie group G in Diff(M)might be obstructed by the Euler characteristic χ(M),i.e.,the negativity of χ(M)would imply the non-existence of a G-action such as the case of compact Riemann surfaces.Unfortunately,it is inappropriate to use χ(M)since if we let M=S2×Σgfor g ≥ 2,then M admits a circle action on the first factor but M satisfies χ(M)＜ 0.Therefore,instead of χ,we use some geometric structure as follows.Let us consider a G-invariant volume form ω on Σg.Then ω is a symplectic form1A differential two-form ω on a manifold M is called symplectic if it is closed(dω =0)and non-degenerate(ωp:TpM ×TpM →R is a non-degenerate bilinear form for every p∈M).and thus(Σg,ω)is a symplectic manifold2A symplectic manifold is a pair(M,ω)which consists of a smooth manifold M and a symplectic form ω on M..Hence we may think of G as a subgroup of the symplectomorphism group

Let J be an ω-compatible almost complex structure on Σg,i.e., ω(·,·)= ω(J·,J·)and ω(·,J·)is a Riemannian metric on M.Note that for any symplectic manifold(M,ω),ω-compatible almost complex structure J always exists.In fact,the space J(M,ω)of ω-compatible almost complex structures is a contractible space(see[8]),which implies that(TM,J)and(TM,J′)are isomorphic as a complex vector bundle for any J and J′in J(M,ω).Hence the first Chern class of(TM,J)does not depend on the choice of a ω-compatible almost complex structure J,and we denote by for any J ∈ J(M,ω).

Since ω represents a non-zero element[ω]∈ H2(Σg;R)R,there exists a constant λ ∈ R such that

Also,the Chern number 〈c1(Σg,ω),[Σg]〉is the same as the Euler characteristic χ(Σg)=2−2g of Σg,where[Σg]∈ H2(Σg,Z)is the fundamental homology class of Σg.Hence we may conclude as follows:

(1)If(Σg,ω)is a closed symplectic surface such that 〈c1(Σg,ω),[Σg]〉＜ 0,then there is no compact connected Lie group action preserving ω.

In fact,we can say more about the G-action on(Σg,ω)in the symplectic setting.For a given symplectic manifold(M,ω)with the symplectomorphism group Symp(M,ω),we say that φ ∈ Symp(M,ω)is a Hamiltonian diffeomorphism if there exists an isotopy φtfor 0 ≤ t ≤ 1 such that

(1)φ0=id,

(2)φ1= φ,and

(3)iXtω = ω(Xt,·)=dHtfor some family of smooth functions{Ht:M → R}0≤t≤1,where Xtis a time-dependent vector field on M such that

We denote by Ham(M,ω)the set of all Hamiltonian diffeomorphisms on(M,ω)and it forms a group under the composition.In fact,the Hamiltonian diffeomorphism group Ham(M,ω)is path-connected and it is a normal subgroup of Symp(M,ω),see[8]for more details.

Now suppose that G is a compact connected Lie group acting on(M,ω)effectively,and g is the Lie algebra of G.By definition of Symp(M,ω)and by the surjectivity of the exponential map exp:TeG → G,we can easily show that G is a subgroup of Symp(M,ω)if and only if each one-parameter subgroup generated by each element X ∈ g preserves ω,i.e.,

for every X∈TeG,whereis the vector field generated by X,i.e.,

for each p∈M.In particular,if iXω is exact for every X ∈g,then iω=dHXfor some smooth function HX:M → R for each X ∈ g.Then,the family{exptX}0≤t≤1is an isotopy which connects id=exp(0·X)with exp(1·X),and it satisfies Equation(1.1),which means that expX ∈ Ham(M,ω).By surjectivity of exp:g→ G(since G is compact),we can deduce that G is a subgroup of Ham(M,ω)if iω is exact for every X ∈ TeG.Conversely,if G is a subgroup of Ham(M,ω),then one can easily check that iω is exact for every X ∈g.

definition 1.1LetGbe a compact connected Lie group acting on(M,ω).We say that aG-action is Hamiltonian ifiωis exact for everyX ∈g.Equivalently,aG-action is Hamiltonian ifGacts onMas a subgroup ofHam(M,ω).

Hence if the G-action is Hamiltonian,there exists a smooth map H:M →g∗(called a moment map)such that〈H,X〉=HXwith iω =dHXfor every X ∈ g,where〈,〉:g∗×g→ R is the usual pairing of the Lie algebra g with its dual.Note that for each X∈g,the set of all critical points of HXcoincides with the set of all points fixed by the one-parameter subgroup generated by X by the non-degeneracy of ω.Note that if M is compact,then any smooth function on M has at least two critical points attaining its extrema.Hence if the G-action is Hamiltonian,the one-parameter subgroup{exp(tX)}t∈Rhas at least two fixed points for every X∈g.Therefore,we have the following proposition as follows.

Proposition 1.1Let(Σg,ω)be a closed two-dimensional symplectic manifold with genusgsuch thatc1(Σg,ω)= λ ·[ω]and letGbe a compact connected Lie group.Suppose that theGaction is effective and it preservesω.Then

(1)Ifλ＞0(g=0),then theG-action is Hamiltonian.

(2)Ifλ=0(g=1),then theG-action is non-Hamiltonian,and

(3)Ifλ ＜0(g＞ 1),thenG={1}.

Proof If g=0,then Σ0S2is simply connected.In particular,we have H1(Σ0;R)=0 so that diω =0 if and only if iω is exact,which implies that any symplectic G-action on Σ0is automatically Hamiltonian.For the second statement,recall that SO(2)× SO(2)acts on Σ1freely and hence the G-action on Σ1is also free.In particular,every one-parameter subgroup of G has no fixed point so that the action is non-Hamiltonian.The last statement comes from the fact that the isometry group of Σgwith g ≥ 2 is finite(see p.2).

In this point of view,we may think of the generalization of the closed Riemann surface case to the symplectic category.Note that if(M,ω)is a symplectic manifold such that c1(M,ω)= λ·[ω],then we call(M,ω)a monotone symplectic manifold if λ ＞ 0,a symplectic Calabi-Yau manifold if λ =0,and a negatively monotone symplectic manifold if λ ＜ 0.In this article,we will regard those three families of symplectic manifolds as a generalization of closed Riemann surfaces,and we prove the following theorem,which is a generalization of Proposition 1.1 to the higher dimensional cases.

Theorem 1.1Let(M,ω)be any smooth closed symplectic manifold such thatc1(M,ω)=λ·[ω]for someλ ∈ R.LetGbe a compact connected Lie group which acts on(M,ω)effectively and preservesω.Then

(1)Ifλ＞0,then theG-action is Hamiltonian.

(2)Ifλ=0,then theG-action is non-Hamiltonian.

(3)Ifλ＜0,thenGis trivial.

Note that Theorem 1.1 is not new.Theorem 1.1(1)was already proved independently by Atiyah-Bott[1]and Lupton-Oprea[5].Also Theorem 1.1(2)–(3)was proved by Ono[10].The proofs given by Atiyah-Bott[1]and Ono[10]are based on the equivariant cohomology theory,and the proof of Lupton-Oprea[5]is based on the homotopy theory,in particular the theory of Gottlieb groups.We do not refer to the details of their proofs,instead we give much more elementary and simple proof of Theorem 1.1 which can be obtained as a corollaryof the following series of the propositions.

Proposition 1.2Let(M,ω)be a smooth closed symplectic manifold equipped with a smoothS1-action preservingω.Suppose that[ω]is a rational class inH2(M;Q).Then the action is non-Hamiltonian if and only if there exists anS1-equivariant symplectic embedding of2-torusi:T2→ M,where the circle acts freely on the left factor ofT2S1× S1.In particular,the normal bundle ofi(T2)inMis trivial so that〈c1(M,ω),[i(T2)]〉=0.

Proposition 1.3Suppose that(M,ω)is a smooth closed symplectic manifold equipped with a Hamiltonian circle action.Then there exist a two-sphereSinMwith positive symplectic area satisfying〈c1(M,ω),[S]〉＞ 0.

Proof of Theorem 1.1 First,suppose that the action is Hamiltonian.If G is not trivial,then there exists a maximal subtorus T with positive dimension in G.For any choice of a circle subgroup S1of T,there exists a two sphere S in M such that〈c1(M,ω),[S]〉= λ ·〈[ω],[S]〉＞ 0 by Proposition 1.3.Since S has positive symplectic area,we have 〈[ω],[S]〉＞ 0 so that λ must be positive.Secondly,if the action is non-Hamiltonian,we can easily show that it is non-Hamiltonian with respect to kω for any positive real number k∈R.Also,k can be chosen so that[kω]∈ H2(M)is rational since[ω]is proportional to c1(M,ω)by our assumption.By Proposition 1.2,there exists a symplectic two torus T in(M,kω)such that

where the first equality comes from the fact that J(M,ω)=J(M,kω)for k ＞ 0.Since〈[kω],[T]〉=k ·〈[ω],[T]〉＞ 0 and k ＞ 0,we have λ =0.

Note that if there is an effective symplectic S1-action on(M,ω),then Propositions 1.2–1.3 imply that λ ≥ 0.Thus if λ is negative,then there is no effective symplectic S1-action on(M,ω).In other words,G must be trivial by the compactness and the connectivity of G.

We make the following two remarks.First,the reason why we only prove Proposition 1.2 for the rational case is that our proof relies on the existence of a generalized moment map[7]which can be defined only when[ω]is rational.If[ω]is not rational,since the non-degeneracy of ω is an open condition,we can always perturb a given ω slightly to another symplectic form ω′such that ω′is G-invariant and[ω′]is rational.Hence if we apply Proposition 1.2 to(M,ω′),then there exists a symplectic embedding i:T2M with respect to the new symplectic structure ω′.But there is no guarantee that the T2-embedding i is symplectic with respect to ω.The authors could not fi ll the gap of the proof of Proposition 1.2 in the case when ω is not rational.

Secondly,unlike Proposition 1.2 which gives a necessary and sufficient condition for the existence of non-Hamiltonian G-action,Proposition 1.3 gives only the sufficient condition for the action to be Hamiltonian.The authors do not know whether the converse of Proposition 1.3 holds or not.

The organization of this paper is as follows.We give an introduction to the theory of Lie group actions on symplectic manifolds in Section 2,and we give the complete proof of Propositions 1.2–1.3 in Section 3.

2 Symplectic Circle Actions

In this section,we give a brief introduction to symplectic circle actions.Most of this section is contained in[2]or[8],but we give a complete proof for readers who are not familiar with symplectic geometry.Let M be a 2n-dimensional smooth closed manifold.A differential 2-form ω is called a symplectic form if ω is closed and non-degenerate,i.e.,

(1)dω=0,and

(2)ωnis nowhere vanishing.

Let us assume that G is a compact connected Lie group acting effectively on M.The G-action on(M,ω)is called symplectic if Lω=0 for every X ∈TeG,whereis the fundamental vector field generated by X,i.e.,G preserves a symplectic form ω.Equivalently,G-action is symplectic if and only if iω is closed.In particular,a G-action is called Hamiltonian if iω is exact for every X∈TeG.

Now suppose that the unit circle group S1acts on(M,ω)symplectically,and let X be a fixed generator of TeS1R.If the action is Hamiltonian,then there exists a smooth function H:M→R such that and we call H a moment map for the S1-action.Note that LXi.e.,a moment map H is S1-invariant.

If the S1-action is symplectic but non-Hamiltonian,then a moment map does not exist.Nevertheless,there exists an R/Z-valued functionµ:M→R/Z which locally looks like a moment map when[ω]∈ H2(M;R)is an integral class.We use the notation R/Z instead of S1to avoid confusion with the acting group S1.

definition 2.1(see[7])Let(M,ω)be a smooth closed symplectic manifold such thatωrepresents an integral cohomology class inH2(M;Z).Suppose that there is a symplectic non-HamiltonianS1-action on(M,ω).Fix a pointx0∈ M,and define anR/Z-valued mapµ:M→R/Zsuch that

whereγxis any pathγx:[0,1] → Msuch thatγx(0)=x0andγx(1)=x.We callµanR/Z-valued moment map(or a generalized moment map)for the action.

By a direct computation,we can easily check that[iω]is an integral class in H1(M;Z)so that a generalized moment map given in definition 2.1 is well-defined.Note thatµdepends on the choice of a base point x0as in definition 2.1.Consider two distinct points p and q on M,and letµp,respectively µq,be the R/Z-valued moment map with base point p,respectively q.For any point x ∈ M,letbe a path from q to p andbe a path from p to x respectively.Then

In other words,µ is unique up to a constant in R/ZS1.In particular,dµ is independent of the choice of a base point.Since dµ :TM → TS1S1×R,we may regard dµ as a differential 1-form on M.

Proposition 2.1(see[7])Letµ:M→R/Zbe anR/Z-valued moment map for a symplectic non-Hamiltonian circle action on(M,ω).Thenµsatisfies

Proof Let x∈M be any point and let U be a contractible open neighborhood of x.Since iω is closed,it is locally exact by Poincar´e lemma so that there exists a smooth function f:U→R such that iω=df on U.Letµbe an R/Z-valued moment map with a base point x0∈ U.Let γxbe a path from x0to x lying on U.Then

so that dµ(x)=df(x)=i(x)ωxfor all x ∈ U.Since x is chosen arbitrarily,we can conclude that dµ=iω on M.

It is an immediate consequence of Proposition 2.1 thatµis S1-invariant,since

Now,let us consider a critical point of a moment map H,i.e.,dH(x)=Since ω is non-degenerate on M,x is a critical point of H if and only if(x)=0,i.e.,x is a fixed point of given S1-action.It is also true for a non-Hamiltonian case,i.e.,x is a critical point of an R/Z-valued moment mapµif and only if x is a fixed point of the action by Proposition 2.1.Hence we have the following proposition.

Proposition 2.2(see[2])Let(M,ω)be a smooth closed symplectic manifold equipped with a symplectic,respectively Hamiltonian,circle action.Ifµ,respectivelyH,is anR/Z-valued moment map,respectively moment map,of given action,thenx∈Mis a critical point ofµ,respectivelyH,if and only ifxis a fixed point of the action.

One of the most important property of symplectic geometry is that,for every point p∈M,there exists a neighborhood Upof p with a local coordinate system(R2n,x1,y1,···,xn,yn)such that ω|Up=Pdxi∧ dyi,i.e.,a local symplectic structure for each point is isomorphic to the standard symplectic structure of R2nso that symplectic geometry is locally the same as a linear symplectic geometry on R2nwith the standard symplectic structurePdxi∧dyi.This is known as the Darboux theorem.Similarly,there is an equivariant version of the Darboux theorem as follows.

Theorem 2.1(Equivariant Darboux Theorem)Let(M,ω)be a symplectic manifold and letGbe a compact Lie group.Suppose that there is a symplecticG-action on(M,ω).For each fixed pointp,there exists a neighborhoodUptogether with a local coordinate system(x1,y1,···,xn,yn)such that

(2)G-action is linear with respect to(z1,···,zn).In particular ifG=S1,then there is a sequence of integersλ1,···,λnsuch that the action is expressed as

for everyt∈S1.

Now,let p be a fixed point of symplectic S1-action on 2n-dimensional symplectic manifold(M,ω).By the equivariant Darboux theorem,there exists a local coordinate system(Up,z1,···,zn)centered at p and a sequence of integers λ1,···,λnsuch that the S1-action is expressed as

for every t∈ S1.By solving iXω =dH on Upwithwe get

Therefore,we have the following corollary.

Corollary 2.1(see[2])LetH:M → Rbe a moment map on(M,ω).ThenHis a Morse-Bott function.Similarly,ifµ:M→R/Zis anR/Z-valued moment map,thenµis anR/Z-valued Morse-Bott function.In either case,a Morse index of any critical submanifold ofHorµis even.

Proof We need to show two things:(1)The critical point set is an embedded submanifold of(M,ω),and(2)The Hessian of H at p is non-degenerate along the normal direction of a critical submanifold containing p.The first claim is obvious since a sub-coordinate system gives a coordinate system of a critical submanifold near p.The Hessian of H at p is a diagonal matrix is given by

so that it finishes the proof.

Now,recall some basic Morse-Bott theory as follows.Let f:M→R be a Morse-Bott function on a closed manifold M,and let Mt={p∈M|f(p)≤t}for every t∈R.Suppose that a and b are regular values of f such that there exists a unique critical value c with a＜c＜b.Let C1,···,Crbe connected components of the critical submanifold lying on H−1(c).According to classical Morse-Bott theory,Mbis homotopy equivalent to

where ν−(Cj)is a negative normal bundle over Cj,D(ν−(Cj))is a disk bundle of ν−(Cj),and φjis an attaching map from a sphere bundle S(ν−(Cj))= ∂D(ν−(Cj))to H−1(a).Note that each S(ν−(Cj))is an Skj−1-bundle over Cj,where kj=ind(Cj)is a Morse index of Cj.In particular,S(ν−(Cj))is connected if and only if kj1.

Proposition 2.3(see[2])LetHbe a moment map on a(possibly non-compact)connected symplectic manifold(M,ω).Then every level set ofHis empty or connected.

Proof For the sake of simplicity,let M(a,b):=H−1((a,b))for a,b∈R.Let us choose any S1-invariant metric 〈·,·〉on M so that the gradient fl ow ▽H of H is defined as

for every smooth vector field X on M.Suppose that there exists a regular value r∈R such that H−1(r)is non-empty and disconnected.By the connectivity of M,there exists a smallest s∈R+such that M(r−s,r+s)is connected.Note that r+s or r−s is a critical value of H,otherwise M(r−s,r+s)is diffeomorphic to M(r−(s− ∈),r+(s−∈))along the gradient fl ow of H for a sufficiently small∈＞ 0 so that it contradicts our assumption “smallest s”.

Without loss of generality,we may assume that c=r+s is a critical value of H.Let C1,···,Crbe connected components of the critical submanifold lying on H−1(c).Then there exists some Cjsuch that D(ν−(Cj))connects two disconnected components of H−1(c− ∈)via the attaching map φj:S(ν−(Cj)) → H−1(c− ∈),i.e.,the index of Cjshould equal to one.Since every critical submanifold of H has even index by Corollary 2.1,such Cjdoes not exist.

Similarly,if c=r−s is a critical value of H,then there exists some Cjof co-index one,but there is no such Cjby Corollary 2.1.Hence it completes the proof.

Proposition 2.4(see[2])Let(M,ω)be a2n-dimensional closed connected symplectic manifold equipped with a symplectic non-Hamiltonian circle action.Suppose[ω]is an integral class inH2(M;Z),and letµ:M→R/Zbe anR/Z-valued moment map defined in definition2.1.Then there is no critical submanifold of index zero nor co-index zero.In particular,every level set is non-empty and the number of connected components ofµ−1(t)is constant for allt∈S1.

ProofLet r∈R/Z be a regular value ofµ:M → R/Z and let N= µ−1(R/Z−{r})which is an open subset in M.With the induced S1-action on N,we may regard N as a noncompact Hamiltonian S1-manifold with a moment map H= µ|N:N → R/Z−{r}(0,1).Let N1,N2,···,Nkbe connected components of N and we denote by Hj:Nj→ (0,1)the restriction of H onto Njso that Hjis a moment map on(Nj,ω|Nj).By Proposition 2.3,every level set of Hjis empty or connected for every j=1,2,···,k.

Firstly,we claim that each Hjis surjective.If not,H(Nj)is either a half-closed interval of the form[s,1)or(0,s]for some s∈(0,1)or a closed interval[a,b]⊂(0,1).If H(Nj)=[a,b],then Njis also a connected component of H−1([a,b])so that Njis both open and closed itself in M so that Nj=M by the connectivity of M,which contradicts the assumption that the action is non-Hamiltonian.If H(Nj1)=[s1,1)for some j1∈ {1,2,···,k},then let us consider the closurewhose boundaryis some connected component,namely B1,ofµ−1(r).Since r is regular,there is a connected component Nj2of N such that(t)attains B1as t→0.If the image of Nj2for Hj2is a half-closed interval of the form(0,s2]for some s2∈(0,1),then∪is connected and both open and closed in M so that we getThen the moment mapµfactors such that

where Hj1,j2maps x∈Nj1to Hj1(x),y∈µ−1(r)to 1,and z∈Nj2to 1+Hj2(z).Then=d(/Z)◦dHj1,j2,but d(/Z)is the identity map so that dHj1,j2=i.e.,the action is Hamiltonian which contradicts to our assumption.Hence H(Nj2)=(0,1),i.e.,Hj2is surjective so that(t)attains some connected component B2B1ofµ−1(r)as t→ 1.Take Nj3such that(t)attains B2as t→0.Then we can easily show that Hj3is surjective by a similar reason.Hence we get a in finite sequence of pairwise distinct connected components B1,B2,···,but it contradicts that the number of connected components ofµ−1(r)is finite by the compactness of M.Therefore,Hjis surjective for every j.In particular,there is no critical submanifold of index 0 nor co-index 0.

To complete the proof,recall that when a level set ofµpasses through some critical value c∈ S1such thatµ−1(c)does not contain a critical submanifold of index 0,1,co-index 0,nor co-index 1,then the number of connected components does not change,i.e.,µ−1(c+ ∈)andµ−1(c− ∈)have the same number of connected components,see the proof of Proposition 2.3.Since an index and co-index of any critical component is even and there is no critical component of index 0 nor co-index 0,every level set has the same number of connected components.

3 Proof of the Main Theorem

Let G be a compact Lie group,and suppose(M,ω)is a closed symplectic manifold equipped with an effective symplectic G-action.In this section,we prove Theorem 1.1.To determine whether a given G-action is Hamiltonian or not,it is enough to check it for every circle subgroup of G since every element g∈G is contained in some maximal torus of G.The following proposition characterizes a non-Hamiltonian circle action in terms of equivariant symplectic embedding of two-torus.

Proof of Proposition 1.2 Let T2=S1×S1be a two-torus and assume that an S1-action on T2is given by

for any t∈S1and(t1,t2)∈T2.We denote the vector field on T2generated by the S1-action by X.Suppose that there exists an S1-equivariant symplectic embedding i:T2→ M.Then the vector field generated by the given S1-action on M is i∗(X).If the given S1-action on(M,ω)is Hamiltonian with a moment map H:M→R,then for any smooth vector field Y on T2,we have

which follows from the fact that di(Y)=i∗Y.Thus the S1-action on(T2,i∗ω)is Hamiltonian with a moment map i∗H=H ◦i:T2→ R.But it contradicts the assumption that S1-action on T2is free,because there must be at least two fixed points,namely the maximum and the minimum of i∗H.Hence the S1-action on(M,ω)cannot be Hamiltonian.

Conversely,suppose that the given symplectic S1-action on(M,ω)is non-Hamiltonian.By our assumption,there exists a natural number N big enough so that N ·ω is integral and we still denote by ω the new symplectic form N ·ω.Obviously,the given S1-action is symplectic with respect to the new symplectic form ω so that there exists an R/Z-valued moment mapµ:M →R/Z satisfying iXω=dµby Proposition 2.1,whereis the vector on M generated by the S1-action.Let M(1)be the set of all points in M with the trivial isotropy subgroup.Now,suppose that there exists a smoothly embedded loop σ :S1=[0,n]/0∼n→ M(1)for some n∈N satisfying the following conditions:

(a)µ?σ(r)?=[r]∈ R/Z for each r∈ [0,n],

(b)σ(0)= σ(n),i.e.,the image of σ is a loop,and

(c)for rr′with(r,r′)(0,n),t·σ(r)σ(r′)for any t∈ S1.

If such σ exists,then we can define a smooth embedding of 2-torus as

It is straightforward that i is S1-equivariant for the S1-action on T2by t·(t1,t2)=(t·t1,t2).To show that i(T2)is a symplectic submanifold with respect to the induced symplectic structure,let us define a smooth vector field Y on i(T2)as

Since σ has no critical point,it is straightforward thathas no zero andspan the tangent space Tpi(T2)for every p=i(t1,t2)∈i(T2).Also,

So,ω is non-degenerate on i(T2),i.e.,i is a symplectic embedding.

Now,we need to prove that such smoothly embedded loop σ in M actually exists.

Lemma 3.1 M(1)is path-connected and open dense inM.

ProofLet Znbe the cyclic subgroup of S1of order n,and we denote by MZnthe set of all points in M fixed by Zn.Since M is compact,there are at most finitely many n’s,say n1,n2,···,nksuch that MZnØ (see[3,Proposition IV.1.2]).Let MS1be the set of all points in M fixed by S1.Since MS1⊂MZnfor every integer n＞1,we have

Furthermore,for each n＞1,MZnis closed symplectic submanifolds of M with the induced symplectic form by the equivariant Darboux theorem 2.1.Thus the setis the union of closed submanifolds with codimensions at least two,in particular M(1)is path-connected and open dense in M.

Corollary 3.1For any regular valuet0∈ R/Zofµ,the subsetM(1)∩µ−1(t0)is open dense inµ−1(t0).

Proof The openness is obvious since M(1)is open.We will show that MZn∩µ−1(t0)is of codimension at least two inµ−1(t0)for each n＞ 1,which implies thatis of codimension at least two in µ−1(t0)so that

is dense in µ−1(t0).We already know that MZnis of codimension at least two in M for each n＞1,since MZnis a(proper)symplectic submanifold of M.Also,we know that t0is a regular value of the restriction mapµ|MZn:MZn→ R/Z,since there is no fixed point in(µ|MZn)−1(t0)=MZn∩ µ−1(t0).Thus we have

Also,we have dimµ−1(t0)=dimM −1 which implies that the codimension of MZn∩µ−1(t0)inµ−1(t0)is equal to dimM −dimMZn≥2.This finishes the proof.

Remark 3.1 The set M(1)∩µ−1(t0)is not necessarily path-connected in Corollary 3.1.

Without loss of generality,we may assume that 0∈ R/Z is a regular value ofµ.For our convenience,we use the following terminology:For a fixed S1-invariant metric h on M,we say that a smooth path γ :[a,b]→ M(1)winds along R/Z if h(▽µγ(t),γ′(t))＞ 0 for every t∈ [a,b],which means that the vector field generated by γ is a gradient-like vector field ofµ with respect to h.We call such a path γ a winding path.Also,we say that two points x,y ∈ M(1)are winding path-connected,if there exists a path γ :[a,b] → M(1)from γ(a)=x to γ(b)=y which winds along R/Z.

Lemma 3.2Forx∈ M(1)∩µ−1(0),there exists somey∈ M(1)∩µ−1(0)such thatxandyare winding path-connected.

Proof Let J be an S1-invariant almost complex structure compatible1We say that an almost complex structure J on(M,ω)is compatible with ω if(1) ω(·,·)= ω(J·,J·)and(2)ω(·,J·)is a Riemannian metric.Such J always exists,see[8]for the details.with ω.We denote by 〈·,·〉= ω(·,J·)the induced S1-invariant metric on M so that the gradient vector field ▽µon M with respect to 〈·,·〉is defined as dµ = 〈▽µ,·〉.Then for any vector field Y on M,we have so that we obtain▽µ=

Note that ▽µ commutes with the S1-action,since J, 〈·,·〉,and µ are chosen to be S1-invariant.Thus the one-parameter group action generated by▽µpreserves their isotropy subgroups,which means that if x∈M is fixed by some subgroup H⊂S1,then any point y in the orbit{(expt▽µ)·x}t∈Ris fixed by H.In particular,the one-parameter group action generated by▽µacts on M(1).

Now,x∈M(1)∩µ−1(0)and consider the integral curve along▽µpassing through x

wheredenotes the one-parameter subgroup of Diff(M)generated by the vector field▽µ.If γxis a winding path from x to some point y ∈ M(1)∩ µ−1(0),then there is nothing to prove.

Since=▽µγx(t)and▽µp=0 if and only if p∈MS1,if γxis not a winding path from x to any point in M(1),thenfor some t0∈ R/Z,which is equivalent to saying that

for some fixed point p∈MS1.By the equivariant Darboux Theorem 2.1,there exists a local coordinate system(Up,z1,···,zn)centered at p such that

and

for every t∈ S1,where λ1,···,λnare weights of the tangential S1-representation on TpM.Let Cpbe the fixed connected component containing p and let ν+and ν−be subsets of Upgiven by

(1) ν+={(z1,···,zn)∈ Up|zj=0 if λj≤ 0},and

(2) ν−={(z1,···,zn)∈ Up|zj=0 if λj≥ 0}.

Then γx(t)is lying on ν−for any sufficiently large t＞ 0.Note that{(z1,···,zn) ∈ Up|zj0 if λj0} ⊂ M(1)since the S1-action is effective.Then we may perturb γxsmoothly to a new gradient-like fl ow eγxon Up∩ M(1)connecting the gradient fl ow γxto γqfor some q ∈ Up∩ M(1)and q/∈ ν−,where γqis the integral curve along ▽µ passing through q(see Figure 1).

Figure 1 Perturbing γxto getx.

Ifis a winding path from x to some point y ∈ M(1)∩ µ−1(0),then it is done.If not,we apply the same procedure whenever our perturbed gradient-like fl ow converges to some fixed point.Then in finite steps,we can get a winding path connecting x to M(1)∩ µ−1(0)as is desired.

Lemma 3.3If a pointxinM(1)∩ µ−1(0)is winding path-connected to some pointyin a connected componentCofM(1)∩µ−1(0),thenxis winding path-connected to every point inC.

Proof For x ∈ M(1)∩µ−1(0),let γxbe a winding path connecting x to y ∈ M(1)∩µ−1(0).Let Cybe the connected component of M(1)∩ µ−1(0)containing y.Fix z ∈ Cyand let σ :[0,1]→ M(1)∩ µ−1(0)be a smooth path with σ(0)=y and σ(1)=z.Since 0 is chosen to be a regular value ofµ,the gradient vector field▽µis non-zero onµ−1(0).Also,we can find a sufficiently small∈such that any r∈ [−∈,0]⊂ S1is a regular value ofµ.

Figure 2 Winding path.

Letbe the intersection of M(1)∩ µ−1(−∈)and the trajectory of σ under the infinitesimal action generated by ▽µ.Let y′be the preimage of y for the infinitesimal action in M(1)∩µ−1(−∈).Then we can easily see that there exists a homotopy in M(1)∩ µ−1([−∈,0])from γxto a winding path eγxconnecting y′and z(see Figure 2).Then the pathdefined by

is a winding path which connects x to z.It completes the proof.

To complete the proof of Proposition 1.2,pick a point x1in a connected component C1of M(1)∩µ−1(0).By Lemma 3.2,x1is winding path-connected to some point x2in some connected component C2of M(1)∩µ−1(0).If C1=C2,then x1is winding path-connected to x1itself by Lemma 3.3 and it satisfies the conditions(a)–(c)automatically.If C1C2,then we can find a point x3in some connected component C3of M(1)∩µ−1(0)which is winding path-connected to x2.If C1=C3,respectively C2=C3,then we can take x3=x1,respectively x3=x2,so that we can obtain a smooth winding loop satisfying(a)–(c).Applying the above process inductively,we obtain a smooth winding loop satisfying(a)–(c).Then the following lemma finishes the proof.

Lemma 3.4LetEbe anS1-equivariant complex vector bundle of rankkoverT2such that the inducedS1-action on the zero-section is free.ThenEis trivial.

Proof Let ZT2be the zero section of E.Note that if the induced S1-action on the zero section Z is free,then the given S1-action on the total space E is free.Now,let us consider a following diagram:

Since the action is free,π′:E/S1→ Z/S1S1is a complex vector bundle of rank k over S1and the quotient map q is a bundle morphism.Note that any complex vector bundle over S1is trivial,since the structure group U(n)is connected.Therefore E/S1S1×Ckand hence Eq′∗(E/S1)Z × Ck.

Now,let us consider the Hamiltonian case.The following proposition characterizes a Hamiltonian circle action in terms of equivariant symplectic embedding of two-spheres.

Proof of Proposition 1.3 Let H:M→R be a moment map of the circle action.Let Zmin,respectively Zmax,be a critical submanifold which attains the minimum,respectively maximum,of H.Let g be an S1-invariant Riemannian metric defined by g(X,Y)=ω(X,JY),where J is an ω-compatible S1-invariant almost complex structure.Let ▽H be the gradient vector field with respect to g,i.e.,dH=g(▽H,·).Since H is a Morse-Bott function by Corollary 2.1,the unstable submanifold of Zmin

is open and dense in M.Similarly,let Ws(Zmax)be the stable submanifold of Zmax.Since both Wu(Zmin)and Ws(Zmax)are open dense subsets,their intersection Wu(Zmin)∩Ws(Zmax)is also open and dense in M,in particular it is non-empty.Now,pick a point p∈Wu(Zmin)∩Ws(Zmax)and let

Then it is straightforward that the closureis homeomorphic to a two-sphere whose north pole,respectively south pole,is in Zmin,respectively Zmax.

Now,letbe the fundamental vector field generated by the S1-action.Since the S1-action and the gradient fl ow are smooth,it is obvious that M(p)is a smoothly embedded two-sphere punctured at the two poles{zN,zS}.In particular,a tangent space TqM(p)for every q∈M(p)is generated byand Jsincenever vanishes on M(p).Since we have chosen J such thatis a smoothly embedded symplectic two sphere punctured at{zN,zS}.Then the closure M(p)is homeomorphic to S2and it defines a homology classin H2(M)even though the closureis not smooth at zNand zSin general.Obviously,the symplectic areais positive.

McDuff and Tolman proved thatin[9,Lemma 2.2].We sketch their idea as follows.Consider

where S1acts on S3⊂ C2by t·(z1,z2)=(tz1,tz2)for t∈ S1.Then E is an M-bundle over S2with sections

Let cvert∈ H2(E)be the first Chern class of the vertical subbundle1of TE.Then cvert(σN)is the first Chern number of the complex vector bundle

over S2.Note that TzNwhere{α1,···,αn}is the set of weights of the S1-representation on TzNM and Cαiis the one-dimensional S1-representation with weight αi.Therefore,

where πiis the complex line bundle over σNwhose first Chern number equals αi.Thus cvert([σN])=mNwhereis the sum of weights of the S1-representation on TzNM.Similarly,we have cvert([σS])=mS,where mSis the sum of weights of the S1-representation on TzSM so that

since every nonzero weight of the S1-representation at the maximum zN(respectively at the minimum zS)is positive(respectively negative).Thus it is enough to show that[M(p)]=[σN]− [σS].

define uω∈Ω2(S3×M)by

where θ is a connection 1-form on S3.Then we can easily check that LXuω=iXuω=0 so that uωinduces a two-form,which we still denote by uω,on S3×S1M.McDuff and Tolman’s idea is that

holds.And if ω′is another symplectic form invariant under the S1-action and Hω′is a corresponding moment map,then

holds.Since the set of cohomology classes represented as S1The vertical subbundle of TE is the kernel of the bundle map dπ :TE → TS2induced by π.-invariant symplectic forms generates whole H2(M;R)as a vector space,we may conclude that

for every β ∈ H2(M;R),and hence we get[M(p)]=[σzN]− [σzS].It completes the proof.

Remark 3.2 In[9],they defined a Hamiltonian action with the following sign convention

while we use the equation iω=dH.In particular,the gradient vector field is▽H=−Jin[9]while▽H=Jin our paper,see[9,p.12].

There are two effects of their sign convention on our proof of Proposition 1.3:

(1)In[9],ω(,J)is negative so that ourhas an opposite orientation,and

(2)uωshould be defined as w − d(H ·θ)so that 〈[uω],[σz]〉= −H(z)for any fixed point z∈MS1.

After identifying our notation with the notation in[9]as above,we can easily check that our argument used in the proof of Proposition 1.3 coincides with Lemma 2.2 in[9].

AcknowledgementsThe authors would like to thank the referee for carefully reading this manuscript and pointing out that there was an error in the previous version of the manuscript.

[1]Atiyah,M.F.and Bott,R.,The moment map and equivariant cohomology,Topology,23(1),1984,1–28.

[2]Audin,M.,The Topology of Torus Actions on Symplectic Manifolds,Progress in Mathematics,93,Birkh¨auser,Basel,1991.

[3]Bredon,G.E.,Introduction to Compact Transformation Groups,Academic Press,New York,London,1972.

[4]Farb,B.and Margalit,D.,A Primer on Mapping Class Groups,Princeton University Press,Princeton,NJ,2012.

[5]Lupton,G.and Oprea,J.,Cohomologically symplectic spaces:Toral actions and the Gottlieb group,Trans.Amer.Math.Soc.,347(1),1995,261–288.

[6]Mann,L.N.,Gaps in the dimensions of isometry groups of Riemannian manifolds,J.Differential.Geom.,11,1976,293–298.

[7]McDuff,D.,The moment map for circle actions on symplectic manifolds,J.Geom.Phys.,5(2),1988,149–160.

[8]McDuff,D.and Salamon,D.,Introduction to Symplectic Topology,Oxford Mathematical Monographs,The Clarendon Press,Oxford University Press,New York,1995.

[9]McDuff,D.and Tolman,S.,Topological properties of Hamiltonian circle actions,Int.Math.Res.Pap.,72826,2006,1–77.

[10]Ono,K.,Some remarks on group actions in symplectic geometry,J.Fac.Sci.Univ.Tokyo Sect.IA Math.,35,1988,431–437.

[11]Ono,K.,Equivariant projective embedding theorem for symplectic manifolds,J.Fac.Sci.Univ.Tokyo Sect.IA Math.,35,1988,381–392.