# On the Geometry of Almost𝒮-Manifolds

Authors: Sean Fitzpatrick
Journal: ISRN Geometry
 (2011)

---

## Abstract

Anf-structure on a manifold
M is an endomorphism field
φ satisfying φ3+φ=0. We call an f-structure
regular if the distribution T=kerφ is involutive and regular, in the sense of
Palais. We show that when a regular f-structure on a compact
manifold M is an almost 𝒮-structure, it determines a torus fibration of
M over a symplectic manifold. When rank
T=1, this result reduces to the Boothby-Wang
theorem. Unlike similar results for manifolds with 𝒮-structure or 𝒦-structure, we do not assume that the
f-structure is normal. We also show that given an almost
𝒮-structure, we obtain an associated Jacobi
structure, as well as a notion of symplectization.


---

## Body

## 1. Introduction

Let(M,η) be a cooriented contact manifold. The Boothby-Wang
theorem [1] tells us that if the Reeb field
ξ corresponding to the contact form
η is regular (in the sense of Palais [2]), then M is a prequantum circle bundle
π:M→N over a symplectic manifold
(N,ω), where π*ω=-dη and η may be identified with the connection 1-form.
Conversely, let M be a prequantum circle bundle over a symplectic
manifold (N,ω), and let η be a connection 1-form. Given a choice of
compatible almost complex structure J for ω, let G(X,Y)=ω(JX,Y) be the associated Riemannian metric on
N, and let π̃ denote the horizontal lift of vector fields defined
by η. We can then define an endomorphism field
φ∈Γ(M,End(TM)) by(1.1)φX=π̃Jπ*X,and a Riemannian metricg by g=π*G+η⊗η. If we let ξ be the vertical vector field satisfying
η(ξ)=1, then (φ,ξ,η,g) defines a contact metric structure on
M [3]. In
particular, we note that φ is an f-structure on M. By construction, we have φ2=-IdTM+η⊗ξ, from which it follows that
φ3+φ=0.In [4, 5], Blair et al. consider compact Riemannian manifolds equipped with a
regular normal f-structure φ and show that such manifolds are the total space of
a principal torus bundle over a complex manifold N, and that in addition, N is a Kähler manifold if the fundamental 2-form
of the f-structure is closed (i.e., if
M is a 𝒦-manifold). Saenz argued in [6] that if this 𝒦-structure is an 𝒮-structure, then the symplectic form of the
Kähler manifold N is integral.While the results in [5, 6] provide us with a generalization of the Boothby-Wang
theorem, the proofs in [5] (and by extension,
the argument in [6]) rely in several places
on the assumption that the f-structure φ is normal. Since this assumption is not required in
the original Boothby-Wang theorem, it is natural to ask what can be said if this
assumption is dropped for f-structures of higher corank. In this note, we use a
theorem of Tanno [7] to show that if
M is a compact almost 𝒮-manifold, in the sense of [8], then M is a principal torus bundle over a symplectic
manifold whose symplectic form is integral. (More precisely, the symplectic form
will be a real multiple of an integral symplectic form.) Not surprisingly, this
tells us that requiring φ to be normal is the same as demanding that the base
of our torus bundle be Kähler.This “generalized Boothby-Wang theorem’’ is one of a number of
similarities between manifolds with almost𝒮-structure and contact manifolds. In the final
section of this paper we demonstrate two more. First, there is a natural notion of
symplectization: given an almost 𝒮-manifold M, there is an open, conic, symplectic submanifold of
T*M whose base is M. Second, a choice of one-form (expressed in terms
of the almost 𝒮-structure) allows us to define a Jacobi bracket on
the algebra of smooth functions on M, giving us in particular a notion of Hamiltonian
vector field on manifolds with almost 𝒮-structure.


## 2. Preliminaries

### 2.1. Regular Involutive Distributions

LetF⊂TM be an involutive distribution of rank
k. We briefly recall the notion of a regular
distribution in the sense of Palais and refer the reader to [2] for the details. Roughly speaking, the
involutive distribution F is regular if each point
p∈M has a coordinate neighbourhood
(U,x1,…xn) such that (2.1){(∂∂x1)p,…,(∂∂xk)p}forms a basis forFp⊂TpM, and such that the integral submanifold of
F through p intersects U in only one k-dimensional slice. When
F is regular, the leaf space
ℱ=M/F is a smooth Hausdorff manifold, and the
quotient mapping πF:M→ℱ is smooth and closed. When
M is compact and connected, the leaves of
F are compact and isomorphic and are the fibres
of the smooth fibration πF:M→ℱ.In particular, a vector fieldX on M is regular if each p∈M has a neighbourhood U through which the integral curve of
X through p passes only once. If M is compact, the integral curves of a regular
vector field are thus diffeomorphic to circles. Applying this fact to the Reeb
vector field of a contact manifold gives part of the proof of the Boothby-Wang
theorem.


### 2.2.f-Structures

Anf-structure on
M is an endomorphism field
φ∈Γ(M,EndTM) such that (2.2)φ3+φ=0. Such structures were introduced by Yano in [9]; many of the facts regarding
f-structures are collected in the book [10]. By a result of Stong [11], every f-structure is of constant rank. If
rankφ=dimM, then φ is an almost complex structure on
M, while if rankφ=dimM-1, then φ determines an almost contact structure on
M.It is easy to check that the operatorsl=-φ2 and m=φ2+IdTM are complementary projection operators; letting
E=l(TM)=imφ and T=m(TM)=kerφ, we obtain the splitting(2.3)TM=E⊕T=imφ⊕ker⁡φ of the tangent bundle. Since
(φ|E)2=-IdE, φ is necessarily of even rank. When the corank of
φ is equal to one, the distribution
T is automatically trivial and involutive.
However, if rankT>1, this need not be the case, and one often makes
additional simplifying assumptions about T. An f-structure such that T is trivial is called an
f-structure with parallelizable
kernel (or f·pk-structure for short) in [8]. We will assume that an
f·pk-structure includes a choice of a
trivializing frame {ξi} and corresponding coframe
{ηi} for T*, with(2.4)ηi(ξj)=δji,φ(ξi)=ηj∘φ=0,φ2=-Id+∑ηi⊗ξi.(This is known as anf-structure with complemented
frames in [4]; such a choice
of frame and coframe always exists.) Given an f·pk-structure, it is always possible
[10] to find a Riemannian metric
g that is compatible with
(φ,ξi,ηj) in the sense that, for all
X,Y∈Γ(M,TM), we have(2.5)g(X,Y)=g(φX,φY)+∑i=1kηi(X)ηi(Y). Following [8], we will call the 4-tuple (φ,ξi,ηj,g) a metricf·pk structure. Given a
metric f·pk-structure (φ,ξi,ηj,g), we can define the fundamental 2-formΦg∈𝒜2(M) by (2.6)Φg(X,Y)=g(φX,Y).Remark 2.1.
Our definition ofΦg is chosen to agree with our preferred sign
conventions in symplectic geometry; however, many authors place
φ in the second slot, so our convention here
uses the opposite sign of that found for example in [5, 8].We will call anf-structure φ  regular if the
distribution T=kerφ is regular in the sense of Palais [2]. An f·pk-structure is regular if the vector fields
ξi are regular and independent. An
f·pk-structure is called
normal [4] if the
tensor N defined by (2.7)N=[φ,φ]+∑i=1kdηi⊗ξi vanishes identically. Here
[φ,φ] denotes the Nijenhuis torsion of
φ, which is given by(2.8)[φ,φ](X,Y)=φ2[X,Y]+[φX,φY]-φ[φX,Y]-φ[X,φY].Whenφ is normal, the +i-eigenbundle of φ (extended by ℂ linearity to TℂM) defines a CR structure
E1,0⊂TℂM. Regular normal f-structures are studied in [5], where it is proved that a compact
manifold with regular normal f-structure is a principal torus bundle over a
complex manifold N. If the fundamental 2-form
Φg of a normal f-structure is closed, then the
f-structure is called a 𝒦-structure, and M a 𝒦-manifold. For a compact regular
𝒦-manifold M, the base N of the torus fibration is a Kähler
manifold. A special case of a 𝒦-manifold is an 𝒮-manifold. On an 𝒮 manifold, there exist constants
α1,…,αk such that dηi=-αiΦg for i=1,…,k. Two commonly considered cases are the case
αi=0 for all i, and the case αi=1 for all i. In the language of CR geometry, the former
case is analogous to a “Levi-flat’’ CR manifold,while the latter defines an analogue of a strongly pseudoconvex CR manifold
(typically, strongly pseudoconvex CR manifolds are assumed to be of
“hypersurface type,’’ meaning that the complementary
distributionT has rank one; see [12]).A refinement of the notion of𝒮-structure was introduced in [8]: a metric f·pk-structure (φ,ξi,ηj,g) which is not necessarily normal is called an
almost𝒮-structure if dηi=-Φg for each i=1,…,k. An f-structure φ is called CR-integrable in [8] if the +i-eigenbundle E1,0⊂TℂM of φ is involutive (and hence, defines a CR
structure). It is shown in [8] that an
f·pk-structure is CR-integrable if and only if the
tensor N given by (2.7) satisfies N(X,Y)=0 for all X,Y∈Γ(M,E), where E=imφ, whereas for a normal f·pk-structure, N must vanish for all X,Y∈Γ(M,TM). In [13]
it is proved that a CR-integrable almost 𝒮-manifold admits a canonical connection
analogous to the Tanaka-Webster connection of a strongly pseudoconvex CR
manifold. For the relationship between this connection and the
∂¯b operator of the corresponding tangential
Cauchy-Riemann complex, as well as an application of this relationship to
defining an analogue of geometric quantization for almost
𝒮-manifolds, see [14].In this paper, we will define an almost𝒦-structure to be a metric
f·pk-structure for which dΦg=0, and we will define an almost
𝒮-structure more generally to be an almost
𝒦-structure such that dηi=-αiΦg for constants αi∈ℝ, for i=1,…,k.


## 2.1. Regular Involutive Distributions

LetF⊂TM be an involutive distribution of rank
k. We briefly recall the notion of a regular
distribution in the sense of Palais and refer the reader to [2] for the details. Roughly speaking, the
involutive distribution F is regular if each point
p∈M has a coordinate neighbourhood
(U,x1,…xn) such that (2.1){(∂∂x1)p,…,(∂∂xk)p}forms a basis forFp⊂TpM, and such that the integral submanifold of
F through p intersects U in only one k-dimensional slice. When
F is regular, the leaf space
ℱ=M/F is a smooth Hausdorff manifold, and the
quotient mapping πF:M→ℱ is smooth and closed. When
M is compact and connected, the leaves of
F are compact and isomorphic and are the fibres
of the smooth fibration πF:M→ℱ.In particular, a vector fieldX on M is regular if each p∈M has a neighbourhood U through which the integral curve of
X through p passes only once. If M is compact, the integral curves of a regular
vector field are thus diffeomorphic to circles. Applying this fact to the Reeb
vector field of a contact manifold gives part of the proof of the Boothby-Wang
theorem.


## 2.2.f-Structures

Anf-structure on
M is an endomorphism field
φ∈Γ(M,EndTM) such that (2.2)φ3+φ=0. Such structures were introduced by Yano in [9]; many of the facts regarding
f-structures are collected in the book [10]. By a result of Stong [11], every f-structure is of constant rank. If
rankφ=dimM, then φ is an almost complex structure on
M, while if rankφ=dimM-1, then φ determines an almost contact structure on
M.It is easy to check that the operatorsl=-φ2 and m=φ2+IdTM are complementary projection operators; letting
E=l(TM)=imφ and T=m(TM)=kerφ, we obtain the splitting(2.3)TM=E⊕T=imφ⊕ker⁡φ of the tangent bundle. Since
(φ|E)2=-IdE, φ is necessarily of even rank. When the corank of
φ is equal to one, the distribution
T is automatically trivial and involutive.
However, if rankT>1, this need not be the case, and one often makes
additional simplifying assumptions about T. An f-structure such that T is trivial is called an
f-structure with parallelizable
kernel (or f·pk-structure for short) in [8]. We will assume that an
f·pk-structure includes a choice of a
trivializing frame {ξi} and corresponding coframe
{ηi} for T*, with(2.4)ηi(ξj)=δji,φ(ξi)=ηj∘φ=0,φ2=-Id+∑ηi⊗ξi.(This is known as anf-structure with complemented
frames in [4]; such a choice
of frame and coframe always exists.) Given an f·pk-structure, it is always possible
[10] to find a Riemannian metric
g that is compatible with
(φ,ξi,ηj) in the sense that, for all
X,Y∈Γ(M,TM), we have(2.5)g(X,Y)=g(φX,φY)+∑i=1kηi(X)ηi(Y). Following [8], we will call the 4-tuple (φ,ξi,ηj,g) a metricf·pk structure. Given a
metric f·pk-structure (φ,ξi,ηj,g), we can define the fundamental 2-formΦg∈𝒜2(M) by (2.6)Φg(X,Y)=g(φX,Y).Remark 2.1.
Our definition ofΦg is chosen to agree with our preferred sign
conventions in symplectic geometry; however, many authors place
φ in the second slot, so our convention here
uses the opposite sign of that found for example in [5, 8].We will call anf-structure φ  regular if the
distribution T=kerφ is regular in the sense of Palais [2]. An f·pk-structure is regular if the vector fields
ξi are regular and independent. An
f·pk-structure is called
normal [4] if the
tensor N defined by (2.7)N=[φ,φ]+∑i=1kdηi⊗ξi vanishes identically. Here
[φ,φ] denotes the Nijenhuis torsion of
φ, which is given by(2.8)[φ,φ](X,Y)=φ2[X,Y]+[φX,φY]-φ[φX,Y]-φ[X,φY].Whenφ is normal, the +i-eigenbundle of φ (extended by ℂ linearity to TℂM) defines a CR structure
E1,0⊂TℂM. Regular normal f-structures are studied in [5], where it is proved that a compact
manifold with regular normal f-structure is a principal torus bundle over a
complex manifold N. If the fundamental 2-form
Φg of a normal f-structure is closed, then the
f-structure is called a 𝒦-structure, and M a 𝒦-manifold. For a compact regular
𝒦-manifold M, the base N of the torus fibration is a Kähler
manifold. A special case of a 𝒦-manifold is an 𝒮-manifold. On an 𝒮 manifold, there exist constants
α1,…,αk such that dηi=-αiΦg for i=1,…,k. Two commonly considered cases are the case
αi=0 for all i, and the case αi=1 for all i. In the language of CR geometry, the former
case is analogous to a “Levi-flat’’ CR manifold,while the latter defines an analogue of a strongly pseudoconvex CR manifold
(typically, strongly pseudoconvex CR manifolds are assumed to be of
“hypersurface type,’’ meaning that the complementary
distributionT has rank one; see [12]).A refinement of the notion of𝒮-structure was introduced in [8]: a metric f·pk-structure (φ,ξi,ηj,g) which is not necessarily normal is called an
almost𝒮-structure if dηi=-Φg for each i=1,…,k. An f-structure φ is called CR-integrable in [8] if the +i-eigenbundle E1,0⊂TℂM of φ is involutive (and hence, defines a CR
structure). It is shown in [8] that an
f·pk-structure is CR-integrable if and only if the
tensor N given by (2.7) satisfies N(X,Y)=0 for all X,Y∈Γ(M,E), where E=imφ, whereas for a normal f·pk-structure, N must vanish for all X,Y∈Γ(M,TM). In [13]
it is proved that a CR-integrable almost 𝒮-manifold admits a canonical connection
analogous to the Tanaka-Webster connection of a strongly pseudoconvex CR
manifold. For the relationship between this connection and the
∂¯b operator of the corresponding tangential
Cauchy-Riemann complex, as well as an application of this relationship to
defining an analogue of geometric quantization for almost
𝒮-manifolds, see [14].In this paper, we will define an almost𝒦-structure to be a metric
f·pk-structure for which dΦg=0, and we will define an almost
𝒮-structure more generally to be an almost
𝒦-structure such that dηi=-αiΦg for constants αi∈ℝ, for i=1,…,k.


## 3. Properties of Almost𝒦 and Almost 𝒮-Structures

Let(φ,ξi,ηi) be an f·pk-structure on a compact, connected manifold
M. Let g be a Riemannian metric satisfying the compatibility
condition (2.5), and let
Φg denote the corresponding fundamental 2-form. Let
E=imφ, and T=kerφ denote the distribution spanned by the
ξi. It is easy to check that the distributions
E and T are orthogonal with respect to
g, and that the restriction of
Φg to E⊗E is nondegenerate, from which we have the following
lemma.Lemma 3.1.
X
∈
Γ
(
M
,
T
) if and only if ι(X)Φg=0.Proposition 3.2.
Let(φ,ξi,ηi,g) be a metric f·pk-structure. Then
T=kerφ is involutive whenever
dΦg=0.Proof.
LetX,Y∈Γ(M,T), and let Z∈Γ(M,TM). Then, using Lemma 3.1 above, we have (3.1)dΦg(X,Y,Z)=X⋅Φg(Y,Z)+Y⋅Φg(Z,X)+Z⋅Φg(X,Y)-Φg([X,Y],Z)-Φg([Y,Z],X)-Φg([Z,X],Y)=-Φg([X,Y],Z). Therefore, if dΦg=0, then ι([X,Y])Φg=0, and thus [X,Y]∈Γ(M,T), which proves the proposition.Let us now suppose that(φ,ξi,ηi,g) is an almost 𝒮-structure, so that the 1-forms
ηi satisfy dηi=-αiΦg for constants αi, some of which may be zero. The following results
were proved in [8] in the case that
αi=1 for all i; we easily see that the results remain true in our
more general setting.Proposition 3.3.
If(φ,ξi,ηj,g) is an almost 𝒮-structure, then ℒ(ξi)ξj=[ξi,ξj]=0 for all i,j=1,…,k.Proof.
Since the fundamental 2-formΦg of an almost 𝒮-structure is closed, the distribution
T is involutive. Thus we may write
[ξi,ξj]=∑cijaξa. But for any a,i,j∈{1,…,k}, we have (3.2)cija=ηa([ξi,ξj])=ξi⋅ηa(ξj)-ξj⋅ηa(ξi)-dηa(ξi,ξj)=αaΦg(ξi,ξj)=0.Proposition 3.4.
If(φ,ξi,ηj,g) is an almost 𝒮-structure, then ℒ(ξi)ηj=0 for all i,j=i,…,k.Proof.
We haveℒ(ξ)ηj=d(ηj(ξi))+ι(ξi)dηj=-αj(ι(ξi)Φg)=0.We remark that several other results from [8]
hold in this more general setting, but they are not needed here. To conclude this
section, we state a theorem due to Tanno [7].Theorem 3.5.
For a regular and proper vector fieldX on a manifold M, the following are equivalent. (i)
The period functionλX of X is constant.(ii)
There exists a 1-formη such that η(X)=1 and ℒ(X)η=0.(iii)
There exists a Riemannian metricg such that g(X,X)=1 and ℒ(X)g=0.In the above theorem, the period functionλX:M→ℝ is defined by(3.3)λX(p)=inf⁡{t>0∣exp⁡(tX)⋅p=p}.IfM is noncompact, the value λX(p)=∞ is possible. Part (iii) of the above tells us that
X is a unit Killing field for the metric
g. Using this result, Tanno was able to give a simple
proof (which is reproduced in [3]) of the
Boothby-Wang theorem [1].


## 4. The Structure of Regular Almost𝒮-Manifolds

As noted above, from [5], a compact manifold
with regular normal f-structure is a principal torus bundle over a
complex manifold N, and N is Kähler if M is a 𝒦-manifold. If M is an 𝒮-manifold with Φg=-dηi for each i, then by [6], the symplectic form on N is integral. We now dispense with the requirement
that the f-structure on M be normal, and state a similar result for almost
𝒮-manifolds.Theorem 4.1.
LetM be a compact manifold of dimension
2n+k equipped with a regular almost
𝒮-structure (φ,ξ̃i,η̃i,g̃) of
rank  2n. Then there exists an almost
𝒮-structure (φ,ξi,ηi,g) on M for which the vector fields
ξ1,…,ξk are the infinitesimal generators of a free and
effective 𝕋k-action on M. Moreover, the quotient
N=M/𝕋k is a smooth symplectic manifold of dimension
2n, and if the αi such that dη̃i=-αiΦg̃ are not all zero, then the symplectic form on
N is a real multiple of an integral symplectic
form.Proof.
By assumption, the vector fieldsξ̃1,…,ξ̃k are regular, independent, and proper, and by
Proposition 3.2, the
distribution T=span{ξ̃1,…,ξ̃k} is involutive. Thus, by the results of Palais,
N=M/T is a smooth manifold, and
π:M→N is a smooth fibration whose fibres are the
leaves of the distribution T. Since M is compact, the fibres are compact and
isomorphic [2]. For each
i=1,…,k, we have η̃i(ξ̃i)=1 and ℒ(ξ̃i)η̃i=0. Thus, by Theorem 3.5, the period functions λi=λξ̃i are constant. We rescale by setting
ξi=λiξ̃i and ηi=(1/λi)η̃i. We still have ηi(ξj)=δji, and note that the associated metric
g for which (φ,ξi,ηi,g) is an almost 𝒮-structure differs from
g̃ only along T, so that Φg=Φg̃. Each ξi now has period 1, and since the vector fields
ξi all commute, they are the generators of a free
and effective 𝕋k-action on M. The argument for local triviality is the same
as in [5], so we do not repeat it here.
Thus, we have that M is a principal 𝕋k-bundle over N=M/T. The infinitesimal action of
ℝk is given by (4.1)X=(t1,…,tk)⟼XM=∑tiξi, from which we see that η=(η1,…,ηk) is a connection 1-form on
M: we have ι(XM)η=X and ℒ(XM)η=0 for all X∈ℝk.
Now, we note that the fundamental 2-formΦg is horizontal and invariant, since
ι(X)Φg=ℒ(X)Φg=0 for all X∈Γ(M,T), and thus there exists a 2-form
Ω on N such that π*Ω=Φg. Since π*dΩ=dΦg=0, Ω is closed, and since π*Ωn=Φgn≠0, Ω is nondegenerate, and hence symplectic.
Finally, let us suppose that one of theαi is nonzero; without loss of generality, let us
say α1≠0. By the same argument as above, the vector
fields ξ2,…,ξk generate a free 𝕋k-1-action on M, giving us a fibration
p:M→P. Now, since ℒ(ξi)ξ1=ℒ(ξi)η1=0 for i=2,…,k, the vector field ξ1 and 1-form η1 are invariant under the
𝕋k-1-action. We can thus define a 1-form
η on P by η(X)=η1(p̃X), where p̃X denotes the horizontal lift of
X with respect to the connection 1-form defined
by η2,…,ηk, and a vector field ξ on P by ξ=p*ξ1. Note that dη(X,Y)=dη1(p̃X,p̃Y). We then have η(ξ)=1, and ℒ(ξ)η=ι(ξ1)dη1=0, so that Theorem 3.5 applies to the pair (η,ξ). It follows that ξ generates a free action of
S1=ℝ/ℤ on P, giving us the 𝕋1-bundle structure q:P→N. Since π=q∘p, it follows that(4.2)dη(X,Y)=dη1(p̃X,p̃Y)=-α1λ1(π*Ω)(p̃X,p̃Y)=-α1λ1q*Ω(X,Y). Thus, P is a Boothby-Wang fibration over
(N,(α1/λ1)Ω), from which it follows that the symplectic form
(α1/λ)Ω must be integral (see [15]), and hence Ω is a real multiple of an integral symplectic
form.Remark 4.2.
Note that since the last part of the argument is valid for any pair of nonzero
constantsαi,αj, from which it follows that for each
i,j for which αi and αj are nonzero, we must have
αi/λi·λj/αj∈ℚ.Conversely, we have the following theorem.Theorem 4.3.
Suppose thatM is a principal 𝕋k-bundle over a symplectic manifold
(N,ω), equipped with connection 1-form
η=(η1,…,ηk) such that there exist constants
α1,…,αk for which dηi=-αiπ*ω. Then M admits an almost 𝒮-structure.Proof.
The proof is essentially the same as the proof given in [4] when N is Kähler, if we omit the proof of
normality. Given a choice of compatible almost complex structure
J and associated metric G, we can define an f-structure φ by φX=π̃Jπ*X, where π̃ denotes the horizontal lift with respect to
η. If we let ξ1,…,ξk denote vertical vectors such that
ηi(ξj)=δji, and define the metric
g by (4.3)g(X,Y)=π*G(X,Y)+∑ηi(X)ηi(Y), then it is straightforward to check that the data
(φ,ξi,ηj,g) defines an almost 𝒮-structure on M. (Note that Φg=π*ω, so that dηi=-αiΦg.)Remark 4.4.
We can also use the results of Tanno [7]
to show that the vector fields ξ1,…,ξk of an almost 𝒮-structure are Killing. Let
π̃ denote the horizontal lift defined by
η. Then we can define a Riemannian metric
G on N by G(X,Y)=g(π̃X,π̃Y) for any X,Y∈Γ(N,TN), where g is the metric of the almost
𝒮-structure on M. It follows that g=π*G+∑ηi⊗ηi, whence g(ξi,ξi)=1 and ℒ(ξi)g=0 for i=1,…,k. Moreover, the endomorphism field
J∈Γ(N,End(TN)) defined by JX=π*φπ̃X is easily seen to be an almost complex
structure on N that is compatible with
G, and the symplectic form
Ω then satisfies Ω(X,Y)=G(X,JY).Remark 4.5.
IfM is only an almost 𝒦-manifold, it is not clear that we can expect
any analogous result to hold, since the proof in [5] for a 𝒦-manifold does not work without normality, and
Tanno’s theorem cannot be applied if ℒ(ξi)ηj≠0 for all i,j, and this need not hold if
dηj is not a multiple of Φg.Remark 4.6.
IfM is noncompact, then as noted below the
statement of Tanno’s theorem, the period λi of one of the ξi could be infinite, in which case
ξi generates an ℝ-action on M instead of an S1-action.


## 5. Symplectization and Jacobi Structures

We conclude this paper with a discussion of the relationship between almost𝒮-structures and related geometries intended to
reinforce the view that almost 𝒮-structures deserve to be viewed as higher corank
analogues of contact structures. (However, see also [16] for the notion of k-contact structures, which, from the point of view
of Heisenberg calculus, are also deserving of the title of higher corank contact
structure. From this perspective, almost 𝒮-structures are perhaps more analogous to contact
metric structures, or even strongly pseudoconvex CR structures, although they are
not CR-integrable in general.)Recall that a stable complex structure on a manifoldM is a complex structure defined on the fibres of
TM⊕ℝk for some k. Given an f·pk-structure (φ,ξi,ηj) on M, we obtain a stable complex structure
J∈Γ(M,End(TM⊕ℝk)) by setting JX=φX for X∈Γ(M,E), and defining Jξi=τi and Jτi=-ξi, where τ1,…,τk is a basis for ℝk. As explained in [17], a stable complex structure determines a Spinc-structure on M.Alternatively, (and with some abuse of notation), we can think of the above complex
structure on each fibreTxM×ℝk as coming from an almost complex structure on
M×ℝk obtained from the f-structure φ. With this point of view, we note that it is
possible to define a “symplectization’’ analogous to the
symplectization of a cooriented contact manifold, provided that our
f·pk-structure is an almost 𝒮-structure, with at least one of the
αj (such that dηj=-αjΦg) nonzero. As above, we let
TM=E⊕T denote the splitting of the tangent bundle
determined by the f-structure, and let E0≅T*=span{ηi}≅M×ℝk denote the annihilator of E. It is then possible to find an open connected
symplectic submanifold E+0 of T*M whose tangent bundle is TxM×ℝk. For concreteness, let us use the identification
E0≅M×ℝk, and with respect to coordinates
(x,t1,…,tk), let (5.1)α=∑i=1ktiηi,and define ω=-dα. (We are abusing notation here slightly;
technically we should write π*ηi in place of ηi, where π:M×ℝk→M is the projection onto the first factor.) Using the
fact that dηi=-αiΦg for each i, we have (5.2)ω=∑ηj∧dtj+(∑tjαj)Φg.Define τ∈C∞(E0) to be the function given in coordinates by
τ=∑αjtj. Note that since ηi∧ηi=dti∧dti=0, we have (5.3)(∑i=1kηj∧dtj)k=k!η1∧dt1∧⋯∧ηk∧dtk.We also note that Φgm=0 for m>n. Thus, using the binomial theorem, we find that the
top-degree form ωn+k has only one nonzero term; namely, (5.4)ωn+k=(n+k)!n!η1∧dt1∧⋯∧ηk∧dtk∧(τΦg)n.Thus,ωn+k is a volume form on the open subset
E+0 of E0 defined by τ>0, and hence ω is a symplectic form on E+0.Next, we will show that for certain choices of sectionη∈Γ(M,E0) we obtain a Jacobi structure on
M defined in a manner analogous to the Jacobi
structure associated to a choice of contact form on a contact manifold. We recall
that a Jacobi structure on M is given by a Lie bracket {·,·} on C∞(M) such that for any f,g∈C∞(M) the support of {f,g} is contained in the intersection of the supports of
f and g. Jacobi structures were introduced independently by
Kirillov [18] and Lichnerowicz [19]; a good introduction can be found in [20].Again, we assumeM is equipped with an almost
𝒮-structure with the constants
αj such that dηj=-αjΦg not all zero. Our first goal is to define a notion
of a Hamiltonian vector field Xf associated to each function
f∈C∞(M). To begin with, let ξ=∑bjξj be an arbitrary section of
T=kerφ, and let η=∑cjηj be an arbitrary section of
E0≅T*. We will narrow down the possibilities for
ξ and η as we consider the properties we wish the vector
fields Xf to satisfy. The idea is to generalize the approach
used to define Hamiltonian vector fields on a contact manifold
(M,η). Recall that on manifold equipped with a contact
form η, where we define Φ=-dη, the Reeb vector field ξ is defined by ι(ξ)η=1 and ι(ξ)Φ=0. A contact Hamiltonian vector field
Xf satisfies the equations ι(Xf)η=f and ι(Xf)Φ=df-(ξ·f)η. Lichnerowicz showed in [21] that these are the necessary and sufficient conditions for
each Xf to be an infinitesimal symmetry of the contact
structure: it follows that for each f∈C∞(M), ℒ(Xf)η=(ξ·f)η.We wish to impose similar conditions onξ, η and (the yet to be defined)
Xf in the case of almost 𝒮-manifolds. We already know that
ι(ξ)Φg=0, by Lemma 3.1, so we begin by adding the requirement that
η(ξ)=∑bjcj=1. Next, we give our definition of a Hamiltonian
vector field.Definition 5.1.
Letη and ξ be as above. For any f∈C∞(M), we define the Hamiltonian vector
field associated to f by the equations (5.5)ι(Xf)ηj=αjf,forj=1,…,k,(5.6)ι(Xf)Φg=df-(ξ⋅f)η.Remark 5.2.
Note that the above equations uniquely defineXf, by the nondegeneracy of the restriction of
Φ to E=imφ. The constants αj are the same ones such that
dηj=-αjΦg. One can check that if we began with
aj in place of the αj, we would be forced to take
aj=αj for consistency reasons. (In particular, this
will be necessary if the bracket we define below is to be a Lie bracket.)
Moreover, this gives us the identity (5.7)L(Xf)ηj=αj(ξ⋅f)η for each j=1,…,k; we would otherwise have an unwanted term of
the form (aj-αj)df. Note that on the right-hand side of the above
equation we have η and not ηj; this is unavoidable with our definition of
Xf.We can fix the coefficients ofξ by requiring that ξ be the Hamiltonian vector field associated to the
constant function 1, as is standard for Jacobi structures (see [20]). It is easy to see that (5.5) then immediately forces us to
take ξ=∑αjξj; that is, the coefficients
bj are equal the constants αj. Thus, ξ is essentially determined by the almost
𝒮-structure, although η is constrained only by the condition
η(ξ)=1, so the Jacobi structure we define below cannot be
considered entirely canonical (as one might expect). From the requirement that
η(ξ)=1, it follows that for each f∈C∞(M), we have (5.8)L(Xf)η=∑cjL(Xf)ηj=∑cjαj(ξ⋅f)η=(ξ⋅f)η,again in analogy with the contact case. Note that the
normalization η(ξ)=1 also implies that dη=-Φg. We are now ready to define our bracket on
C∞(M).Definition 5.3.
LetM be a manifold with almost
𝒮-structure, with constants
αj not all zero. Let ξ=∑αjξj, and let η be a section of E0 such that η(ξ)=1. We then define a bracket on
C∞(M) by (5.9){f,g}=ι([Xf,Xg])η.The bracket is clearly antisymmetric, and one checks (using the identityι([X,Y])=[ℒ(X),ι(Y)]) that(5.10){f,g}=Xf⋅g-Xg⋅f+Φg(Xf,Xg)=Xf⋅g-(ξ⋅f)g.Note that since the definition of the Hamiltonian vector fields depended on the
choice ofη, the bracket depends on η, even though η no longer appears explicitly in either of the above
expressions for the bracket. From the latter equality we see that the support of
{f,g} is contained in the support of
g, and by antisymmetry it must be contained in the
support of f as well. Thus, the bracket given by (5.9) is a Jacobi bracket provided we
can verify the Jacobi identity. Since the Jacobi identity is valid for the Lie
bracket on vector fields, it suffices to prove the following proposition.Proposition 5.4.
Let{f,g} be the bracket on C∞(M) given by (5.9). Then the vector field X{f,g} corresponding to the function
{f,g} is given by X{f,g}=[Xf,Xg].Lemma 5.5.
For eachi=1,…,k, we have [ξi,Xf]=Xξi·f.Proof.
From Propositions3.3 and 3.4, we know that
[ξi,ξj]=0 and ℒ(ξi)ηj=0 for any i,j∈{1,…,k}; from the latter, it follows easily that
ℒ(ξi)Φg=0 as well. The result then follows from the
uniqueness of Hamiltonian vector fields, since (5.11)ι([ξi,Xf])ηj=[L(ξi),ι(Xf)]ηj=αjξi⋅f,ι([ξi,Xf])Φg=L(ξi)(df-(ξ⋅f)η)=d(ξi⋅f)-(ξ⋅(ξi⋅f))η.Lemma 5.6.
For eachi=1,…,k, we have ξi·{f,g}={ξi·f,g}+{f,ξi·g}.Proof.
We have, using Lemma5.5 and the
fact that [ξi,ξ]=0 in the second line, (5.12)ξi⋅{f,g}=ξi⋅(Xf⋅g)-ξi⋅((ξ⋅f)g)=Xf⋅(ξi⋅g)-(ξ⋅f)(ξi⋅g)+Xξi⋅f⋅g-ξ(ξi⋅f)g={f,ξi⋅g}+{ξi⋅f,g}.Proof of Proposition5.4.
We need to show thatι([Xf,Xg])ηj=αj{f,g} for each j=1,…,k, and that ι([Xf,Xg])Φ=d{f,g}-(ξ·{f,g})η. First, since ι(Xg)η=∑cjαjg=g, we have (5.13)ι([Xf,Xg])ηj=L(Xf)ηj(Xg)-ι(Xg)L(Xf)ηj=αjXf⋅g-ι(Xg)(αjξ⋅f)η=αj{f,g}. From Lemma 5.6, we have ξ·{f,g}={f,ξ·g}-{g,ξ·f}=Xf·(ξ·g)-Xg·(ξ·f), and, thus, (5.14)ι([Xf,Xg])Φg=L(Xf)(dg-(ξ⋅g)η)-ι(Xg)(-d(ξ⋅f)∧η+(ξ⋅f)Φg)=d(Xf⋅g)-Xf⋅(ξ⋅g)-(ξ⋅g)(ξ⋅f)η+Xg⋅(ξ⋅f)η-gd(ξ⋅f)-(ξ⋅f)(dg-(ξ⋅g)η)=d(Xf⋅g-(ξ⋅f)g)-(Xf⋅(ξ⋅g)-Xg⋅(ξ⋅f))η=d{f,g}-ξ⋅{f,g}η.



---

*Source: 879042-2011-12-13.xml*