1. Introduction

Let be a cooriented contact manifold. The Boothby-Wang theorem [1] tells us that if the Reeb field corresponding to the contact form is regular (in the sense of Palais [2]), then is a prequantum circle bundle over a symplectic manifold , where and may be identified with the connection 1-form. Conversely, let be a prequantum circle bundle over a symplectic manifold , and let be a connection 1-form. Given a choice of compatible almost complex structure for , let be the associated Riemannian metric on , and let denote the horizontal lift of vector fields defined by . We can then define an endomorphism field by

and a Riemannian metric by . If we let be the vertical vector field satisfying , then defines a contact metric structure on [3]. In particular, we note that is an -structure on . By construction, we have , from which it follows that .

In [4, 5], Blair et al. consider compact Riemannian manifolds equipped with a regular normal -structure and show that such manifolds are the total space of a principal torus bundle over a complex manifold , and that in addition, is a Kähler manifold if the fundamental 2-form of the -structure is closed (i.e., if is a -manifold). Saenz argued in [6] that if this -structure is an -structure, then the symplectic form of the Kähler manifold 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 -structure 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 -structures of higher corank. In this note, we use a theorem of Tanno [7] to show that if is a compact almost -manifold, in the sense of [8], then 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 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 almost -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 -manifold , there is an open, conic, symplectic submanifold of whose base is . Second, a choice of one-form (expressed in terms of the almost -structure) allows us to define a Jacobi bracket on the algebra of smooth functions on , giving us in particular a notion of Hamiltonian vector field on manifolds with almost -structure.

2. Preliminaries

2.1. Regular Involutive Distributions

Let be an involutive distribution of rank . 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 is regular if each point has a coordinate neighbourhood such that

forms a basis for , and such that the integral submanifold of through intersects in only one -dimensional slice. When is regular, the leaf space is a smooth Hausdorff manifold, and the quotient mapping is smooth and closed. When is compact and connected, the leaves of are compact and isomorphic and are the fibres of the smooth fibration .

In particular, a vector field on is regular if each has a neighbourhood through which the integral curve of through passes only once. If 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. -Structures

An -structure on is an endomorphism field such that Such structures were introduced by Yano in [9]; many of the facts regarding -structures are collected in the book [10]. By a result of Stong [11], every -structure is of constant rank. If , then is an almost complex structure on , while if , then determines an almost contact structure on .

It is easy to check that the operators and are complementary projection operators; letting and , we obtain the splitting of the tangent bundle. Since , is necessarily of even rank. When the corank of is equal to one, the distribution is automatically trivial and involutive. However, if , this need not be the case, and one often makes additional simplifying assumptions about . An -structure such that is trivial is called an -structure with parallelizable kernel (or pk-structure for short) in [8]. We will assume that an ·pk-structure includes a choice of a trivializing frame and corresponding coframe for , with

(This is known as an -structure with complemented frames in [4]; such a choice of frame and coframe always exists.) Given an ·pk-structure, it is always possible [10] to find a Riemannian metric that is compatible with in the sense that, for all , we have Following [8], we will call the 4-tuple a metric ·pk structure. Given a metric ·pk-structure , we can define the fundamental 2-form by

Remark 2.1. Our definition of is chosen to agree with our preferred sign conventions in symplectic geometry; however, many authors place in the second slot, so our convention here uses the opposite sign of that found for example in [5, 8].

We will call an -structure   regular if the distribution is regular in the sense of Palais [2]. An pk-structure is regular if the vector fields are regular and independent. An ·pk-structure is called normal [4] if the tensor defined by vanishes identically. Here denotes the Nijenhuis torsion of , which is given by

When is normal, the -eigenbundle of (extended by linearity to ) defines a CR structure . Regular normal -structures are studied in [5], where it is proved that a compact manifold with regular normal -structure is a principal torus bundle over a complex manifold . If the fundamental 2-form of a normal -structure is closed, then the -structure is called a -structure, and a -manifold. For a compact regular -manifold , the base of the torus fibration is a Kähler manifold. A special case of a -manifold is an -manifold. On an manifold, there exist constants such that for . Two commonly considered cases are the case for all , and the case for all . 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 distribution has rank one; see [12]).

A refinement of the notion of -structure was introduced in [8]: a metric pk-structure which is not necessarily normal is called an almost -structure if for each . An -structure is called CR-integrable in [8] if the -eigenbundle of is involutive (and hence, defines a CR structure). It is shown in [8] that an pk-structure is CR-integrable if and only if the tensor given by (2.7) satisfies for all , where , whereas for a normal pk-structure, must vanish for all . In [13] it is proved that a CR-integrable almost -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 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 -manifolds, see [14].

In this paper, we will define an almost -structure to be a metric pk-structure for which , and we will define an almost -structure more generally to be an almost -structure such that for constants , for .

3. Properties of Almost and Almost -Structures

Let be an pk-structure on a compact, connected manifold . Let be a Riemannian metric satisfying the compatibility condition (2.5), and let denote the corresponding fundamental 2-form. Let , and denote the distribution spanned by the . It is easy to check that the distributions and are orthogonal with respect to , and that the restriction of to is nondegenerate, from which we have the following lemma.

Lemma 3.1. if and only if .

Proposition 3.2. Let be a metric ·pk-structure. Then is involutive whenever .

Proof. Let , and let . Then, using Lemma 3.1 above, we have Therefore, if , then , and thus , which proves the proposition.

Let us now suppose that is an almost -structure, so that the 1-forms satisfy for constants , some of which may be zero. The following results were proved in [8] in the case that for all ; we easily see that the results remain true in our more general setting.

Proposition 3.3. If is an almost -structure, then for all .

Proof. Since the fundamental 2-form of an almost -structure is closed, the distribution is involutive. Thus we may write . But for any , we have

Proposition 3.4. If is an almost -structure, then for all .

Proof. We have .

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 field on a manifold , the following are equivalent. (i)The period function of is constant.(ii)There exists a 1-form such that and .(iii)There exists a Riemannian metric such that and .

In the above theorem, the period function is defined by

If is noncompact, the value is possible. Part (iii) of the above tells us that is a unit Killing field for the metric . 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 Almost -Manifolds

As noted above, from [5], a compact manifold with regular normal -structure is a principal torus bundle over a complex manifold , and is Kähler if is a -manifold. If is an -manifold with for each , then by [6], the symplectic form on is integral. We now dispense with the requirement that the -structure on be normal, and state a similar result for almost -manifolds.

Theorem 4.1. Let be a compact manifold of dimension equipped with a regular almost -structure of rank  . Then there exists an almost -structure on for which the vector fields are the infinitesimal generators of a free and effective -action on . Moreover, the quotient is a smooth symplectic manifold of dimension , and if the such that are not all zero, then the symplectic form on is a real multiple of an integral symplectic form.

Proof. By assumption, the vector fields are regular, independent, and proper, and by Proposition 3.2, the distribution is involutive. Thus, by the results of Palais, is a smooth manifold, and is a smooth fibration whose fibres are the leaves of the distribution . Since is compact, the fibres are compact and isomorphic [2]. For each , we have and . Thus, by Theorem 3.5, the period functions are constant. We rescale by setting and . We still have , and note that the associated metric for which is an almost -structure differs from only along , so that . Each now has period 1, and since the vector fields all commute, they are the generators of a free and effective -action on . The argument for local triviality is the same as in [5], so we do not repeat it here. Thus, we have that is a principal -bundle over . The infinitesimal action of is given by from which we see that is a connection 1-form on : we have and for all .
Now, we note that the fundamental 2-form is horizontal and invariant, since for all , and thus there exists a 2-form on such that . Since , is closed, and since , is nondegenerate, and hence symplectic.
Finally, let us suppose that one of the is nonzero; without loss of generality, let us say . By the same argument as above, the vector fields generate a free -action on , giving us a fibration . Now, since for , the vector field and 1-form are invariant under the -action. We can thus define a 1-form on by , where denotes the horizontal lift of with respect to the connection 1-form defined by , and a vector field on by . Note that . We then have , and , so that Theorem 3.5 applies to the pair . It follows that generates a free action of on , giving us the -bundle structure . Since , it follows that Thus, is a Boothby-Wang fibration over , from which it follows that the symplectic form must be integral (see [15]), and hence 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 constants , from which it follows that for each for which and are nonzero, we must have .

Conversely, we have the following theorem.

Theorem 4.3. Suppose that is a principal -bundle over a symplectic manifold , equipped with connection 1-form such that there exist constants for which . Then admits an almost -structure.

Proof. The proof is essentially the same as the proof given in [4] when is Kähler, if we omit the proof of normality. Given a choice of compatible almost complex structure and associated metric , we can define an -structure by , where denotes the horizontal lift with respect to . If we let denote vertical vectors such that , and define the metric by then it is straightforward to check that the data defines an almost -structure on . (Note that , so that .)

Remark 4.4. We can also use the results of Tanno [7] to show that the vector fields of an almost -structure are Killing. Let denote the horizontal lift defined by . Then we can define a Riemannian metric on by for any , where is the metric of the almost -structure on . It follows that whence and for . Moreover, the endomorphism field defined by is easily seen to be an almost complex structure on that is compatible with , and the symplectic form then satisfies .

Remark 4.5. If is only an almost -manifold, it is not clear that we can expect any analogous result to hold, since the proof in [5] for a -manifold does not work without normality, and Tanno’s theorem cannot be applied if for all , and this need not hold if is not a multiple of .

Remark 4.6. If is noncompact, then as noted below the statement of Tanno’s theorem, the period of one of the could be infinite, in which case generates an -action on instead of an -action.

5. Symplectization and Jacobi Structures

We conclude this paper with a discussion of the relationship between almost -structures and related geometries intended to reinforce the view that almost -structures deserve to be viewed as higher corank analogues of contact structures. (However, see also [16] for the notion of -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 -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 manifold is a complex structure defined on the fibres of for some . Given an pk-structure on , we obtain a stable complex structure by setting for , and defining and , where is a basis for . As explained in [17], a stable complex structure determines a -structure on .

Alternatively, (and with some abuse of notation), we can think of the above complex structure on each fibre as coming from an almost complex structure on obtained from the -structure . 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 pk-structure is an almost -structure, with at least one of the (such that ) nonzero. As above, we let denote the splitting of the tangent bundle determined by the -structure, and let denote the annihilator of . It is then possible to find an open connected symplectic submanifold of whose tangent bundle is . For concreteness, let us use the identification , and with respect to coordinates , let and define . (We are abusing notation here slightly; technically we should write in place of , where is the projection onto the first factor.) Using the fact that for each , we have Define to be the function given in coordinates by . Note that since , we have We also note that for . Thus, using the binomial theorem, we find that the top-degree form has only one nonzero term; namely,

Thus, is a volume form on the open subset of defined by , and hence is a symplectic form on .

Next, we will show that for certain choices of section we obtain a Jacobi structure on 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 is given by a Lie bracket on such that for any the support of is contained in the intersection of the supports of and . Jacobi structures were introduced independently by Kirillov [18] and Lichnerowicz [19]; a good introduction can be found in [20].

Again, we assume is equipped with an almost -structure with the constants such that not all zero. Our first goal is to define a notion of a Hamiltonian vector field associated to each function . To begin with, let be an arbitrary section of , and let be an arbitrary section of . We will narrow down the possibilities for and as we consider the properties we wish the vector fields to satisfy. The idea is to generalize the approach used to define Hamiltonian vector fields on a contact manifold . Recall that on manifold equipped with a contact form , where we define , the Reeb vector field is defined by and . A contact Hamiltonian vector field satisfies the equations and . Lichnerowicz showed in [21] that these are the necessary and sufficient conditions for each to be an infinitesimal symmetry of the contact structure: it follows that for each , .

We wish to impose similar conditions on , and (the yet to be defined) in the case of almost -manifolds. We already know that , by Lemma 3.1, so we begin by adding the requirement that . Next, we give our definition of a Hamiltonian vector field.

Definition 5.1. Let and be as above. For any , we define the Hamiltonian vector field associated to by the equations

Remark 5.2. Note that the above equations uniquely define , by the nondegeneracy of the restriction of to . The constants are the same ones such that . One can check that if we began with in place of the , we would be forced to take 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 for each ; we would otherwise have an unwanted term of the form . Note that on the right-hand side of the above equation we have and not ; this is unavoidable with our definition of .

We can fix the coefficients of by requiring that 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 ; that is, the coefficients are equal the constants . Thus, is essentially determined by the almost -structure, although is constrained only by the condition , so the Jacobi structure we define below cannot be considered entirely canonical (as one might expect). From the requirement that , it follows that for each , we have again in analogy with the contact case. Note that the normalization also implies that . We are now ready to define our bracket on .

Definition 5.3. Let be a manifold with almost -structure, with constants not all zero. Let , and let be a section of such that . We then define a bracket on by

The bracket is clearly antisymmetric, and one checks (using the identity ) that

Note that since the definition of the Hamiltonian vector fields depended on the choice of , the bracket depends on , even though no longer appears explicitly in either of the above expressions for the bracket. From the latter equality we see that the support of is contained in the support of , and by antisymmetry it must be contained in the support of 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 be the bracket on given by (5.9). Then the vector field corresponding to the function is given by .

Lemma 5.5. For each , we have .

Proof. From Propositions 3.3 and 3.4, we know that and for any ; from the latter, it follows easily that as well. The result then follows from the uniqueness of Hamiltonian vector fields, since

Lemma 5.6. For each , we have .

Proof. We have, using Lemma 5.5 and the fact that in the second line,

Proof of Proposition 5.4. We need to show that for each , and that . First, since , we have From Lemma 5.6, we have , and, thus,

Acknowledgments

The research for this paper was made possible by a postdoctoral fellowship from Natural Sciences and Engineering Research Council of Canada (NSERC), and by the University of California, Berkeley, the host institution for the fellowship. The author would like to thank Alan Weinstein for several useful discussions and suggestions which helped to improve the paper.