1. Introduction

While it is well known (see [1–4]) that every symplectic manifold can be made into an almost Kähler manifold by choosing an almost complex structure that satisfies and the compatibility condition for every (Moreover, for an almost Kähler manifold is required to be a positive definite Riemannian metric on , that is, is required to be tame. If is merely pseudo-Riemannian, then is called an almost pseudo-Kähler manifold.), it is more difficult to find in the literature a concise answer to the corresponding question for almost paracomplex structures.

Definition 1.1. Let be a (almost) symplectic manifold. A bundle automorphism satisfying and for every is called a compatible almost paracomplex structure on .

An introduction to paracomplex geometry can be found in [5–7]. As illustrated by [6, Theorem  6, Proposition  7], compatible almost paracomplex structures on symplectic manifolds correspond on the one hand to almost bi-Lagrangian structures (the eigenbundles of to the eigenvalues are transversal Lagrangian distributions, i.e., and hold) and on the other hand to almost para-Kähler structures (by a neutral metric is defined, which satisfies ).

Definition 1.2. A symplectic manifold endowed with a compatible almost paracomplex structure is called an almost para-Kähler manifold (an almost bi-Lagrangian manifold).

Existence of compatible almost paracomplex structures is characterized by the following theorem.

Theorem 1.3. On a (almost) symplectic manifold of dimension there exists a compatible almost paracomplex structure if and only if the structure group of can be reduced from to the paraunitary group .

The validity of this theorem is mentioned in [6, Section  2.5]. For the convenience of the reader a proof of Theorem 1.3 is given in Section 2. An aim of this paper is to characterize (almost) symplectic manifolds that admit a compatible almost paracomplex structure and a tame compatible almost complex structure such that is valid.

Definition 1.4. A pair of an almost complex structure and an almost paracomplex structure on a manifold is called an almost hyperparacomplex structure if and only if is valid.

Note that on an almost hyperparacomplex manifold the bundle automorphism is another almost paracomplex structure. In analogy to the case of almost hyper-Kähler manifolds where a symplectic manifold is endowed with a pair of two tame compatible almost complex structures satisfying , symplectic manifolds are called almost hyper-para-Kähler manifolds, if the almost (para)complex structures are compatible and is tame.

Definition 1.5. A symplectic manifold endowed with a pair of a compatible almost paracomplex structure and a tame compatible almost complex structure satisfying is called an almost hyper-para-Kähler manifold.

Existence of tame compatible almost hyper-paracomplex structures is characterized by the following theorem.

Theorem 1.6. On a (almost) symplectic manifold of dimension there exists a compatible almost paracomplex structure and a tame compatible almost complex structure such that if and only if the structure group of can be reduced from to .

In the symplectic case, the following corollary is an immediate consequence of Definition 1.5.

Corollary 1.7. A symplectic manifold of dimension can be made into an almost hyper-para-Kähler manifold if and only if the structure group of can be reduced from to .

Note that a reduction of the structure group of from to is always possible and corresponds to the choice of a tame compatible almost complex structure on . Theorem 1.6 is proved in Section 3 and can be viewed as a combination of [6, Theorem  1], where it is shown that the existence of a Lagrangian distribution on implies the existence of infinitely many different Lagrangian distributions, and [8, Corollary  2.1], where a one-to-one correspondence between Lagrangian distributions on and reductions of the structure group of from to is established. Especially, due to existence of compatible almost paracomplex structures on a (almost) symplectic manifold can alternatively be characterized as follows.

Corollary 1.8. On a (almost) symplectic manifold of dimension there exists a compatible almost paracomplex structure if and only if the structure group of can be reduced from to .

In the final section topological and metric properties of almost hyper-para-Kähler manifolds as well as some facts about integrability are discussed and applications are mentioned. Especially, it is shown in Proposition 4.3 and Corollary 4.4 that the Pontrjagin classes of the vector bundles over do not depend on the chosen compatible almost paracomplex structure but only on the symplectic structure. This result may initiate a deeper study of the question of which manifolds admit a symplectic structure with structure group reducible to .

In the appendix a paracomplex analogue of polarization is formulated.

2. Existence of Compatible Almost Paracomplex Structures

In this section the existence of a compatible almost paracomplex structure on a symplectic manifold is characterized. Recall that a bundle automorphism on a manifold is called an almost product structure if (often the trivial case is excluded). Obviously, merely has the eigenvalues , and if the corresponding eigenbundles satisfy , then is called an almost paracomplex structure. In this case, necessarily has even dimension. On an almost symplectic manifold every almost product structure that satisfies the compatibility condition is automatically an almost paracomplex structure.

To prove Theorem 1.3, some information about the frame bundle of is needed. If has dimension , then the fiber of the frame bundle at a point consists of the ordered bases (frames) of , and is a principal -bundle. The choice of an almost symplectic form on , that is, a nondegenerate (but not necessarily closed) 2-form , corresponds to a reduction of the structure group of from to by selecting only those frames with and for , that is, has the matrix representation in these so-called symplectic frames.

The following proof of Theorem 1.3 shows that the choice of a compatible almost paracomplex structure on corresponds to a reduction of the structure group of from to the paraunitary group where is used as symbol for the paracomplex numbers , , , and is considered as (almost) paracomplex structure on .

Proof of Theorem 1.3. As already mentioned in the introduction, compatible almost paracomplex structures correspond to almost bi-Lagrangian structures by assigning to the eigenbundles to the eigenvalues , and conversely to an almost bi-Lagrangian structure the unique almost product structure which has as eigenbundles to the eigenvalue .
For a given almost bi-Lagrangian structure on , select only those symplectic frames at for which is a base of and is a base of . If , respectively, , is another base of , respectively, , then there exist matrices with , respectively, , and from we conclude , that is, . Therefore, the frames and are related by the matrix . Thus, the selected frames define a reduction of the structure group of from to .
Conversely, if the structure group of is reduced from to , then two transversal distributions can be defined by assigning to a frame at the subspace and . Note that does not depend on the chosen frame because if is a different frame, then is related to by a matrix and is related to by a . Especially, and are valid. Further, is Lagrangian as for every , and therefore are transversal Lagrangian distributions.
Thus, almost bi-Lagrangian structures (and hence compatible almost paracomplex structures) are in one-to-one correspondence with reductions of the structure group of from to .

Although it seems that Theorem 1.3 completely characterizes the existence of compatible almost paracomplex structures on symplectic manifolds, there is a small gap in this characterization. In fact, the analytic conditions required from a symplectic manifold , that is, closedness of , may already imply that the structure group of can be reduced from to . However, this is not the case as there are many symplectic manifolds that do not admit a compatible almost paracomplex structure, see also [6, Section  2.5].

Example 2.1. The 2-sphere is an example of a symplectic manifold that does not admit any compatible almost paracomplex structure, see also [9, Corollary  2.5]. In fact, the 2-form on given in polar coordinates by the surface area is nondegenerate and closed, that is, a symplectic form on , but there does not exist a Lagrangian distribution on because else would split into two one-dimensional bundles, contradicting nontriviality of the bundle over .

3. Existence of Almost Hyper-Para-Kähler Structures

Given a (almost) symplectic manifold the question arises whether a compatible almost paracomplex structure and a tame compatible almost complex structure exist such that holds. Hereby, is called tame if is positive definite.

Recall that the choice of a tame almost complex structure on is always possible and corresponds to a reduction of the structure group of from to . In fact, if the structure group of has already been reduced from to , that is, if has been endowed with an almost symplectic form , then it can further be reduced to , and this reduction corresponds to the choice of a tame compatible almost complex structure on by selecting only those symplectic frames that additionally satisfy for , that is, has the matrix representation in these so-called unitary frames. Consequently, the positive definite Riemannian metric defined by has in unitary frames the matrix representation . For the convenience of the reader and later reference let us give a short proof of the existence of a compatible almost complex structure on an almost symplectic manifold (see also [1–4]).

Lemma 3.1. On every almost symplectic manifold there exists a tame compatible almost complex structure .

Proof. Choose an arbitrary positive definite Riemannian metric on and define a bundle automorphism by , which represents with respect to . Let be the unique polar decomposition of into a positive definite symmetric and an orthogonal with respect to . Then the -tensor defined by is positive definite symmetric and satisfies , Further, as is skew symmetric w.r.t due to the bundle automorphisms and obtained by polar decomposition commute, that is, also and (or ) commute. Thus, not only holds by orthogonality of , but symmetry of also implies that is, . Hence, is valid and compatibility of follows from

As already stated in the introduction, Theorem 1.6 can be considered as a combination of [8, Corollary  2.1] and [6, Theorem  1]. The following two lemmata are reformulations of these results.

Lemma 3.2. On an almost symplectic manifold of dimension there exists a Lagrangian distribution if and only if the structure group of can be reduced from to .

Proof. Due to Lemma 3.1 without restriction it can be assumed that the structure group of has already been reduced from to by choosing a tame compatible almost complex structure and the corresponding positive definite Riemannian metric on .
For a given Lagrangian distribution select only those unitary frames at for which is an orthonormal base of with respect to . If is another base of that is orthonormal w.r.t. , then there exists a real orthogonal matrix such that , and due to the corresponding frames are related by the matrix . Thus, the selected frames define a reduction of the structure group of from to the subgroup .
Conversely, if the structure group of is reduced from to , then by assigning to a frame at the subspace a Lagrangian distribution can be defined. Note that does not depend on the chosen frame because if is a different frame, then the equation is valid with an orthogonal matrix , and especially . Further, is Lagrangian as for every , and therefore is a Lagrangian distribution.

Remark 3.3. The proof of Lemma 3.2 even shows that there is a one-to-one correspondence of Lagrangian distributions and different reductions of the bundle of unitary frames on to a principal -bundle.

Lemma 3.4. Let be a (almost) symplectic manifold. If there exists a Lagrangian distribution on , then there exists a tame compatible almost complex structure and a compatible almost paracomplex structure having as eigenbundle to the eigenvalue 1 and satisfying .

Proof. By Lemma 3.1 there exists a tame compatible almost complex structure on . Denote by the corresponding positive definite Riemannian metric. Let , let be the orthogonal complement of w.r.t. , and let be the the almost product structure with as eigenbundles to the eigenvalues . Then due to for every and . Thus, not only is Lagrangian but also , as holds for . Hence, is a compatible almost paracomplex structure with as eigenbundle to the eigenvalue 1, and holds due to

Remark 3.5. The proof of Lemma 3.4 even shows that to a tame compatible almost complex structure and a Lagrangian distribution on there exists a unique compatible almost paracomplex structure such that is the eigenbundle of to the eigenvalue 1 and is the eigenbundle to −1. Further, this unique satisfies . Especially, every compatible almost paracomplex structure on an almost Kähler manifold can be changed to a unique compatible almost paracomplex structure having the same eigenbundle but satisfying additionally .

The two former lemmata directly imply Theorem 1.6 and Corollary 1.8.

Proof of Theorem 1.6 respectively Corollary 1.8. If there exists an almost hyper-para-Kähler structure (resp., a compatible almost paracomplex structure ) on , then the eigenbundle of to the eigenvalue 1 is Lagrangian and by Lemma 3.2 the structure group of can be reduced from to .
Conversely, if the structure group of can be reduced from (or ) to , then by Lemma 3.2 there exists a Lagrangian distribution on , and by Lemma 3.4 there exists a hyper-para-Kähler structure (resp., a compatible almost paracomplex structure ) on .

Theorem 1.6 shows that tame compatible almost hyperparacomplex structures on an almost symplectic manifold correspond to a reduction of the structure group of from to . In the corresponding frames is represented by the matrix as the condition implies due to with the eigenbundles of to the eigenvalues . Especially, the neutral metric defined by has the representation in these frames.

4. Properties of Almost Hyper-Para-Kähler Manifolds

4.1. Topological Properties

In Lemma 3.1 polarization w.r.t. an arbitrary positive definite Riemannian metric was used to associate with an almost symplectic form on a tame compatible almost complex structure . Especially, the space of all tame compatible almost complex structures is contractible. In fact, the space of all positive definite Riemannian metrics is contractible, and composition of the mappings (where the positive definite Riemannian metric is defined by ) and (where is obtained from polarization w.r.t. ) is the identity . As a consequence, the Chern classes associated with the complex vector bundle over do not depend on the choice of but only on . Therefore, the Chern classes can be used to formulate topological obstructions to the existence of a (almost) symplectic form on a manifold , but also to the existence of compatible almost paracomplex structures.

Proposition 4.1. A necessary condition for the existence of a compatible almost paracomplex structure on a symplectic manifold is that the odd Chern classes of vanish.

Proof. By Corollary 1.8 a compatible almost paracomplex structure exists on if and only if the structure group of can be reduced from to . In this case the Chern classes are not only real but vanish for odd because the Chern polynomial is odd for , as implies

Example 4.2. The symplectic sphere of Example 2.1 can be identified with . Thus, it admits a (integrable) compatible almost complex structure . Further, the Chern class does not vanish. This again shows that the symplectic sphere does not admit any compatible almost paracomplex structure .

While on a symplectic manifold the Chern classes of the complex vector bundle do not depend on the choice of the tame compatible almost complex structure , it is a priori not clear whether the Pontrjagin classes of the eigenbundles of to the eigenvalues depend on the choice of the compatible almost paracomplex structure . This is not the case as the following proposition and its corollary show that the Pontrjagin classes of do not depend on the choice of but only on the symplectic structure.

Proposition 4.3. On an almost hyper-para-Kähler manifold the odd Chern classes vanish and the even Chern classes are related to the Pontrjagin classes of the eigenbundles of to the eigenvalues by

Proof. Because satisfies , the eigenbundles of satisfy and . Thus is a bundle isomorphism and therefore holds. Moreover, the tangential bundle of can be identified via the bundle isomorphism with the complexification , and hence holds.

Corollary 4.4. On an almost para-Kähler manifold the Pontrjagin classes of the eigenbundles of are identical and do not depend on the choice of but only on the symplectic structure.

Proof. By Remark 3.5 for a chosen tame compatible almost complex structure on the compatible almost paracomplex structure can be changed to a compatible almost paracomplex structure with the same eigenbundle to 1 such that is an almost hyper-para-Kähler manifold. Thus, by Proposition 4.3 the Pontrjagin classes of are related to the Chern classes of by . Especially, depends only on the symplectic structure of . The same argument applied to shows .

Because polarization implies the independence of the Chern classes of of the chosen tame compatible complex structure , the question arises whether there is a paracomplex analogue of polarization. This question is discussed in the appendix.

4.2. Metric Properties

As already mentioned in the introduction, on a (almost) symplectic manifold endowed with a compatible almost paracomplex structure a neutral metric can be defined by , and satisfies . Recall that a nondegenerate symmetric -tensor on a manifold is called a pseudo-Riemannian metric and if has signature , then is said to be a neutral metric. If additionally is a compatible almost complex structure on and is the associated metric, then by definition of and the equation is valid. On an almost hyper-para-Kähler manifold moreover is symmetric w.r.t. and is symmetric w.r.t. .

Lemma 4.5. On an almost hyper-para-Kähler manifold with associated metrics to , respectively, to the compatible almost complex structure is symmetric with respect to and the compatible almost paracomplex structure is symmetric with respect to .

Proof. Symmetry of with respect to follows from and symmetry of with respect to holds due to

In applications it may be worthwhile to calculate the signature of the restriction of the neutral metric to a Lagrangian submanifold of as parts of with different signature of may be interpreted as different “phases” of a mechanical systems with state space modeled by and configuration space given by , and a change of signature of may indicate a kind of “phase transition.”

Example 4.6. If the almost bi-Lagrangian structure is integrable (see Section 4.3) and given by , , in local canonical coordinates with , then . Thus, if is a Lagrangian submanifold locally given by with the derivative of a function , then the pullback of to by is Therefore, is positive (resp., negative) definite if and only if is convex (resp., concave), and the signature of changes along those hypersurfaces where the second-order derivative of does not have full rank.

Associated with and are the corresponding Levi-Cita connections and , but there are also other useful connections (possibly with torsion) like the almost Kähler connection uniquely determined by , and or the almost para-Kähler connection uniquely determined by , and for , respectively, . For a study of connections on almost para-Kähler manifolds and their curvature see [5–7] and the references therein.

4.3. Integrability

A compatible almost paracomplex structure on a symplectic manifold is said to be integrable if the eigenbundles of to the eigenvalues are involutive. Symplectic manifolds endowed with such a structure were first studied by [10], see also [11, Chapter 10]. Recall that each is a Lagrangian distribution by compatibility of . An involutive Lagrangian distribution is also called a real polarization and induces by Frobenius’ theorem a foliation of into Lagrangian submanifolds. Therefore, if a compatible almost paracomplex structure on is integrable, then the eigenbundles induce two transversal Lagrangian foliations and is called a bi-Lagrangian manifold.

Note that with equal right could be called a para-Kähler manifold. In fact, is integrable on if and only if the Levi-Cita connection associated with the unique neutral metric satisfying does not only parallelize but also (and thus ), that is, , , and are valid, see [6, Theorem  6] or [11, Definition  10.2]. Another possibility to test the integrability of a compatible almost paracomplex structure on a symplectic manifold is to use the -tensor defined by for vector fields on , which is called the Nijenhuis tensor of . In fact, is integrable if and only if the Nijenhuis tensor of vanishes, that is, if and only if holds.

In the case that the structure group of the tangential bundle of a symplectic manifold (endowed with a tame compatible almost complex structure ) can be reduced from to (resp., from to ), the existence of a compatible almost paracomplex structure is guaranteed by Theorem 1.3, but by no means has to be integrable. For example, [12] shows that there exist symplectic manifolds that do not admit any polarization, regardless whether they are real, complex, or of mixed type. Further, there also are manifolds that admit an integrable complex polarization but not any real Lagrangian distribution, see Example 4.2.

For an almost hyper-para-Kähler manifold it may happen that neither the almost complex structure nor the almost paracomplex structure is integrable. Similarly, integrability of does not imply integrability of , and conversely from integrability of it does not follow that is integrable. However, if and are integrable, then also the almost paracomplex structure is integrable, and in this case is called a hyper-para-Kähler manifold. Such manifolds are, for example, studied in the context of supersymmetry, see [13].

Proposition 4.3 shows that in the chain of proper inclusions (where a manifold is called -symplectic if it is symplectic and its structure group can be reduced to ) topologically the second inclusion does not depend on the choice of . In the complex case the analogous chain of inclusions is widely used to study topological obstructions to the existence of symplectic forms on manifolds. The corresponding chain of inclusions for symplectic manifolds, whose structure group is reducible to , does not seem to be intensively studied in the literature. However, see [14], where topological obstructions to the existence of compatible almost paracomplex structures are given by means of the Euler class.

Another possible application of compatible almost paracomplex structures is geometric quantization, where symplectic manifolds with integral cohomology class are considered, because only in this case there exists a complex line bundle of . However, in geometric quantization not every section of such a line bundle is considered as a wave function of the quantized system, but only those sections that vanish along a polarization. Now an integrable compatible almost paracomplex structure just defines two transversal real polarizations, that is, intrinsically a dual real polarization is given, while there is only one real polarization in the ordinary setting. There are some efforts to generalize geometric quantization with complex polarizations, that is, Kähler quantization, to almost Kähler quantization, see [15, 16], and it may be worthwhile to study in analogy almost para-Kähler quantization.

5. Conclusion

In this paper the existence of compatible almost paracomplex structures (almost bi-Lagrangian structures) and almost hyper-para-Kähler structures on a symplectic manifold was characterized. Further, topological and metric properties of such manifolds were discussed. Especially, the result that the second inclusion in (where a manifold is called -symplectic if it is symplectic and its structure group can be reduced to ) is topologically independent of the choice of may initiate a deeper study of the topological obstructions to the existence of compatible almost paracomplex structures on symplectic manifolds.

Appendix

A Paracomplex Analogue of Polarization

In this appendix it is discussed whether there is a paracomplex analogue of polarization. Note that the polarization of a skew symmetric representing the (almost) symplectic form via w.r.t. a chosen positive definite Riemannian metric on is obtained from the (complex) eigenvalue decomposition with the eigenbundles of to the eigenvalues , , by and . It is simple to see that the complex linear automorphisms and of are in fact real, that is, they are induced by real linear automorphisms on denoted again by and allow a decomposition on .

A paracomplex analogue is the decomposition with and of , where denotes conjugation on and maps onto , respectively, onto . Note that has the real eigenvalues , that is, is neutral, while satisfies . However, and are merely real linear automorphisms on and not complex linear, that is, they are not induced by real linear automorphisms and on .