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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 535101, 13 pages
http://dx.doi.org/10.5402/2012/535101
Research Article

On Almost Hyper-Para-KΓ€hler Manifolds

Institute of Mathematics, University of Rostock, Ulmenstr. 69 (Haus 3), 18057 Rostock, Germany

Received 28 November 2011; Accepted 27 December 2011

Academic Editors: P.Β Aluffi and M.Β Rosenbaum

Copyright Β© 2012 Jochen Merker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper it is shown that a 2𝑛-dimensional almost symplectic manifold (𝑀,πœ”) can be endowed with an almost paracomplex structure 𝐾, 𝐾2=Id𝑇𝑀, and an almost complex structure 𝐽, 𝐽2=βˆ’Id𝑇𝑀, satisfying πœ”(𝐽𝑋,π½π‘Œ)=πœ”(𝑋,π‘Œ)=βˆ’πœ”(𝐾𝑋,πΎπ‘Œ) for 𝑋,π‘Œβˆˆπ‘‡π‘€, πœ”(𝑋,𝐽𝑋)>0 for 𝑋≠0 and 𝐾𝐽=βˆ’π½πΎ, if and only if the structure group of 𝑇𝑀 can be reduced from 𝑆𝑝(2𝑛) (or π‘ˆ(𝑛)) to 𝑂(𝑛). In the symplectic case such a manifold (𝑀,πœ”,𝐽,𝐾) is called an almost hyper-para-KΓ€hler manifold. Topological and metric properties of almost hyper-para-KΓ€hler manifolds as well as integrability of (𝐽,𝐾) are discussed. It is especially shown that the Pontrjagin classes of the eigenbundles 𝑃± of 𝐾 to the eigenvalues Β±1 depend only on the symplectic structure and not on the choice of 𝐾.

1. Introduction

While it is well known (see [1–4]) that every symplectic manifold (𝑀,πœ”) can be made into an almost KΓ€hler manifold by choosing an almost complex structure π½βˆΆπ‘‡π‘€β†’π‘‡π‘€ that satisfies 𝐽∘𝐽=βˆ’Id𝑇𝑀 and the compatibility condition πœ”(𝐽𝑋,π½π‘Œ)=πœ”(𝑋,π‘Œ) for every 𝑋,π‘Œβˆˆπ‘‡π‘€ (Moreover, for an almost KΓ€hler manifold 𝑔(𝑋,π‘Œ)∢=πœ”(𝑋,π½π‘Œ) is required to be a positive definite Riemannian metric on 𝑀, that is, 𝐽 is required to be tame. If 𝑔 is merely pseudo-Riemannian, then (𝑀,πœ”,𝐽) is called an almost pseudo-KΓ€hler 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 (𝑀,πœ”) be a (almost) symplectic manifold. A bundle automorphism πΎβˆΆπ‘‡π‘€β†’π‘‡π‘€ satisfying 𝐾∘𝐾=Id𝑇𝑀 and πœ”(𝐾𝑋,πΎπ‘Œ)=βˆ’πœ”(𝑋,π‘Œ) for every 𝑋,π‘Œβˆˆπ‘‡π‘€ is called a compatible almost paracomplex structure on (𝑀,πœ”).

An introduction to paracomplex geometry can be found in [5–7]. As illustrated by [6, Theorem  6, Proposition  7], compatible almost paracomplex structures 𝐾 on symplectic manifolds (𝑀,πœ”) correspond on the one hand to almost bi-Lagrangian structures (the eigenbundles π‘ƒΒ±βŠ‚π‘‡π‘€ of 𝐾 to the eigenvalues Β±1 are transversal Lagrangian distributions, i.e., 𝑃+βŠ•π‘ƒβˆ’=𝑇𝑀 and πœ”|𝑃±×𝑃±=0 hold) and on the other hand to almost para-KΓ€hler structures (by β„Ž(𝑋,π‘Œ)∢=πœ”(𝐾𝑋,π‘Œ) a neutral metric is defined, which satisfies β„Ž(𝐾𝑋,πΎπ‘Œ)=βˆ’β„Ž(𝑋,π‘Œ)).

Definition 1.2. A symplectic manifold (𝑀,πœ”) endowed with a compatible almost paracomplex structure 𝐾 is called an almost para-KΓ€hler 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 (𝑀,πœ”) of dimension 2𝑛 there exists a compatible almost paracomplex structure 𝐾 if and only if the structure group of 𝑇𝑀 can be reduced from Sp(2𝑛) to the paraunitary group π‘ˆ(𝑛,𝔸).

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 (𝑀,πœ”) that admit a compatible almost paracomplex structure 𝐾 and a tame compatible almost complex structure 𝐽 such that 𝐾∘𝐽=βˆ’π½βˆ˜πΎ is valid.

Definition 1.4. A pair (𝐽,𝐾) of an almost complex structure π½βˆΆπ‘‡π‘€β†’π‘‡π‘€ and an almost paracomplex structure πΎβˆΆπ‘‡π‘€β†’π‘‡π‘€ on a manifold 𝑀 is called an almost hyperparacomplex structure if and only if 𝐾∘𝐽=βˆ’π½βˆ˜πΎ is valid.

Note that on an almost hyperparacomplex manifold (𝑀,𝐽,𝐾) the bundle automorphism 𝐽∘𝐾 is another almost paracomplex structure. In analogy to the case of almost hyper-KΓ€hler manifolds where a symplectic manifold (𝑀,πœ”) is endowed with a pair (𝐼,𝐽) of two tame compatible almost complex structures satisfying 𝐽∘𝐼=βˆ’πΌβˆ˜π½, symplectic manifolds are called almost hyper-para-KΓ€hler manifolds, if the almost (para)complex structures 𝐽,𝐾 are compatible and 𝐽 is tame.

Definition 1.5. A symplectic manifold (𝑀,πœ”) endowed with a pair (𝐽,𝐾) of a compatible almost paracomplex structure 𝐾 and a tame compatible almost complex structure 𝐽 satisfying 𝐾∘𝐽=βˆ’π½βˆ˜πΎ is called an almost hyper-para-KΓ€hler manifold.

Existence of tame compatible almost hyper-paracomplex structures (𝐽,𝐾) is characterized by the following theorem.

Theorem 1.6. On a (almost) symplectic manifold (𝑀,πœ”) of dimension 2𝑛 there exists a compatible almost paracomplex structure 𝐾 and a tame compatible almost complex structure 𝐽 such that 𝐾∘𝐽=βˆ’π½βˆ˜πΎ if and only if the structure group of 𝑇𝑀 can be reduced from Sp(2𝑛)(orπ‘ˆ(𝑛)) to 𝑂(𝑛).

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

Corollary 1.7. A symplectic manifold (𝑀,πœ”) of dimension 2𝑛 can be made into an almost hyper-para-KΓ€hler manifold if and only if the structure group of 𝑇𝑀 can be reduced from 𝑆𝑝(2𝑛)(orπ‘ˆ(𝑛)) to 𝑂(𝑛).

Note that a reduction of the structure group of 𝑇𝑀 from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛) is always possible and corresponds to the choice of a tame compatible almost complex structure 𝐽 on (𝑀,πœ”). 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 (𝑀,πœ”) implies the existence of infinitely many different Lagrangian distributions, and [8, Corollary  2.1], where a one-to-one correspondence between Lagrangian distributions on (𝑀,πœ”,𝐽) and reductions of the structure group of 𝑇𝑀 from π‘ˆ(𝑛) to 𝑂(𝑛) is established. Especially, due to π‘ˆ(𝑛)βˆ©π‘ˆ(𝑛,𝔸)=𝑂(𝑛) existence of compatible almost paracomplex structures on a (almost) symplectic manifold (𝑀,πœ”) can alternatively be characterized as follows.

Corollary 1.8. On a (almost) symplectic manifold (𝑀,πœ”) of dimension 2𝑛 there exists a compatible almost paracomplex structure 𝐾 if and only if the structure group of 𝑇𝑀 can be reduced from Sp(2𝑛)(orπ‘ˆ(𝑛)) to 𝑂(𝑛).

In the final section topological and metric properties of almost hyper-para-KΓ€hler 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 𝑃± over 𝑀 do not depend on the chosen compatible almost paracomplex structure 𝐾 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 𝑂(𝑛).

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 𝐾 on a symplectic manifold (𝑀,πœ”) is characterized. Recall that a bundle automorphism πΎβˆΆπ‘‡π‘€β†’π‘‡π‘€ on a manifold 𝑀 is called an almost product structure if 𝐾∘𝐾=Id𝑇𝑀 (often the trivial case 𝐾=Β±Id𝑇𝑀 is excluded). Obviously, 𝐾 merely has the eigenvalues Β±1, and if the corresponding eigenbundles 𝑃± satisfy dim(𝑃+)=dim(π‘ƒβˆ’), then 𝐾 is called an almost paracomplex structure. In this case, necessarily 𝑀 has even dimension. On an almost symplectic manifold (𝑀,πœ”) every almost product structure 𝐾 that satisfies the compatibility condition πœ”(𝐾𝑋,πΎπ‘Œ)=βˆ’πœ”(𝑋,π‘Œ) is automatically an almost paracomplex structure.

To prove Theorem 1.3, some information about the frame bundle 𝐺𝑙(𝑇𝑀) of 𝑇𝑀 is needed. If 𝑀 has dimension 2𝑛, then the fiber of the frame bundle 𝐺𝑙(𝑇𝑀) at a point π‘šβˆˆπ‘€ consists of the ordered bases (frames) (𝑋1,…,𝑋2𝑛) of π‘‡π‘šπ‘€, and 𝐺𝑙(𝑇𝑀) is a principal 𝐺𝑙(2𝑛)-bundle. The choice of an almost symplectic form πœ” on 𝑀, that is, a nondegenerate (but not necessarily closed) 2-form πœ”, corresponds to a reduction of the structure group of 𝑇𝑀 from 𝐺𝑙(2𝑛) to 𝑆𝑝(2𝑛) by selecting only those frames (𝑋1,…,𝑋𝑛,π‘Œ1,…,π‘Œπ‘›) with πœ”(𝑋𝑖,𝑋𝑗)=0=πœ”(π‘Œπ‘–,π‘Œπ‘—) and πœ”(𝑋𝑖,π‘Œπ‘—)=𝛿𝑖𝑗 for 𝑖,𝑗=1,…,𝑛, that is, πœ” has the matrix representation ξ€·0βˆ’IdId0ξ€Έ in these so-called symplectic frames.

The following proof of Theorem 1.3 shows that the choice of a compatible almost paracomplex structure 𝐾 on (𝑀,πœ”) corresponds to a reduction of the structure group of 𝑇𝑀 from Sp(2𝑛) to the paraunitary group ⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽ0ξ€·π΄π‘ˆ(𝑛,𝔸)∢=𝐴0βˆ—ξ€Έβˆ’1⎞⎟⎟⎠⎫βŽͺ⎬βŽͺβŽ­βˆ£π΄βˆˆπΊπ‘™(𝑛,ℝ),(2.1) where 𝔸=ℝ[π‘˜] is used as symbol for the paracomplex numbers π‘Ž+π‘˜π‘, π‘˜2=1, π‘Ž,π‘βˆˆβ„, and ξ€·Id00βˆ’Idξ€Έ is considered as (almost) paracomplex structure on ℝ2𝑛.

Proof of Theorem 1.3. As already mentioned in the introduction, compatible almost paracomplex structures 𝐾 correspond to almost bi-Lagrangian structures 𝑃± by assigning to 𝐾 the eigenbundles 𝑃± to the eigenvalues Β±1, and conversely to an almost bi-Lagrangian structure 𝑃± the unique almost product structure 𝐾 which has 𝑃± as eigenbundles to the eigenvalue Β±1.
For a given almost bi-Lagrangian structure 𝑃± on (𝑀,πœ”), select only those symplectic frames (𝑋1,…,𝑋𝑛,π‘Œ1,…,π‘Œπ‘›) at π‘šβˆˆπ‘€ for which (𝑋1,…,𝑋𝑛) is a base of 𝑃+ and π‘Œ1,…,π‘Œπ‘› is a base of π‘ƒβˆ’. If (𝑋1𝑋,…,𝑛), respectively, (ξ‚π‘Œ1ξ‚π‘Œ,…,𝑛), is another base of 𝑃+, respectively, π‘ƒβˆ’, then there exist matrices 𝐴,π΅βˆˆπΊπ‘™(𝑛) with 𝑋𝑖=βˆ‘π‘—π‘Žπ‘–π‘—π‘‹π‘—, respectively, ξ‚π‘Œπ‘–=βˆ‘π‘—π‘π‘–π‘—π‘Œπ‘—, and from πœ”(𝑋𝑖,π‘Œπ‘—)=𝛿𝑖𝑗𝑋=πœ”(𝑖,ξ‚π‘Œπ‘—) we conclude π΅βˆ—π΄=Id, that is, 𝐡=(π΄βˆ—)βˆ’1. Therefore, the frames (𝑋1,…,𝑋𝑛,π‘Œ1,…,π‘Œπ‘›) and (𝑋1𝑋,…,𝑛,ξ‚π‘Œ1ξ‚π‘Œ,…,𝑛) are related by the matrix 𝐴00(π΄βˆ—)βˆ’1. Thus, the selected frames define a reduction of the structure group of 𝑇𝑀 from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛,𝔸).
Conversely, if the structure group of 𝑇𝑀 is reduced from 𝑆𝑝(𝑛) to π‘ˆ(𝑛,𝔸), then two transversal distributions 𝑃± can be defined by assigning to a frame (𝑋1,…,𝑋𝑛,π‘Œ1,…,π‘Œπ‘›) at π‘šβˆˆπ‘€ the subspace 𝑃+(π‘š)∢=span(𝑋1,…,𝑋𝑛) and π‘ƒβˆ’(π‘š)∢=span(π‘Œ1,…,π‘Œπ‘›). Note that 𝑃± does not depend on the chosen frame because if (𝑋1𝑋,…,𝑛,ξ‚π‘Œ1ξ‚π‘Œ,…,𝑛) is a different frame, then (𝑋1𝑋,…,𝑛) is related to (𝑋1,…,𝑋𝑛) by a matrix π΄βˆˆπΊπ‘™(𝑛) and (ξ‚π‘Œ1ξ‚π‘Œ,…,𝑛) is related to (π‘Œ1,…,π‘Œπ‘›) by a (π΄βˆ—)βˆ’1. Especially, 𝑋span(1𝑋,…,𝑛)=span(𝑋1,…,𝑋𝑛) and ξ‚π‘Œspan(1ξ‚π‘Œ,…,𝑛)=span(π‘Œ1,…,π‘Œπ‘›) are valid. Further, 𝑃± is Lagrangian as πœ”(𝑋𝑖,𝑋𝑗)=0=πœ”(π‘Œπ‘–,π‘Œπ‘—) for every 𝑖,𝑗=1,…,𝑛, and therefore 𝑃± 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 𝑇𝑀 from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛,𝔸).

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 (𝑀,πœ”), that is, closedness of πœ”, may already imply that the structure group of 𝑇𝑀 can be reduced from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛,𝔸). 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 𝑆2 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 πœ” on 𝑆2 given in polar coordinates (πœ™,πœƒ)∈(βˆ’πœ‹,πœ‹)Γ—(βˆ’πœ‹/2,πœ‹/2) by the surface area πœ”=cos(πœƒ)π‘‘πœ™βˆ§π‘‘πœƒ(2.2) is nondegenerate and closed, that is, a symplectic form on 𝑆2, but there does not exist a Lagrangian distribution on 𝑆2 because else 𝑇𝑆2 would split into two one-dimensional bundles, contradicting nontriviality of the bundle 𝑇𝑆2 over 𝑆2.

3. Existence of Almost Hyper-Para-KΓ€hler Structures

Given a (almost) symplectic manifold (𝑀,πœ”) the question arises whether a compatible almost paracomplex structure 𝐾 and a tame compatible almost complex structure 𝐽 exist such that K∘𝐽=βˆ’π½βˆ˜πΎ holds. Hereby, 𝐽 is called tame if 𝑔(𝑋,π‘Œ)∢=πœ”(𝑋,π½π‘Œ) is positive definite.

Recall that the choice of a tame almost complex structure 𝐽 on 𝑀 is always possible and corresponds to a reduction of the structure group of 𝑇𝑀 from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛). In fact, if the structure group of 𝑇𝑀 has already been reduced from 𝐺𝑙(2𝑛) to 𝑆𝑝(2𝑛), that is, if 𝑀 has been endowed with an almost symplectic form πœ”, then it can further be reduced to π‘ˆ(𝑛), and this reduction corresponds to the choice of a tame compatible almost complex structure 𝐽 on (𝑀,πœ”) by selecting only those symplectic frames (𝑋1,…,𝑋𝑛,π‘Œ1,…,π‘Œπ‘›) that additionally satisfy π‘Œπ‘–=𝐽𝑋𝑖 for 𝑖=1,…,𝑛, that is, 𝐽 has the matrix representation ξ€·0βˆ’IdId0ξ€Έ in these so-called unitary frames. Consequently, the positive definite Riemannian metric 𝑔 defined by 𝑔(𝑋,π‘Œ)∢=πœ”(𝑋,π½π‘Œ) has in unitary frames the matrix representation ξ€·Id00Idξ€Έ. 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 (𝑀,πœ”) there exists a tame compatible almost complex structure 𝐽.

Proof. Choose an arbitrary positive definite Riemannian metric βŸ¨β‹…,β‹…βŸ© on 𝑀 and define a bundle automorphism π΄βˆΆπ‘‡π‘€β†’π‘‡π‘€ by πœ”(𝑋,π‘Œ)=βŸ¨π΄π‘‹,π‘ŒβŸ©, which represents πœ” with respect to βŸ¨β‹…,β‹…βŸ©. Let 𝐴=𝐺∘𝐽 be the unique polar decomposition of 𝐴 into a positive definite symmetric 𝐺 and an orthogonal 𝐽 with respect to βŸ¨β‹…,β‹…βŸ©. Then the (0,2)-tensor 𝑔 defined by 𝑔(𝑋,π‘Œ)∢=βŸ¨πΊπ‘‹,π‘ŒβŸ© is positive definite symmetric and satisfies 𝑔(𝐽𝑋,π‘Œ)=πœ”(𝑋,π‘Œ), Further, as 𝐴 is skew symmetric w.r.t βŸ¨β‹…,β‹…βŸ© due to βŸ¨π΄π‘‹,π‘ŒβŸ©=πœ”(𝑋,π‘Œ)=βˆ’πœ”(π‘Œ,𝑋)=βˆ’βŸ¨π΄π‘Œ,π‘‹βŸ©=βˆ’βŸ¨π‘‹,π΄π‘ŒβŸ©,(3.1) the bundle automorphisms 𝐺 and 𝐽 obtained by polar decomposition commute, that is, also 𝐴 and 𝐺 (or πΊβˆ’1) commute. Thus, not only π½βˆ—=π½βˆ’1 holds by orthogonality of 𝐽, but symmetry of πΊβˆ’1 also implies ξ«βŸ¨π‘‹,π½π‘ŒβŸ©=𝑋,πΊβˆ’1ξ¬ξ«π΄π‘Œ=βˆ’π΄πΊβˆ’1𝐺𝑋,π‘Œ=βˆ’βˆ’1𝐴𝑋,π‘Œ=βˆ’βŸ¨π½π‘‹,π‘ŒβŸ©,(3.2) that is, π½βˆ—=βˆ’π½. Hence, 𝐽2=βˆ’Id𝑇𝑀 is valid and compatibility of 𝐽 follows from ξ€·π½πœ”(𝐽𝑋,π½π‘Œ)=𝑔2𝑋,π½π‘Œ=βˆ’π‘”(𝑋,π½π‘Œ)=βˆ’π‘”(π½π‘Œ,𝑋)=βˆ’πœ”(π‘Œ,𝑋)=πœ”(𝑋,π‘Œ).(3.3)

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 (𝑀,πœ”) of dimension 2𝑛 there exists a Lagrangian distribution π‘ƒβŠ‚π‘‡π‘€ if and only if the structure group of 𝑇𝑀 can be reduced from Sp(2𝑛)(orπ‘ˆ(𝑛)) to 𝑂(𝑛).

Proof. Due to Lemma 3.1 without restriction it can be assumed that the structure group of 𝑇𝑀 has already been reduced from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛) by choosing a tame compatible almost complex structure 𝐽 and the corresponding positive definite Riemannian metric 𝑔 on (𝑀,πœ”).
For a given Lagrangian distribution 𝑃 select only those unitary frames (𝑋1,…,𝑋𝑛,𝐽𝑋1,…,𝐽𝑋𝑛) at π‘šβˆˆπ‘€ for which (𝑋1,…,𝑋𝑛) is an orthonormal base of π‘ƒπ‘šβŠ‚π‘‡π‘šπ‘€ with respect to 𝑔. If (𝑋1𝑋,…,𝑛) is another base of π‘ƒπ‘š that is orthonormal w.r.t. 𝑔, then there exists a real orthogonal matrix π΄βˆˆπ‘‚(𝑛) such that 𝑋𝑖=βˆ‘π‘—π‘Žπ‘–π‘—π‘‹π‘—, and due to 𝐽𝑋𝑖=βˆ‘π‘—π‘Žπ‘–π‘—π½π‘‹π‘— the corresponding frames are related by the matrix 𝐴00𝐴. Thus, the selected frames define a reduction of the structure group of 𝑇𝑀 from ⎧βŽͺ⎨βŽͺβŽ©βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ βŽ«βŽͺ⎬βŽͺβŽ­π‘ˆ(𝑛)∢=π΄βˆ’π΅π΅π΄βˆ£π΄+π‘–π΅βˆˆπ‘ˆ(𝑛,β„‚).(3.4) to the subgroup {𝐴00𝐴|π΄βˆˆπ‘‚(𝑛)}.
Conversely, if the structure group of 𝑇𝑀 is reduced from π‘ˆ(𝑛) to 𝑂(𝑛), then by assigning to a frame (𝑋1,…,𝑋𝑛,𝐽𝑋1,…,𝐽𝑋𝑛) at π‘šβˆˆπ‘€ the subspace π‘ƒπ‘šβˆΆ=span(𝑋1,…,𝑋𝑛) a Lagrangian distribution 𝑃 can be defined. Note that π‘ƒπ‘š does not depend on the chosen frame because if (𝑋1𝑋,…,𝑛,𝐽𝑋1,…,𝐽𝑋𝑛) is a different frame, then the equation 𝑋𝑖=βˆ‘π‘—π‘Žπ‘–π‘—π‘‹π‘— is valid with an orthogonal matrix π΄βˆˆπ‘‚(𝑛), and especially 𝑋span(1𝑋,…,𝑛)=span(𝑋1,…,𝑋𝑛). Further, π‘ƒπ‘š is Lagrangian as πœ”(𝑋𝑖,𝑋𝑗)=0 for every 𝑖,𝑗=1,…,𝑛, and therefore 𝑃 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 π‘ˆ(𝑇𝑀) of unitary frames on (𝑀,πœ”,𝐽) to a principal 𝑂(𝑛)-bundle.

Lemma 3.4. Let (𝑀,πœ”) be a (almost) symplectic manifold. If there exists a Lagrangian distribution 𝑃 on (𝑀,πœ”), then there exists a tame compatible almost complex structure 𝐽 and a compatible almost paracomplex structure 𝐾 having 𝑃 as eigenbundle to the eigenvalue 1 and satisfying 𝐾∘𝐽=βˆ’π½βˆ˜πΎ.

Proof. By Lemma 3.1 there exists a tame compatible almost complex structure 𝐽 on (𝑀,πœ”). Denote by 𝑔 the corresponding positive definite Riemannian metric. Let 𝑃+∢=𝑃, let π‘ƒβˆ’βˆΆ=π‘ƒβŸ‚ be the orthogonal complement of 𝑃 w.r.t. 𝑔, and let 𝐾 be the the almost product structure with 𝑃± as eigenbundles to the eigenvalues Β±1. Then 𝐽𝑃+=π‘ƒβˆ’ due to 𝑔(𝐽𝑋,π‘Œ)=πœ”(𝑋,π‘Œ)=0 for every 𝑋,π‘Œβˆˆπ‘ƒ=𝑃+ and dim(𝑃+)=dim(π‘ƒβˆ’). Thus, not only 𝑃+=𝑃 is Lagrangian but also π‘ƒβˆ’, as πœ”(𝐽𝑋,π½π‘Œ)=πœ”(𝑋,π‘Œ)=0 holds for 𝑋,π‘Œβˆˆπ‘ƒ=𝑃+. Hence, 𝐾 is a compatible almost paracomplex structure with 𝑃 as eigenbundle to the eigenvalue 1, and 𝐾∘𝐽=βˆ’π½βˆ˜πΎ holds due to 𝐽𝐾𝑃+=𝐽𝑃+=π‘ƒβˆ’=βˆ’πΎπ‘ƒβˆ’=βˆ’πΎπ½π‘ƒ+,π½πΎπ‘ƒβˆ’=βˆ’π½π‘ƒβˆ’=βˆ’π½π½π‘ƒ+=𝑃+=𝐾𝑃+=βˆ’πΎπ½π½π‘ƒ+=βˆ’πΎπ½π‘ƒβˆ’.(3.5)

Remark 3.5. The proof of Lemma 3.4 even shows that to a tame compatible almost complex structure 𝐽 and a Lagrangian distribution 𝑃 on (𝑀,πœ”) there exists a unique compatible almost paracomplex structure 𝐾 such that 𝑃 is the eigenbundle of 𝐾 to the eigenvalue 1 and 𝐽𝑃 is the eigenbundle to βˆ’1. Further, this unique 𝐾 satisfies 𝐾∘𝐽=βˆ’π½βˆ˜πΎ. Especially, every compatible almost paracomplex structure 𝐾 on an almost KΓ€hler manifold (𝑀,πœ”,𝐽) can be changed to a unique compatible almost paracomplex structure 𝐾 having the same eigenbundle 𝑃+ but satisfying additionally 𝐾∘𝐽=βˆ’π½βˆ˜πΎ.

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-KΓ€hler structure (𝐽,𝐾) (resp., a compatible almost paracomplex structure 𝐾) on (𝑀,πœ”), then the eigenbundle 𝑃+ of 𝐾 to the eigenvalue 1 is Lagrangian and by Lemma 3.2 the structure group of 𝑇𝑀 can be reduced from 𝑆𝑝(2𝑛)(orπ‘ˆ(𝑛)) to 𝑂(𝑛).
Conversely, if the structure group of 𝑇𝑀 can be reduced from 𝑆𝑝(2𝑛) (or π‘ˆ(𝑛)) to 𝑂(𝑛), then by Lemma 3.2 there exists a Lagrangian distribution 𝑃 on 𝑀, and by Lemma 3.4 there exists a hyper-para-KΓ€hler structure (𝐽,𝐾) (resp., a compatible almost paracomplex structure 𝐾) on (𝑀,πœ”).

Theorem 1.6 shows that tame compatible almost hyperparacomplex structures (𝐽,𝐾) on an almost symplectic manifold (𝑀,πœ”) correspond to a reduction of the structure group of 𝑇𝑀 from 𝑆𝑝(2𝑛) to 𝑂(𝑛). In the corresponding frames 𝐾 is represented by the matrix ξ€·Id00βˆ’Idξ€Έ as the condition 𝐾∘𝐽=βˆ’π½βˆ˜πΎ implies 𝐽𝑃+=π‘ƒβˆ’ due to 𝐾(𝐽𝑃+)=βˆ’π½πΎπ‘ƒ+=βˆ’π½π‘ƒ+ with the eigenbundles 𝑃± of 𝐾 to the eigenvalues Β±1. Especially, the neutral metric β„Ž defined by β„Ž(𝑋,π‘Œ)∢=πœ”(𝐾𝑋,π‘Œ) has the representation ξ€·0IdId0ξ€Έ in these frames.

4. Properties of Almost Hyper-Para-KΓ€hler Manifolds

4.1. Topological Properties

In Lemma 3.1 polarization w.r.t. an arbitrary positive definite Riemannian metric βŸ¨β‹…,β‹…βŸ© was used to associate with an almost symplectic form πœ” on 𝑀 a tame compatible almost complex structure 𝐽. Especially, the space of all tame compatible almost complex structures 𝐽 is contractible. In fact, the space of all positive definite Riemannian metrics is contractible, and composition of the mappings 𝐽↦𝑔 (where the positive definite Riemannian metric 𝑔 is defined by 𝑔(𝑋,π‘Œ)∢=πœ”(𝑋,π½π‘Œ)) and βŸ¨β‹…,β‹…βŸ©β†¦π½ (where 𝐽 is obtained from polarization w.r.t. βŸ¨β‹…,β‹…βŸ©) is the identity 𝐽↦𝐽. As a consequence, the Chern classes associated with the complex vector bundle (𝑇𝑀,𝐽) over 𝑀 do not depend on the choice of 𝐽 but only on (𝑀,πœ”). Therefore, the Chern classes can be used to formulate topological obstructions to the existence of a (almost) symplectic form on a manifold 𝑀, 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 𝐾 on a symplectic manifold (𝑀,πœ”) is that the odd Chern classes of (𝑀,πœ”) vanish.

Proof. By Corollary 1.8 a compatible almost paracomplex structure 𝐾 exists on (𝑀,πœ”) if and only if the structure group of 𝑇𝑀 can be reduced from π‘ˆ(𝑛) to 𝑂(𝑛). In this case the Chern classes are not only real but vanish for odd π‘˜ because the Chern polynomial is odd for π΄βˆˆπ”¬(𝑛), as 𝐴𝑇=βˆ’π΄ implies ξ‚€1detπœ†Idβˆ’π΄ξ‚ξ‚€12πœ‹π‘–=detπœ†Id+𝐴2πœ‹π‘–π‘‡ξ‚ξ‚€1=detπœ†Id+𝐴.2πœ‹π‘–(4.1)

Example 4.2. The symplectic sphere 𝑆2 of Example 2.1 can be identified with β„‚βˆͺ{∞}. Thus, it admits a (integrable) compatible almost complex structure 𝐽. Further, the Chern class 𝑐1(𝑇𝑆2,𝐽)=βˆ’2 does not vanish. This again shows that the symplectic sphere 𝑆2 does not admit any compatible almost paracomplex structure 𝐾.

While on a symplectic manifold (𝑀,πœ”) the Chern classes of the complex vector bundle (𝑇𝑀,𝐽) do not depend on the choice of the tame compatible almost complex structure 𝐽, it is a priori not clear whether the Pontrjagin classes of the eigenbundles 𝑃± of 𝐾 to the eigenvalues Β±1 depend on the choice of the compatible almost paracomplex structure 𝐾. This is not the case as the following proposition and its corollary show that the Pontrjagin classes of 𝑃± do not depend on the choice of 𝐾 but only on the symplectic structure.

Proposition 4.3. On an almost hyper-para-KΓ€hler manifold (𝑀,πœ”,𝐽,𝐾) the odd Chern classes vanish and the even Chern classes 𝑐2π‘˜(𝑇𝑀,𝐽) are related to the Pontrjagin classes π‘π‘˜(𝑃±) of the eigenbundles 𝑃± of 𝐾 to the eigenvalues Β±1 by (βˆ’1)π‘˜π‘2π‘˜(𝑇𝑀)=π‘π‘˜ξ€·π‘ƒ+ξ€Έ=π‘π‘˜ξ€·π‘ƒβˆ’ξ€Έ.(4.2)

Proof. Because 𝐾 satisfies 𝐾∘𝐽=βˆ’π½βˆ˜πΎ, the eigenbundles 𝑃± of 𝐾 satisfy 𝐽𝑃+=π‘ƒβˆ’ and π½π‘ƒβˆ’=𝑃+. Thus π½βˆΆπ‘ƒ+β†’π‘ƒβˆ’ is a bundle isomorphism and therefore π‘π‘˜(𝑃+)=π‘π‘˜(π‘ƒβˆ’) holds. Moreover, the tangential bundle 𝑇𝑀 of 𝑀 can be identified via the bundle isomorphism (𝑃+)β„‚βˆ‹π‘‹+π‘–π‘ŒβŸΌπ‘‹+π½π‘Œβˆˆπ‘‡π‘€(4.3) with the complexification (𝑃+)β„‚, and hence (βˆ’1)π‘˜π‘2π‘˜(𝑇𝑀)=π‘π‘˜(𝑃+) holds.

Corollary 4.4. On an almost para-KΓ€hler manifold (𝑀,πœ”,𝐾) the Pontrjagin classes of the eigenbundles 𝑃± of 𝐾 are identical and do not depend on the choice of 𝐾 but only on the symplectic structure.

Proof. By Remark 3.5 for a chosen tame compatible almost complex structure 𝐽 on (𝑀,πœ”) the compatible almost paracomplex structure 𝐾 can be changed to a compatible almost paracomplex structure 𝐾 with the same eigenbundle 𝑃+ to 1 such that (𝑀,πœ”,𝐽,𝐾) is an almost hyper-para-KΓ€hler manifold. Thus, by Proposition 4.3 the Pontrjagin classes of 𝑃+ are related to the Chern classes of (𝑇𝑀,𝐽) by (βˆ’1)π‘˜π‘2π‘˜(𝑇𝑀)=π‘π‘˜(𝑃+). Especially, π‘π‘˜(𝑃+) depends only on the symplectic structure of (𝑀,πœ”). The same argument applied to βˆ’πΎ shows (βˆ’1)π‘˜π‘2π‘˜(𝑇𝑀)=π‘π‘˜(π‘ƒβˆ’).

Because polarization implies the independence of the Chern classes of (𝑇𝑀,𝐽) of the chosen tame compatible complex structure 𝐽, 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 (𝑀,πœ”) endowed with a compatible almost paracomplex structure 𝐾 a neutral metric β„Ž can be defined by β„Ž(𝑋,π‘Œ)∢=πœ”(𝐾𝑋,π‘Œ), and β„Ž satisfies β„Ž(𝐾𝑋,πΎπ‘Œ)=βˆ’β„Ž(𝑋,π‘Œ). Recall that a nondegenerate symmetric (0,2)-tensor β„Ž on a manifold 𝑀 is called a pseudo-Riemannian metric and if β„Ž has signature (𝑛,𝑛), then β„Ž is said to be a neutral metric. If additionally 𝐽 is a compatible almost complex structure on (𝑀,πœ”) and 𝑔(𝑋,π‘Œ)∢=πœ”(𝑋,π½π‘Œ) is the associated metric, then by definition of 𝑔 and β„Ž the equation𝑔(𝐾𝑋,π‘Œ)=πœ”(𝐾𝑋,π½π‘Œ)=β„Ž(𝑋,π½π‘Œ)(4.4) is valid. On an almost hyper-para-KΓ€hler manifold (𝑀,πœ”,𝐽,𝐾) moreover 𝐽 is symmetric w.r.t. β„Ž and 𝐾 is symmetric w.r.t. 𝑔.

Lemma 4.5. On an almost hyper-para-KΓ€hler manifold (𝑀,πœ”,𝐽,𝐾) with associated metrics 𝑔 to 𝐽, respectively, β„Ž to 𝐾 the compatible almost complex structure 𝐽 is symmetric with respect to β„Ž and the compatible almost paracomplex structure 𝐾 is symmetric with respect to 𝑔.

Proof. Symmetry of 𝐽 with respect to β„Ž follows from β„Ž(𝑋,π½π‘Œ)=πœ”(𝐾𝑋,π½π‘Œ)=βˆ’πœ”(𝐽𝐾𝑋,π‘Œ)=πœ”(𝐾𝐽𝑋,π‘Œ)=βˆ’πœ”(𝐽𝑋,πΎπ‘Œ)=πœ”(πΎπ‘Œ,𝐽𝑋)=β„Ž(π‘Œ,𝐽𝑋)=β„Ž(𝐽𝑋,π‘Œ),(4.5) and symmetry of 𝐾 with respect to β„Ž holds due to 𝑔(𝐾𝑋,π‘Œ)=πœ”(𝐾𝑋,π½π‘Œ)=βˆ’πœ”(𝐽𝐾𝑋,π‘Œ)=πœ”(𝐾𝐽𝑋,π‘Œ)=βˆ’πœ”(𝐽𝑋,πΎπ‘Œ)=πœ”(πΎπ‘Œ,𝐽𝑋)=𝑔(πΎπ‘Œ,𝑋)=𝑔(𝑋,πΎπ‘Œ).(4.6)

In applications it may be worthwhile to calculate the signature of the restriction of the neutral metric β„Ž to a Lagrangian submanifold 𝐿 of (𝑀,πœ”) as parts of 𝐿 with different signature of β„Ž may be interpreted as different β€œphases” of a mechanical systems with state space modeled by (𝑀,πœ”) and configuration space given by πΏβŠ‚π‘€, and a change of signature of β„Ž may indicate a kind of β€œphase transition.”

Example 4.6. If the almost bi-Lagrangian structure 𝑃± is integrable (see Section 4.3) and given by 𝑃+=span(πœ•/πœ•π‘žπ‘˜), π‘ƒβˆ’=span(πœ•/πœ•π‘π‘˜), in local canonical coordinates (π‘ž,𝑝) with βˆ‘πœ”=π‘˜π‘‘π‘žπ‘˜βˆ§π‘‘π‘π‘˜, then βˆ‘β„Ž=π‘˜π‘‘π‘žπ‘˜βŠ—symπ‘‘π‘π‘˜. Thus, if 𝐿 is a Lagrangian submanifold locally given by 𝑝=𝑏(π‘ž) with the derivative 𝑏 of a function π‘„βˆ‹π‘žβ†¦πœ™(π‘ž)βˆˆβ„, then the pullback of β„Ž to 𝑄 by π‘‘πœ™βˆΆπ‘žβ†¦(π‘ž,𝑏(π‘ž)) is (π‘‘πœ™)βˆ—ξ“β„Ž=π‘˜π‘—πœ•2πœ™πœ•π‘žπ‘˜πœ•π‘žπ‘—π‘‘π‘žπ‘˜βŠ—symπ‘‘π‘žπ‘—.(4.7) Therefore, β„Ž is positive (resp., negative) definite if and only if πœ™ is convex (resp., concave), and the signature of β„Ž changes along those hypersurfaces where the second-order derivative of πœ™ does not have full rank.

Associated with β„Ž and 𝑔 are the corresponding Levi-Cita connections βˆ‡β„Ž and βˆ‡π‘”, but there are also other useful connections βˆ‡ (possibly with torsion) like the almost KΓ€hler connection uniquely determined by βˆ‡πœ”=0, βˆ‡π½=0 and Torβˆ‡(𝑋,π‘Œ)=(1/4)[𝐽,𝐽] or the almost para-KΓ€hler connection uniquely determined by βˆ‡πœ”=0, βˆ‡πΎ=0 and Torβˆ‡(𝑋,π‘Œ)=0 for 𝑋,π‘Œβˆˆπ‘ƒ+, respectively, 𝑋,π‘Œβˆˆπ‘ƒβˆ’. For a study of connections on almost para-KΓ€hler manifolds and their curvature see [5–7] and the references therein.

4.3. Integrability

A compatible almost paracomplex structure 𝐾 on a symplectic manifold (𝑀,πœ”) is said to be integrable if the eigenbundles 𝑃± of 𝐾 to the eigenvalues Β±1 are involutive. Symplectic manifolds endowed with such a structure were first studied by [10], see also [11, Chapter 10]. Recall that each 𝑃± is a Lagrangian distribution by compatibility of 𝐾. An involutive Lagrangian distribution is also called a real polarization and induces by Frobenius’ theorem a foliation of (𝑀,πœ”) into Lagrangian submanifolds. Therefore, if a compatible almost paracomplex structure 𝐾 on (𝑀,πœ”) is integrable, then the eigenbundles 𝑃± induce two transversal Lagrangian foliations and (𝑀,πœ”,𝐾) is called a bi-Lagrangian manifold.

Note that with equal right (𝑀,πœ”,𝐾) could be called a para-KΓ€hler manifold. In fact, 𝐾 is integrable on (𝑀,πœ”) if and only if the Levi-Cita connection βˆ‡β„Ž associated with the unique neutral metric β„Ž satisfying β„Ž(𝐾𝑋,π‘Œ)=πœ”(𝑋,π‘Œ) does not only parallelize β„Ž but also 𝐾 (and thus πœ”), that is, βˆ‡β„Žβ„Ž=0, βˆ‡β„ŽπΎ=0, and βˆ‡β„Žπœ”=0 are valid, see [6, Theorem  6] or [11, Definition  10.2]. Another possibility to test the integrability of a compatible almost paracomplex structure 𝐾 on a symplectic manifold (𝑀,πœ”) is to use the (1,2)-tensor defined by[][]𝐾,𝐾(𝑋,π‘Œ)=𝐾𝑋,πΎπ‘Œ+𝐾2[][][]𝑋,π‘Œβˆ’πΎπΎπ‘‹,π‘Œβˆ’πΎπ‘‹,πΎπ‘Œ(4.8) for vector fields 𝑋,π‘Œ on 𝑀, which is called the Nijenhuis tensor of 𝐾. In fact, 𝐾 is integrable if and only if the Nijenhuis tensor of 𝐾 vanishes, that is, if and only if [𝐾,𝐾](𝑋,π‘Œ)=[𝐾𝑋,πΎπ‘Œ]+[𝑋,π‘Œ]βˆ’πΎ[𝐾𝑋,π‘Œ]βˆ’πΎ[𝑋,πΎπ‘Œ]=0 holds.

In the case that the structure group of the tangential bundle 𝑇𝑀 of a symplectic manifold (𝑀,πœ”) (endowed with a tame compatible almost complex structure 𝐽) can be reduced from 𝑆𝑝(2𝑛) to π‘ˆ(𝑛,𝔸) (resp., from π‘ˆ(𝑛) to 𝑂(𝑛)), the existence of a compatible almost paracomplex structure 𝐾 is guaranteed by Theorem 1.3, but by no means 𝐾 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-KΓ€hler manifold (𝑀,πœ”,𝐽,𝐾) it may happen that neither the almost complex structure 𝐽 nor the almost paracomplex structure 𝐾 is integrable. Similarly, integrability of 𝐽 does not imply integrability of 𝐾, and conversely from integrability of 𝐾 it does not follow that 𝐽 is integrable. However, if 𝐽 and 𝐾 are integrable, then also the almost paracomplex structure 𝐽∘𝐾 is integrable, and in this case (𝑀,πœ”,𝐽,𝐾) is called a hyper-para-KΓ€hler manifold. Such manifolds are, for example, studied in the context of supersymmetry, see [13].

Proposition 4.3 shows that in the chain of proper inclusionshyper-para-KΓ€hlerβŠŠπ‘‚(𝑛)βˆ’symplectic⊊almosthyper-paracomplex,(4.9) (where a manifold is called 𝑂(𝑛)-symplectic if it is symplectic and its structure group can be reduced to 𝑂(𝑛)) topologically the second inclusion does not depend on the choice of (𝐽,𝐾). In the complex case the analogous chain of inclusionsK̈ahler⊊symplectic⊊almostcomplex(4.10) 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 𝑂(𝑛), 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 (𝑀,πœ”) with integral cohomology class [πœ”]∈𝐻2(𝑀,β„€) are considered, because only in this case there exists a complex line bundle of 𝑀. 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 𝐾 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, KΓ€hler quantization, to almost KΓ€hler quantization, see [15, 16], and it may be worthwhile to study in analogy almost para-KΓ€hler quantization.

5. Conclusion

In this paper the existence of compatible almost paracomplex structures 𝐾 (almost bi-Lagrangian structures) and almost hyper-para-KΓ€hler structures (𝐽,𝐾) on a symplectic manifold (𝑀,πœ”) was characterized. Further, topological and metric properties of such manifolds were discussed. Especially, the result that the second inclusion in̈hyper-para-KahlerβŠŠπ‘‚(𝑛)βˆ’symplectic⊊almosthyper-paracomplex,(5.1) (where a manifold is called 𝑂(𝑛)-symplectic if it is symplectic and its structure group can be reduced to 𝑂(𝑛)) is topologically independent of the choice of (𝐽,𝐾) 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 𝐴=𝐺∘𝐽 of a skew symmetric 𝐴 representing the (almost) symplectic form πœ” via πœ”(𝑋,π‘Œ)=βŸ¨π΄π‘‹,π‘ŒβŸ© w.r.t. a chosen positive definite Riemannian metric βŸ¨β‹…,β‹…βŸ© on 𝑀 is obtained from the (complex) eigenvalue decomposition βˆ‘π΄=π‘˜(π‘–πœ†π‘˜Idπ‘‰π‘˜)βŠ•(βˆ’π‘–πœ†π‘˜Idπ‘‰π‘˜) with the eigenbundles π‘‰π‘˜βŠ‚π‘‡π‘€β„‚ of 𝐴 to the eigenvalues π‘–πœ†π‘˜, πœ†π‘˜>0, by βˆ‘πΊβˆΆ=π‘˜πœ†π‘˜Idπ‘‰π‘˜+π‘‰π‘˜ and βˆ‘π½βˆΆ=π‘˜(𝑖Idπ‘‰π‘˜)βŠ•(βˆ’π‘–Idπ‘‰π‘˜). It is simple to see that the complex linear automorphisms 𝐺 and 𝐽 of 𝑇𝑀ℂ are in fact real, that is, they are induced by real linear automorphisms on 𝑇𝑀 denoted again by 𝐺,𝐽 and allow a decomposition 𝐴=𝐺∘𝐽 on 𝑇𝑀.

A paracomplex analogue is the decomposition 𝐾𝐴=𝐻∘ with ξ‚βˆ‘π»βˆΆ=π‘˜(βˆ’π‘–πœ†π‘˜β‹…|π‘‰π‘˜)βŠ•(π‘–πœ†π‘˜β‹…|π‘‰π‘˜) and ξ‚βˆ‘πΎβˆΆ=π‘˜β‹…|π‘‰π‘˜βŠ•β‹…|π‘‰π‘˜ of 𝑇𝑀ℂ, where β‹… denotes conjugation on 𝑇𝑀ℂ and maps π‘‰π‘˜ onto π‘‰π‘˜, respectively, π‘‰π‘˜ onto π‘‰π‘˜. Note that 𝐻 has the real eigenvalues Β±πœ†π‘˜, that is, 𝐻 is neutral, while 𝐾 satisfies ξ‚ξ‚πΎβˆ˜πΎ=Id𝑇𝑀ℂ. However, 𝐻 and 𝐾 are merely real linear automorphisms on 𝑇𝑀ℂ and not complex linear, that is, they are not induced by real linear automorphisms 𝐻 and 𝐾 on 𝑇𝑀.

Nevertheless, with a Lagrangian distribution 𝑃 on (𝑀,πœ”,𝐽) a real neutral 𝐻, respectively, a real 𝐾 on 𝑇𝑀 can be associated such that the complexification of 𝐻, respectively, 𝐾 coincides with 𝐻, respectively, 𝐾 on 𝑃+𝑖𝐽𝑃. In fact, let 𝑃+∢=𝑃 and π‘ƒβˆ’βˆΆ=𝐽𝑃+, then the real dimension of (𝑃++π‘–π‘ƒβˆ’)βˆ©π‘‰π‘˜ is the same as the complex dimension of π‘‰π‘˜ because if 𝑣 is an eigenvector of 𝐽 to 𝑖 and 𝑣=𝑣1+𝑣2∈(𝑃++π‘–π‘ƒβˆ’)βŠ•(π‘ƒβˆ’+𝑖𝑃+)=𝑇𝑀ℂ, then due to 𝐽𝑃+=π‘ƒβˆ’, π½π‘ƒβˆ’=𝑃+ the decomposition 𝐽𝑣2+𝐽𝑣1=𝐽𝑣=𝑖𝑣=𝑖𝑣2+𝑖𝑣1βˆˆξ€·π‘ƒ++π‘–π‘ƒβˆ’ξ€ΈβŠ•ξ€·π‘ƒβˆ’+𝑖𝑃+ξ€Έ(A.1) implies 𝐽𝑣1=𝑖𝑣1, 𝐽𝑣2=𝑖𝑣2. Especially, 𝑣1∈(𝑃++π‘–π‘ƒβˆ’) is an eigenvector of 𝐽 to 𝑖, and as the eigenspace of 𝐽 to 𝑖 is the sum of the π‘‰π‘˜, the real subspace (𝑃++π‘–π‘ƒβˆ’)βˆ©π‘‰π‘˜ of 𝑇𝑀ℂ is nonempty and dimℝ((𝑃++π‘–π‘ƒβˆ’)βˆ©π‘‰π‘˜)=dimβ„‚(π‘‰π‘˜). Thus, associated with 𝑃 there are unique real linear automorphisms 𝐻 and 𝐾 on 𝑇𝑀 such that the complexification of 𝐻 coincides with 𝐻 on (𝑃++π‘–π‘ƒβˆ’)βˆ©π‘‰π‘˜, and the complexification of 𝐾 coincides on (𝑃++π‘–π‘ƒβˆ’)βˆ©π‘‰π‘˜ with 𝐾. As a consequence, the decomposition 𝐴=𝐻∘𝐾 holds, 𝐾 is orthogonal w.r.t. βŸ¨β‹…,β‹…βŸ© and satisfies 𝐾2=Id𝑇𝑀, and a neutral metric β„Ž satisfying β„Ž(𝐾𝑋,π‘Œ)=πœ”(𝑋,π‘Œ) can be defined by β„Ž(𝑋,π‘Œ)∢=βŸ¨π»π‘‹,π‘ŒβŸ©. However, note that the decomposition 𝐴=𝐻∘𝐾 into a nondegenerate neutral symmetric 𝐻 and an orthogonal 𝐾 w.r.t. βŸ¨β‹…,β‹…βŸ© was merely made unique by the choice of 𝑃, in general there are many such decompositions.

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