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Some remarks on cosymplectic 3-structures

In this note we briefly review some recent results of the authors on the topological and geometrical properties of 3-cosymplectic manifolds.


INTRODUCTION
Cosymplectic manifolds were introduced in the frame of quasi-Sasakian manifolds by Blair in [2] as the closest odd-dimensional counterpart of Kähler manifolds. Since then cosymplectic geometry has attracted the interest of many researchers also due to its role in mechanics and physics. Recently, a great deal of work on the topological properties of cosymplectic manifolds was done (see [6,7,9] among others). In particular, in [6] Chinea, de León and Marrero proved several important results for the Betti numbers of a compact cosymplectic manifold.
The notion of 3-cosymplectic manifold is the transposition of the notion of cosymplectic manifold to the setting of 3-structures. Namely, a 3-cosymplectic manifold is a smooth manifold endowed with three distinct cosymplectic structures related to each other by means of some relations formally similar to the quaternionic identities (see Section 2 for more details). This note contains a concise review of the main properties of 3-cosymplectic manifolds, recently obtained by the authors in [4,5]. Especially, we emphasize our results concerning Betti numbers of compact 3-cosymplectic manifolds. Finally, we present a method for constructing non-trivial examples of such compact manifolds.

3-COSYMPLECTIC GEOMETRY
An almost contact manifold is an odd-dimensional smooth manifold M endowed with a tensor field φ of endomorphisms on the tangent spaces, a vector field ξ and a 1-form η satisfying φ 2 = −I + η ⊗ ξ , where I denotes the identity mapping of T M. It is known that there exists a Riemannian metric g which is compatible with the structure, in the sense that for any X ,Y ∈ Γ(T M). When one fixes one compatible metric, the resulting geometric structure (φ , ξ , η, g) is called an almost contact metric structure on M. From (1) it follows that the bilinear form Φ := g(·, φ ·) is in fact a 2-form, called the fundamental 2-form of the almost contact metric manifold. An almost cosymplectic manifold is an almost contact metric manifold (M, φ , ξ , η, g) such that both the 1-form η and the fundamental 2-form Φ are closed. If in addition the structure is normal, that is, if the Nijenhuis tensor field of φ vanishes identically, (M, φ , ξ , η, g) is said to be a cosymplectic manifold. In terms of the covariant derivative of the structure tensor field φ , this condition is equivalent to ∇φ = 0. Now, we come to the main topic of the paper.
for any even permutation (α, β , γ) of the set {1, 2, 3}, is called an almost contact 3-structure on M. Then, the dimension of the manifold is necessarily of the form 4n + 3. This notion was introduced independently by Kuo ( [8]) and Udriste ( [12]). In particular, Kuo proved that one can always find a Riemannian metric g which is compatible with each almost contact structure. If we fix a compatible metric, we speak of almost contact metric 3-structure. Any smooth manifold endowed with an almost contact metric 3-structure carries two orthogonal distributions: the Reeb distribution V := span{ξ 1 , ξ 2 , ξ 3 } and the horizontal distribution H := ker(η 1 ) ∩ker(η 2 ) ∩ker(η 3 ). A remarkable case is when each structure is cosymplectic. In this case we say that M is a 3-cosymplectic manifold. In any 3-cosymplectic manifold the forms η α and Φ α are harmonic. Moreover, the tensors ξ α , η α , φ α , Φ α are all ∇-parallel. In particular, since the Reeb vector fields commute with each other, it follows that the Reeb distribution is integrable and defines a 3-dimensional foliation F 3 of M. As it was proven in [4], F 3 is a Riemannian and transversely hyper-Kähler foliation with totally geodesic leaves. Moreover, since dη α = 0, also the horizontal distribution H is integrable and hence defines a Riemannian, totally geodesic foliation complementary to F 3 .
Another important property of 3-cosymplectic manifolds that should be mentioned is that they are Ricci-flat ( [4]).

THE COHOMOLOGY OF A 3-COSYMPLECTIC MANIFOLD
Let M be a compact 3-cosymplectic manifold of dimension 4n + 3. We will denote by H * dR (M) the usual de Rham cohomology of M. By the Hodge-de Rham theory each vector space H k dR (M) can be identified with the vector space Ω k H (M) of harmonic kforms on M. Recall also that the space of basic k-forms (with respect to F 3 ) is defined by Ω k B (M) := ω ∈ Ω k (M) i ξ α ω = 0, i ξ α dω = 0, for each α = 1, 2, 3 .