Einstein–Weyl structures on almost cosymplectic manifolds

In this article, we study Einstein–Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic (κ,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\kappa ,\mu )$$\end{document}-manifold is Einstein or cosymplectic if it admits a closed Einstein–Weyl structure or two Einstein–Weyl structures. Next for a three dimensional compact almost α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-cosymplectic manifold admitting closed Einstein–Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-cosymplectic admitting two Einstein–Weyl structures is either Einstein or α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compact K-cosymplectic manifold with a closed Einstein–Weyl structure or two special Einstein–Weyl structures is cosymplectic.


Introduction
A Weyl structure W = (D, [g]) on a smooth manifold M is a torsion free affine connection D preserving a conformal structure [g]. Namely there exists a unique 1-form θ such that Dg = −2θ ⊗ g. The concept of Weyl structure goes back to the work of Weyl. He introduced the definition to unify gravitational fields and electromagnetic fields (see [27]). Later on Hitchin [16] in studying 3-dimensional minitwistor theory observed that the minitwistor theory can be generalized over a 3-manifold endowed with a Weyl structure satisfying a certain Ricci tensor condition, called an Einstein-Weyl structure. Refer also to [17]. A Weyl structure W = (D, [g]) is Einstein-Weyl if the symmetrized Ricci tensor is proportional to a metric g representing [g]: Further, if the unique 1-form θ is closed, then W is said to be a closed Einstein-Weyl structure. The Einstein-Weyl condition plays a key role in physics, the pure Einstein theory being too strong as a system model for various physical questions. On the contrary, Einstein-Weyl structures appear naturally as the background of the static Yang-Mills-Higgs theory.
On the other hand, almost contact geometry also provides a natural underlying structure to analyse many problems in physics. For example, Sasakian-Einstein and 3-Sasakian geometry have emerged in the context of dualities of certain supersymmetric conformal field theories [3], and general almost contact structures have also been used to study special magnetic fields [26]. Meanwhile, Matzeu proved that several classes of almost contact manifolds also naturally carry Einstein-Weyl structures [19]. Therefore, Einstein-Weyl structures have received a lot of attentions in the frame work of almost contact metric manifold (see [11,[20][21][22]).
Notice that an Einstein-Weyl structure is a generalization of Einstein metric in terms of affine connection. Recall the Goldberg conjecture [13] that a compact Einstein almost Kähler manifold is Kähler. The conjecture is true if the scalar curvature is non-negative [25]. As is well known, contact metric manifold can be considered as an odd-dimensional analogue to almost Kählar manifold. Boyer and Galicki [2] proposed an odd-dimensional Goldberg conjecture that a compact Einstein K -contact manifold is Sasakian and proved that it is true. As a generalization, Ghosh [12] and Gauduchon and Moroianu [10] simultaneously showed that a compact closed Einstein-Weyl K -contact manifold is also Sasakian using different method.
We also remark that another class of almost contact manifold, called almost cosymplectic manifold, was also paid many attentions (see a survey [5]). The concept was first defined by Goldberg and Yano [14] as an almost contact manifold whose 1-form η and fundamental 2-form ω are closed. An almost cosymplectic manifold is said to be cosymplectic if in addition the almost contact structure is normal (notice that here we adopt "cosymplectic" to represent "coKähler" in [5]). Concerning cosymplectic geometry, we mention the result that locally conformal cosymplectic manifolds admit a naturally defined conformally invariant Weyl structure [20]. Later on, Matzeu proved that every (2n + 1)-dimensional cosymplectic manifold of constant φ-sectional curvature c > 0 admits two Ricci-flat Weyl structures where the 1-forms associated to the metric g ∈ [g] are ±θ = ±λη, where λ = 2c 2n−1 . More recently, she generalized this result by proving that if a compact cosymplectic manifold (M, φ, ξ, η, g) admits a closed Einstein-Weyl structure D, then M is necessarily η-Einstein [21].
Recently, Bazzoni and Goertsches [2] defined a K -cosymplectic manifold, namely an almost cosymplectic manifold whose Reeb vector field is Killing. In [4], in fact it is proved that every compact Einstein K -cosymplectic manifold is necessarily cosymplectic. In addition, Endo [8] defined the notion of almost cosymplectic (κ, μ)-manifold, i.e., the curvature tensor of an almost cosymplectic manifold satisfies for any vector fields X , Y , where κ, μ are constant and h = 1 2 L ξ φ. As the extension of almost cosymplectic manifold, Kenmotsu [18] defined the almost Kenmotsu manifold, which is an almost contact manifold satisfying dη = 0 and dω = 2η ∧ ω. Based on this Kim and Pak [6] introduced the concept of almost α-cosymplectic manifold, i.e., an almost contact manifold satisfying dη = 0 and dω = 2αη ∧ ω for some real number α.
Motivated by the above background, in the present paper we first study an almost cosymplectic (κ, μ)-manifold and an almost α-cosymplectic manifold admitting Einstein-Weyl structures. Finally, we consider a compact K -cosymplectic manifold admitting a closed Einstein-Weyl structure. In order to prove our results, we need to recall some definitions and related conclusions on almost cosymplectic manifolds as well as Weyl structures, which are presented in Sects. 2 and 3, respectively. Starting from Sect. 4, we will state our results and give their proofs.
A Riemannian metric g on M is called compatible with the almost contact structure if for any X , Y ∈ X(M). An almost contact structure together with a compatible metric is called an almost contact metric structure and (M, φ, ξ, η, g) is called an almost contact metric manifold. An almost contact structure (φ, ξ, η) is said to be normal if the corresponding complex structure J on M × R is integrable. Denote by ω the fundamental 2-form on M defined by ω(X , Y ) := g(φ X , Y ) for all X , Y ∈ X(M). An almost α-cosymplectic manifold [6,24] is an almost contact metric manifold (M, φ, ξ, η, g) such that the fundamental form ω and 1-form η satisfy dη = 0 and Let M be an almost α-cosymplectic manifold, we recall that there is an operator h = 1 2 L ξ φ which is a self-dual operator. The Levi-Civita connection is given by (see [24]) for arbitrary vector fields X , Y , where N is the Nijenhuis torsion of M. Then by a simple calculation, we have Using (2.1), a straightforward calculation gives and ∇ ξ φ = 0. Denote by R and Ric the Riemannian curvature tensor and Ricci tensor, respectively. For an almost α-cosymplectic manifold (M 2n+1 , φ, ξ, η, g) the following equations were proved [24]: for any vector fields X , Y on M.

Weyl structures
Now suppose that (M, c) is a conformal manifold with conformal class c. A Weyl connection D in (M, c) is a torsion-free linear connection which preserves the conformal class c. For any metric g in c it carries a 1-form θ , called thee Lee form with respect to g, such that Dg = −2θ ⊗ g. It is related to the Levi-Civita connection ∇ by the following relation: ) is said to be closed, resp. exact if its Lee form is closed, resp. exact with respect to any metric in c. By (3.1), a straightforward computation implies the curvature tensor and Ricci tensor of the Weyl connection D are as follows: and δθ denotes the codifferential of θ with respect to g. Moreover, the following characterization of closed Weyl connection was proved. Proof By Proposition 3.1, we obtain from (3.2) that for all vector fields X , Y , Here we have used the relation the above relation is simplified as Thus by taking Y = ξ , we see that θ(X )η(B) = 0 for all X ∈ D, that means that either B ∈ D or B ∈ Rξ .  (3.4) where σ = δθ − (2n − 1)|θ | 2 + 2 . Furthermore, if M admits two Einstein-Weyl structures with θ and −θ , then the following two equations hold for arbitrary vector fields X , Y in M (Higa [15]): Here r denotes the scalar curvature of M.
Since the Weyl curvature tensor R D and the Weyl Ricci tensor Ric D of closed Einstein-Weyl structures defined on compact manifolds vanish identically (see [9]), from (3.2) and (3.3) we obtain Moreover, using (3.8) we obtain where λ = δθ − (2n − 1)|θ | 2 and Q is the Ricci operator defined by Ric(X , Y ) = g(Q X, Y ) for any vectors X , Y . Thus from (3.9), it is easy to yield Taking (3.9) into account, the formula (3.7) becomes
Using (2.2), it follows from (4.1) that the scalar curvature r = 2nκ and Qξ = 2nκξ. By (4.2), we find easily that κ ≤ 0 and κ = 0 if and only if M is a cosymplectic manifold, thus in the following we always suppose κ < 0.

Theorem 4.1 A (2n+1)-dimensional almost (κ, μ)-cosymplectic manifold admitting a closed Einstein-Weyl structure is an Einstein manifold or a cosymplectic manifold.
Proof By Proposition 3.2, we know that either B ∈ D or B ∈ Rξ . Next we consider these two cases respectively. We first assume that θ = f η for some function f . Since dθ = 0, by (2.3), Eq. (3.4) is rewritten as That is, Applying (4.1) in the above formula gives Replacing X by h X and using (4.2), we have Moreover, by taking an inner product of the foregoing relation with φ X and contracting X over the resulting equation, we get −σ trace(φh) + 2n(2n − 1) f κ = 0.
Thus the relation (2.7) shows that f = 0 and M is Einstein from (4.3).
In the following we consider the case where B ∈ D. In view of (4.1) and (2.3), the Eq.
Putting X = ξ implies 2nκ = σ , so we get h B = 0 by the above formula. Furthermore it yields from (4.2) that κφ 2 B = 0, i.e., B = 0. On the other hand, contracting X over (3.4) we have Because the scalar curvature r = 2nκ, we derive It comes to a contradiction with κ < 0, hence it is impossible. Summing up the above discussion, we complete the proof.
If M admits two Einstein-Weyl structures with θ and −θ , we immediately prove the following result.
If B ∈ Rξ , we may set B = f ξ for some smooth function f on M. The Eq. (3.6) becomes Furthermore, in terms of (3.5), we get Replacing X by h X and Y by φ X and contracting X over the resulting equation, we can prove that f = 0 or h = 0. Therefore M is an Einstein manifold or a cosymplectic manifold by (4.5).

Einstein-Weyl structures on almost˛-cosymplectic manifolds
In this section we study an almost α-cosymplectic manifold admitting Einstein-Weyl structures. First we consider the case of three dimension. (M 3 , , ξ, η, g) be a compact almost α-cosymplectic manifold. Suppose that M admits a closed Einstein-Weyl structure. Then M is Ricci-flat. Proof As before, by Proposition 3.2, B ∈ D or θ = f η where f = θ(ξ). Next we divide into two cases to discuss.

Theorem 5.1 Let
Case I. First we set θ = f η for some function f . By (2.3), we have 2). Using (5.1), the formula (3.9) becomes In terms of (5.1) and using (2.7), we compute λ = δθ − |θ On the other hand, it is well known that the curvature tensor of a 3-dimensional Riemannian manifold is given by Connecting this with (2.8) implies On the other and, differentiating (5.2) along Y and using (2.3), we conclude Thus the Eq. (3.10) becomes By comparing (5.6) and (5.4) with Z = B, we have Now let us put Y = ξ , then the above formula is simplified as Furthermore, by (5.5), the above formula becomes Take X = e 1 ∈ D such that he 1 = νe 1 with ν being a non-zero function, so the above relation becomes − ξ( f )∇ e 1 ξ + 2e 1 (λ)ξ − 2ξ(λ)e 1 = −4 f 2 αe 1 . Case II. When B ∈ D, from (3.12) and (5.3) with n = 1, we follow When X = B, the above relation becomes Contracting X over this equation, we also get |θ | 2 = 0. By (3.8), thus M is Ricci-flat.
Next we consider the case where M admits two Einstein-Weyl structures ±θ and obtain the following result.
Since the Ricci tensor is commuting, Taking X = B gives φ B = 0 or B = 0. Thus we know that B = f ξ , where f = θ(ξ). In terms of (3.5), we get As the proof of Theorem 4.2, replacing X by h X and Y by φ X , contracting X over the resulting equation and using (2.7), we obtain f = 0 or h = 0. Therefore we complete the proof by (5.8).

Einstein-Weyl structures on K -cosymplectic manifolds
Let M be a (2n + 1)-dimensional almost cosymplectic manifold defined in Sect. 2, namely the 1-form η and the fundamental form ω are closed and satisfy η ∧ ω n = 0 at every point of M.
Moreover, it follows from Theorem 3.29 in [5] that That shows that Qξ = 0. In the following we suppose that M admits a closed Einstein-Weyl structure, hence either B ∈ Rξ or B ∈ D by Proposition 3.2.
Summing up the above discussion, we actually proved the following conclusion. For a K -cosymplectic manifold with two Einstein-Weyl structures with ±θ , we also have the following conclusion. Theorem 6.3 Let (M, φ, ξ, η, g) be a (2n + 1)-dimensional, 2n + 1 ≥ 3, K-cosymplectic manifold. Suppose that M admits two Einstein-Weyl structures with ±θ . Then either M is Ricci-flat, or the scalar curvature is non-positive and invariant along the Reeb vector field ξ .
Since any compact Ricci-flat almost cosymplectic manifold is cosymplectic (see [5,Corollary 3.35]), we conclude immediately from the previous theorem the following corollary.

Corollary 6.4 A compact K -cosymplectic manifold admitting two Einstein-Weyl structures
with ±θ = ± f η for some function f is cosymplectic.