From Dual Connections to Almost Contact Structures

Abstract

A dualistic structure on a smooth Riemaniann manifold M is a triple $(M,g,\nabla )$ with g a Riemaniann metric and ∇ an affine connection generally assumed to be torsionless. From g and ∇, dual connection ${\nabla }^{*}$ can be defined. In this work, we give conditions on the basis of this notion for a manifold to admit an almost contact structure and some related structures: almost contact metric, contact, contact metric, cosymplectic, and co-Kähler in the three-dimensional case.

1. Introduction

Finding characteristic obstructions to the existence of structures is a particularly important question arising in mathematics. In this work, we give conditions for an orientable manifold to admit an almost contact structure (almost cosymplectic structure), almost contact metric structure, cosymplectic (symplectic mapping torus) structure, using the notion of dual connections that was introduced in the context of information geometry [1,2]. We also use duality to describe the relationships between the structures on an even dimensional manifold and the corresponding ones on an odd dimensional manifold. Going back to the original paper in [3], given a differentiable manifold M of odd dimensions $2n+1$, an almost contact structure is defined by a triple $(\varphi ,\xi ,\eta )$ with $\varphi \in T_1^1\left(M\right),\xi \in T\left(M\right),\mathrm{and} \eta \in T^{*}\left(M\right)$, such that:

$$\begin{array}{cc}\hfill & \eta \left(\xi \right)=1\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & {\varphi }^2=-Id+\eta \otimes \xi \hfill \end{array}$$

(2)

A manifold with an almost contact structure can also be defined equivalently as one whose structure group is reducible to $U\left(n\right)\times 1$. The most basic example of such a manifold is given by ${\mathbb{R}}^{2n+1}$, with:

$$\varphi =\left(\begin{array}{ccc}0& -Id& 0\\ Id& 0& 0\\ 0& 0& 0\end{array}\right)$$

and $\xi =\partial x_{2n+1},\eta =d{x}_{2n+1}$. It is also a prototype for most of the considered structures in this paper. From the above equation, one can easily deduce the next proposition:

Proposition 1.

$$\begin{array}{cc}\hfill & rank\varphi =2n\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \varphi \xi =0\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \eta \varphi =0\hfill \end{array}$$

(5)

The proof is elementary and relies only on basic linear algebra. In fact, if $p\in M$ and $X\in T_{p}M\ne 0$ is such that $\eta \left(X\right)=0$, then ${\varphi }^2\left(X\right)=-X$, and $X\notin \mathit{ker}\varphi$. From (2), $\eta \left({\varphi }^2\left(X\right)\right)=-\eta \left(X\right)+\eta \left(\xi \right)\eta \left(X\right)=0$ and using the previous remark implies $\eta \left(\varphi \right(X\left)\right)=0$. Since ${\varphi }^2\left(\xi \right)=0$, it emerges at once that $\varphi \left(\xi \right)=0$. A Riemannian metric g on M is adapted to the almost contact structure if it satisfies for all vector fields $X,Y$:

$$g\left(\varphi \left(X\right),\varphi \left(Y\right)\right)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right)$$

(6)

Using (4) and the above definition:

$$\eta \left(X\right)=g(X,\xi )$$

(7)

In turn,

$$g\left(\varphi \left(X\right),\xi \right)=0$$

(8)

Endomorphism $\varphi$ is skew-symetric with respect to an adapted metric:

$$g\left(X,\varphi \left(Y\right)\right)=-g\left(\varphi \left(X\right),Y\right)$$

(9)

and thus gives rise to a canonical 2-form $\mathsf{\Omega }$:

$$\mathsf{\Omega }(X,Y)=g\left(X,\varphi \left(Y\right)\right)$$

(10)

When $\mathsf{\Omega }=d\eta$, the almost contact structure is a contact metric structure. Lastly, if $\xi$ is Killing, then the structure is K-contact. Any 3-dimensionnal oriented Riemannian manifold $(M,g)$ admits an almost contact structure with g as adapted metric [4]. For the classification of almost contact metric structures, see also [5].

In 1969, M. Gromov [6] proved that any almost contact open manifold M admits a contact structure. A similar result was proved in the closed oriented 3-dimensional manifold case by Lutz [7] and Martinet [8], the 5-dimensional case was proved by J. Etnyre [9] and the work of R. Casals, D.M. Pancholi, and F. Presas [10]. In [11], Matthew Strom Borman, Yakov Eliashberg, and Emmy Murphy proved the same result in any dimension.

As an almost contact manifold is a purely topological condition, in dimension 5, it boils down to the vanishing of the third integral Stiefel–Whitney class. In [12], this property was used to classify simply connected almost contact manifolds. An almost cosymplectic manifold (cf. [13,14]) of dimension $2n+1$ is a triple $(M,\omega ,\eta )$, such that the 2-form $\omega$ and the 1-form $\eta$ satisfy ${\omega }^n\wedge \eta \ne 0$. In the language of G-structures, an almost cosymplectic structure can be defined equivalently as an $1\times Sp(n,R)$-structure.

From [14], every almost cosymplectic structure on M induces an isomorphism of $C^{\infty }\left(M\right)$-modules:

$${♭}_{(\eta ,\omega )}:\left\{\begin{array}{cc}& \mathcal{X}\left(M\right)\to {\mathsf{\Omega }}^1\left(M\right)\hfill \\ & X\mapsto i_X\omega +\eta \left(X\right)\eta \hfill \end{array}\right.$$

for every vector field $X\in \mathcal{X}\left(M\right)$. A vector bundle isomorphism (denoted with the same symbol) ${♭}_{(\eta ,\omega )}:TM\to T^{\star }M$ is also induced, and the vector field

$$\xi ={♭}_{(\eta ,\omega )}^{-1}\left(\eta \right)$$

on M is called the Reeb vector field of the almost cosymplectic manifold $(M,\eta ,\omega )$. It is characterized by the following conditions:

$$i_{\xi }\omega =0\mathrm{and}{i}_{\xi }\eta =1$$

Conversely, we have the following characterization of almost cosymplectic manifolds that follows from [14] (Proposition 2).

Proposition 2.

Let M be a manifold endowed with a 1-form η and a 2-form ω such that the map ${♭}_{(\eta ,\omega )}:TM\to T^{\star }M$ is an isomorphism. Assume also that there exists a vector field ξ such that $i_{\xi }\omega =0$ and $\eta \left(\xi \right)=1$. Then, M has an odd dimension, and $(M,\eta ,\omega )$ is an almost cosymplectic manifold with Reeb vector field ξ.

By a cosymplectic manifold, we mean a (2n + 1)-manifold M together with a closed 1-form $\eta$ and a closed 2-form $\omega$ such that $\eta \wedge {\omega }^n$ is a volume form. This was P. Libermann’s definition in 1959 [15], under the name of cosymplectic manifold. The pair $(\eta ,\omega )$ is called a cosymplectic structure on M. In [16], Blair gives an equivalent definition of cosymplectic manifolds, which is more often referred to in the literature, see [17,18,19,20,21,22,23]. From Blair [16] an almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on an odd-dimensional smooth manifold M is cosymplectic if $d\eta =d\mathsf{\Omega }=0$, where $\mathsf{\Omega }$ is the fundamental 2-form.Cosymplectic manifolds can be thought of as odd-dimensional counterparts of symplectic manifolds. ${\mathbb{R}}^{2n+1}$ equipped with the canonical 2-form $\mathsf{\Omega }={\sum }_{i=1}^{n}d{x}_{2i-1}\wedge d{x}_{2i}$ is a cosymplectic manifold with $\eta =d{x}_{2n+1}.$ In fact, on any cosymplectic manifold $(M,\eta ,\omega )$ the so-called horizontal distribution $\mathit{ker}\eta$ is integrable to a symplectic foliation of codimension 1. On the other hand, one has the following result due to Manuel de Léon and Martin Saralegi:

Theorem 1

([24]). Let M be a manifold and ω, η two differential forms on M with degrees 2 and 1 respectively. Consider, on $Y=M\times \mathbb{R}$, the differential 2-form $\mathsf{\Omega }={\pi }^{\star }\omega +{\pi }^{\star }\eta \wedge dt$, where $t\in \mathbb{R}$ and $\pi :Y\to M$ is then canonical projection. Then: $(M,\eta ,\omega )$ is a cosymplectic manifold if and only if $(Y,\mathsf{\Omega })$ is a symplectic manifold.

The Darboux theorem admits an equivalent in cosymplectic structure:

Proposition 3.

Any cosymplectic manifold $(M,\eta ,\omega )$ of dimension $2n+1$ admits around any point local coordinates $(t,q^{\alpha },p_{\alpha }),\alpha =1,...,n$, such that:

$$\omega =\sum _{\alpha =1}^{n}d{q}^{\alpha }\wedge d{p}_{\alpha },\eta =dt,\xi =\frac{\partial }{\partial t}$$

In 2008, Hongjun Li’s main theorem in [17] asserted that cosymplectic manifolds are equivalent to symplectic mapping tori. The main idea of Li’s proof came from a theorem of Tischler [25] stating that a compact manifold admits a nonvanishing closed 1-form if and only if the manifold fibres over a circle. This assertion is also equivalent to a compact manifold being a mapping torus if and only if it admits a nonvanishing closed 1-form. The co-one-dimensional co-orientable foliations defined by the kernel of nowhere-zero closed one-form are termed unimodular foliations. From [26], the existence of a unimodular foliation is equivalent to a vanishing modular class.

Theorem 2

([26]). The first obstruction (modular) class $c_{\mathcal{F}}$ vanishes; identically, the defining one-form η of the foliation $\mathcal{F}$ is closed.

In Section 2, we briefly summarize results of the gauge equation for dual connections. There is no claim of originality here, only a reformulation of the previous results obtained by Pr. M. Boyom [27]. In Section 3, we discuss the relationship between skew-symmetric solutions of maximal rank of the gauge equation and the existence of almost cosympletic and almost contact metric and cosymplectic structures. Lastly, the case of three-dimensional co-Kähler manifolds is examined in Section 3.6.1.

2. Gauge Transformations and Parallelism

In this section, $(M,g)$ is a smooth Riemannian manifold. As usual, for a vector bundle $E\to M$, $\mathsf{\Gamma }\left(E\right)$ denotes the space of smooth sections. For an affine connection ∇, its dual connection ${\nabla }^{*}$ is defined by the relation ${\left({\nabla }_Y^{*}X\right)}^{♭}={\nabla }_Y{X}^{♭}$ or equivalently as satisfying, for any vector fields $X,Y,Z$ in $TM$, the equation:

$$Z\left(g\left(X,Y\right)\right)=g\left({\nabla }_{Z}X,Y\right)+g\left(X,{\nabla }_Z^{*}Y\right)$$

(11)

Equation (11) proves by symmetry that ${\nabla }^{**}=\nabla$.

On 1-forms, the duality relation becomes ${\left({\nabla }_X\omega \right)}^{♯}={\nabla }_X^{*}{\omega }^{♯}$, for any 1-form $\omega$ and vector field X.

Levi-Civita connection ${\nabla }^{lc}$ is self-dual and for any dual pair $\nabla ,{\nabla }^{*}$ without torsion

$$\nabla ={\nabla }^{lc}-\frac{1}{2}D,{\nabla }^{*}={\nabla }^{lc}+\frac{1}{2}D$$

(12)

with $D={\nabla }^{*}-\nabla$ a $(2,1)$-tensor (since the difference of two affine connections is a tensor). In the sequel, $\nabla ,{\nabla }^{*}$ are assumed to be torsionless unless the converse is explicitly stated.

The relationship between the curvatures of dual connections is given by:

$$g(R^{\nabla }(X,Y)V,W)=-g(V,R^{{\nabla }^{*}}(X,Y)W).$$

A connection ∇ in $TM$ is metric if $\nabla g=0$, i.e.,

$$X.\left(g(Y,Z)\right)=g({\nabla }_{X}Y,Z)+g(Y,{\nabla }_{X}Z),\mathrm{for}\mathrm{any}\mathrm{vector}\mathrm{fields}X,Y,Z.$$

Metric connections are not unique, but differ only by torsion. As a consequence of $\nabla g=0$, one has

$$g(R^{\nabla }(X,Y)V,W)=-g(V,R^{\nabla }(X,Y)W).$$

Proposition 4.

D is symmetric in its first two arguments. Furthermore, for any vector fields $X,Y,Z$:

$$g\left(D(Z,X),Y\right)=g\left(X,D(Z,Y)\right)$$

Proof. 

The first claim is a consequence of $\nabla ,{\nabla }^{*}$ being torsionless:

$$D(X,Y)={\nabla }_X^{*}Y-{\nabla }_{X}Y={\nabla }_Y^{*}X+[X,Y]-{\nabla }_{Y}X-[X,Y]=D(Y,X)$$

For the second, the starting point is Equation (11) rewritten with the expressions from Equation (12):

$$Z\left(g\left(X,Y\right)\right)=g\left({\nabla }_Z^{lc}X,Y\right)+g\left(X,{\nabla }_Z^{lc}Y\right)-\frac{1}{2}g\left(D(Z,X),Y\right)+\frac{1}{2}g\left(X,D(Z,Y)\right)$$

Using the defining property of the Levi-Civita connection:

$$g\left(D(Z,X),Y\right)-g\left(X,D(Z,Y)\right)=0$$

and the claim follows. □

Proposition 5.

Tensor

$$T:(X,Y,Z)\mapsto g\left(D(Z,X),Y\right)$$

is totally symmetric. Furthermore, $T(X,Y,Z)=\left({\nabla }_{Z}g\right)\left(X,Y\right)$

Proof. 

The symmetry comes from the one of D. For the second part of the proposition:

$$\begin{array}{cc}\hfill \left({\nabla }_{Z}g\right)\left(X,Y\right)& =Z\left(g\left(X,Y\right)\right)-g\left({\nabla }_{Z}X,Y\right)-g\left(X,{\nabla }_Z,Y\right)\hfill \\ & =g\left({\nabla }_Z^{*}X,Y\right)-g\left({\nabla }_{Z}X,Y\right)\hfill \\ & =g\left(D(Z,X),Y\right).\hfill \end{array}$$

□

For a vector bundle $E\to M$, a gauge transformation is a bundle automorphism, that is, a diffeomorphism from E to E that restricts fiberwise to a linear automorphism. A gauge transformation $\theta$ acts on linear connections by conjugation; thus, it is natural to consider pairs of connections ${\nabla }_1,{\nabla }_2$, such that ${\nabla }_2\theta =\theta {\nabla }_1$. This is the motivation for the next definition.

Definition 1.

Given a torsionless connection ∇, a $(1,1)$-tensor θ satisfies the gauge equation if, for all vector fields $X,Y$,

$${\nabla }_X^{*}\theta Y=\theta {\nabla }_{X}Y$$

(13)

Equivalently, using tensor D, the gauge equation can be rewritten as follows:

$$\begin{array}{cc}\hfill & \nabla \theta =-\left(D\otimes 1\right)\theta \hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\nabla }^{*}\theta =-\left(1\otimes D\right)\theta \hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \left({\nabla }^{lc}+\frac{1}{2}\left(1\otimes D+D\otimes 1\right)\right)\theta =0\hfill \end{array}$$

(16)

with:

$$(D\otimes 1)\left(\theta \right)(X,Y)=D\left(X,\theta Y\right),(1\otimes D)\left(\theta \right)(X,Y)=\theta D(X,Y).$$

When $\nabla ={\nabla }^{lc}$, Equation (16) yields ${\nabla }^{lc}\theta =0$. In this case, Equation (16) indicates that local solutions exist provided that the conditions of [28] are satisfied. In coordinates, the gauge equation becomes with the Einstein convention of summation on repeated indices:

$${\partial }_k{\theta }_i^j={\mathsf{\Gamma }}_{ik}^b{\theta }_b^j-{\mathsf{\Gamma }}_{ak}^j{\theta }_i^a-{\theta }_i^a{D}_{ak}^j$$

(17)

where ${\mathsf{\Gamma }}_{ij}^k$ are the Christoffel symbols of ∇. It is convenient to use an orthonormal frame $\left(X_1,\dots ,X_n\right)$ and its associated coframe $\left({\omega }^1=X_1^{♭},\dots ,{\omega }^n=X_n^{♭}\right)$ to represent tensor D:

$$D_{ij}^k={\mathsf{\Gamma }}_{ij}^k+{\mathsf{\Gamma }}_{ik}^j$$

(18)

where all the coefficients are expressed in the orthonormal frame/coframe, that is,

$$D=D_{ij}^k{X}_k\otimes {\omega }^i\otimes {\omega }^j$$

Definition 2.

Let θ be a $(1,1)$-tensor. Its adjoint ${\theta }^{*}$ is defined, for all vector fields $X,Y$, by the relation:

$$g\left(\theta X,Y\right)=g\left(X,{\theta }^{*}Y\right)$$

Proposition 6.

If θ is a solution of the gauge equation for ∇, then so is its adjoint ${\theta }^{*}$.

Proof. 

For any vector fields $X,Y,Z$:

$$\begin{array}{cc}\hfill g\left(\left({\nabla }_Z^{*}\theta \right)X,Y\right)& =g\left({\nabla }_Z^{*}\left(\theta X\right),Y\right)-g\left(\theta {\nabla }_Z^{*}X,Y\right)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & =Z\left(g\left(\theta X,Y\right)\right)-g\left(\theta X,{\nabla }_Z^Y\right)-g\left(\theta {\nabla }_Z^{*}X,Y\right)\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & =Z\left(g\left(X,{\theta }^{*}Y\right)\right)-g\left(X,{\theta }^{*}{\nabla }_{Z}Y\right)-g\left({\nabla }_Z^{*}X,{\theta }^{*}Y\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & =g\left(X,{\nabla }_Z{\theta }^{*}Y\right)-g\left(X,{\theta }^{*}{\nabla }_{Z}Y\right)\hfill \end{array}$$

(22)

Since $\theta$ satisfies the gauge equation, ${\nabla }_Z^{*}\theta =-\theta D(Z,.)$, thus:

$$g\left(\left({\nabla }_Z^{*}\theta \right)X,Y\right)=-g\left(\theta D(Z,X),Y\right)=-g\left(D(Z,X),{\theta }^{*}Y\right)=-g\left(X,D\left(Z,{\theta }^{*}Y\right)\right)$$

and so:

$$\begin{array}{cc}\hfill 0=& g\left(X,D\left(Z,{\theta }^{*}Y\right)\right)+g\left(X,{\nabla }_Z{\theta }^{*}Y\right)-g\left(X,{\theta }^{*}{\nabla }_{Z}Y\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & =g\left(X,{\nabla }_Z^{*}{\theta }^{*}Y\right)-g\left(X,{\nabla }_Z{\theta }^{*}Y\right)+g\left(X,{\nabla }_Z{\theta }^{*}Y\right)-g\left(X,{\theta }^{*}{\nabla }_Z^{*}Y\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & =g\left(X,{\nabla }_Z^{*}{\theta }^{*}Y\right)-g\left(X,{\theta }^{*}{\nabla }_{Z}Y\right)\hfill \end{array}$$

(25)

This equation in turn implies the required property:

$${\nabla }_Z^{*}{\theta }^{*}Y={\theta }^{*}{\nabla }_{Z}Y$$

□

Remark 1.

This proposition generalizes Theorem 10.3.2 in [29]. It implies that, if a tensor is a solution of the gauge equation, so are its symmetric and skew-symmetric parts.

Proposition 7.

Let θ be a skew-symmetric solution of the gauge equation. Let tensor $p_{\theta }$ be defined for all vector fields $X,Y$ by:

$$p_{\theta }(X,Y)=g\left(\theta X,Y\right)$$

Then, p is ∇ parallel or equivalently, for any vector fields $X,Y,Z$,

$$\left({\nabla }_Z^{*}g\right)(\theta X,Y)=g\left(\left({\nabla }_Z\theta \right)X,Y\right)$$

Proof. 

For any vector fields $X,Y,Z$:

$$\begin{array}{cc}\hfill \left({\nabla }_Z{p}_{\theta }\right)(X,Y)& =Z\left(p_{\theta }(X,Y)\right)-p_{\theta }\left({\nabla }_{Z}X,Y\right)-p_{\theta }\left(X,{\nabla }_{Z}Y\right)\hfill \\ & =g\left({\nabla }_Z^{*}\theta X,Y\right)+g\left(\theta X,{\nabla }_{Z}Y\right)-g\left(\theta {\nabla }_{Z}X,Y\right)-g\left(\theta X,{\nabla }_{Z}Y\right)\hfill \\ & =g\left(\left({\nabla }_Z^{*}\theta -\theta {\nabla }_Z\right)X,Y\right)=0\hfill \end{array}$$

On the other hand,

$$\begin{array}{cc}\hfill \left({\nabla }_Z^{*}g\right)(\theta X,Y)& =Z\left(g\left(\theta X,Y\right)\right)-g\left({\nabla }_Z^{*}\theta X,Y\right)-g\left(\theta X,{\nabla }_Z^{*}Y\right)\hfill \\ & =g\left({\nabla }_Z\theta X,Y\right)-g\left({\nabla }_Z^{*}\theta X,{\nabla }_{Z}Y\right)\hfill \\ & =-g\left(D(Z,\theta X),Y\right)\hfill \end{array}$$

and by the gauge equation:

$$-g\left(D(Z,\theta X),Y\right)=g\left(\left({\nabla }_Z\theta \right)X,Y\right)$$

proving the second assertion. □

Corollary 1.

Let θ be a solution of the gauge equation. Then, the two following conditions are equivalent.

1. 

$\nabla \theta =0$

2. 

∇ is a metric connection for the metric g.

Proof. 

By Proposition 7 we have

$$\left(-{\nabla }_{Z}g\right)(\theta X,Y)=\left({\nabla }_Z^{*}g\right)(\theta X,Y)=g\left(\left({\nabla }_Z\theta \right)X,Y\right)$$

The proposition is demonstrated. □

Remark 2.

In the case of torsionless dual connections, ∇ is exactly the Levi-Civita connection of the metric.

Corollary 2.

Let θ be a solution of the gauge equation of dual torsionless connections. Tensor $p_{\theta }$ is closed, and ∇-coclosed.

Proof. 

For a torsionless connection ∇ and a k-form $\omega$:

$$d{\omega }_{\theta }\left(X_0,\dots ,X_k\right)=\sum _{i=0}^k{(-1)}^i\left({\nabla }_{X_i}\omega \right)\left(X_0,\dots ,{\widehat{X}}_i,\dots ,X_k\right)$$

Since $\nabla p_{\theta }=0$, the previous formula applied to $p_{\theta }$ shows that $d{p}_{\theta }=0$. From [30], the codifferential relative to ∇ is defined as follows:

$${\delta }^{\nabla }\omega =-t{r}_g\nabla \omega$$

the previous formula applied to $p_{\theta }$ shows that ${\delta }^{\nabla }{p}_{\theta }=0,$ thus $p_{\theta }$ is ∇-coclosed. □

3. From Dual Connections to Almost Contact Manifold

3.1. Gauge Equation of Dual Connections

Theorem 3.

The following assertions are equivalent:

1. 

M of dimension $2n+1$ admits an almost cosymplectic structure (almost contact structure),

2. 

The gauge equation of dual connections on M admits a skew-symmetric solution θ such that $rank\theta =2n.$

Proof. 

Let us prove that the necessary part (1) implies (2):

Assuming that M admits an almost contact structure $(\omega ,\eta )$, there exists a vector field $\xi$, such that $i_{\xi }\omega =0\mathrm{and}\eta \left(\xi \right)=1$. For all $p\in M$, there exists an adapted frame $(X_0,X_1,\dots ,X_n,{\widehat{X}}_1,\dots ,{\widehat{X}}_n)$ of $T_{p}M$, such that:

$$X_0={\xi }_p\mathrm{and}(X_1,\dots ,X_n,{\widehat{X}}_1,\dots ,{\widehat{X}}_n)\mathrm{is}\mathrm{a}\mathrm{symplectic}\mathrm{basis}\mathrm{of}H=ker\left(\eta \right).$$

The adapted coframe $\left({\alpha }^0=X_0^{♭},\dots ,{\widehat{\alpha }}^n={\widehat{X}}_n^{♭}\right)$ satisfy:

$${\omega }_p={\alpha }^1\wedge {\widehat{\alpha }}^1+\dots +{\alpha }^n\wedge {\widehat{\alpha }}^n\mathrm{and}{\eta }_x={\alpha }^0.$$

Let $(Y_0,\dots ,Y_n,{\widehat{Y}}_1,\dots ,{\widehat{Y}}_n)$ and $(X_0,\dots ,X_n,{\widehat{X}}_1,\dots ,{\widehat{X}}_n)$ be two adapted frames at p. we have

$$Y_i=C_i^j{X}_j+D_i^j{\widehat{X}}_j\mathrm{and}{\widehat{Y}}_i=-D_i^j{X}_j+C_i^j{\widehat{X}}_j$$

where $C,D\in GL(n,\mathbb{R}).$ Hence, the two frames are related by the $(2n+1)\times (2n+1)$ matrix S:

$$\left(\begin{array}{ccc}C& D& 0\\ -D& C& 0\\ 0& 0& 1\end{array}\right)$$

Since the structure group of M is reducible to $\mathrm{Sp}(n,\mathbb{R})\times 1$, one can find a adapted connection ∇ preserving $\omega ,\xi$:

$$\nabla \xi =0\mathrm{and}\nabla \omega =0.$$

From [16] to an almost cosymplectic structure $(\omega ,\eta )$ there exists an almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on M with the same $\xi$ and $\eta$, whose fundamental 2-form $\mathsf{\Omega }$ coincides with $\omega .$ We define a metric g on M by

$$g(X,Y)=g_H(X,Y),g(X,\xi )=0,g(\xi ,\xi )=1,\forall X,Y\in \mathsf{\Gamma }\left(H\right).$$

where $g_H$ is a metric on H such that

$$\mathsf{\Omega }(X,Y)=g_H(JX,Y)\forall X,Y\in \mathsf{\Gamma }\left(H\right)$$

with $J^2=-I{d}_H$. The $(1,1)$-tensor $\theta :TM⟶TM$ is defined by:

$$\theta X=JX,\theta \xi =0\forall X\in \mathsf{\Gamma }\left(H\right)$$

It comes:

$$\omega (X,Y)=\mathsf{\Omega }(X,Y)=g(\theta X,Y)\mathrm{and}g(\theta X,Y)=-g(X,\theta Y)$$

Then:

$$\begin{array}{cc}& \nabla \omega =\nabla \mathsf{\Omega }=0\hfill \\ & X.\mathsf{\Omega }(Y,Z)-\mathsf{\Omega }({\nabla }_{X}Y,Z)-\mathsf{\Omega }(Y,{\nabla }_{X}Z)=0\hfill \\ & X.g(\theta Y,Z)-g(\theta {\nabla }_{X}Y,Z)-g(\theta Y,{\nabla }_{X}Z)=0\hfill \end{array}$$

Through the duality between $\nabla ,{\nabla }^{*}$, we have

$$g({\nabla }_X^{*}\theta Y,Z)-g(\theta {\nabla }_{X}Y,Z)=0$$

we deduce that

$${\nabla }_X^{*}\theta Y=\theta {\nabla }_{X}Y\mathrm{and}g(\theta X,Y)=-g(X,\theta Y)$$

So, $\theta$ is skew-symmetric solution of the gauge equation, such that $rank\left(\theta \right)=2n.$

Let us now prove that (2) implies (1): Let $\theta$ be a skew-symmetric solution of the gauge equation. Through the assumption that the rank of $\theta$ is 2n, so 2-form $p_{\theta }$ has maximal rank, i.e., $p_{\theta }^n$ vanishes nowhere. Associated to $p_{\theta }$ is its 1-dimensional kernel distribution $ker{p}_{\theta }$. Since M is orientable, by using the Hodge operator ⋆ on M, we define a one-form ${\eta }_{\theta }$, such that: ${\eta }_{\theta }{=}^{\star }{p}_{\theta }^n$. It satisfies $p_{\theta }^n\wedge {\eta }_{\theta }\ne 0.$ The 2-form $p_{\theta }$ defines a line bundle $l_{p_{\theta }}={\cup }_{p\in M}\{p,\mathit{ker}{p}_{\theta }\left(p\right)\}$. Let ${\xi }_{\theta }$ be the unique section of $l_{p_{\theta }}$, such that $i_{{\xi }_{\theta }}{\eta }_{\theta }=1.$ The one-form ${\eta }_{\theta }$ induces an hyperplane distribution by: $H^{{\eta }_{\theta }}=ker{\eta }_{\theta }$, which is everywhere transverse to $l_{p_{\theta }}$. $(p_{\theta },{\eta }_{\theta })$ determines a splitting

$$TM=(l_{p_{\theta }},{\xi }_{\theta })\oplus (H^{{\eta }_{\theta }},{\widehat{p}}_{\theta })$$

of the tangent space of M into a line bundle and an almost-symplectic hyperplane-bundle $(H^{{\lambda }_{\theta }},{\widehat{p}}_{\theta })$, where ${\widehat{p}}_{\theta }$ is the restriction of $p_{\theta }$ to $H^{{\eta }_{\theta }}$. □

Corollary 3.

In an almost cosymplectic manifold $(M,\omega ,\eta )$, with M of dimension $2n+1$, there are always dual connections $(\nabla ,{\nabla }^{*})$ adapted to the distributions $ker\omega$ and $ker\eta$, that is $\nabla {\mathsf{\Gamma }}^{\infty }\left(ker\omega \right)\subset {\mathsf{\Gamma }}^{\infty }\left(ker\omega \right)\mathrm{and}{\nabla }^{*}{\mathsf{\Gamma }}^{\infty }\left(ker\eta \right)\subset {\mathsf{\Gamma }}^{\infty }\left(ker\eta \right).$

Proof. 

Let $(\omega ,\eta )$ be an almost cosymplectic structure on M. There exists ∇, such that:

$$\nabla \omega =0,\nabla \xi =0.$$

Let $Y\in {\mathsf{\Gamma }}^{\infty }\left(ker\omega \right)$. By using identity

$$X.\omega (Y,Z)-\omega ({\nabla }_{X}Y,Z)-\omega (Y,{\nabla }_{X}Z)=0.$$

we have

$$\nabla {\mathsf{\Gamma }}^{\infty }\left(ker\omega \right)\subset {\mathsf{\Gamma }}^{\infty }\left(ker\omega \right).$$

Since $\nabla \xi =0$, the duality relation yields:

$$\left({\nabla }^{*}\eta \right)(X,Y)=g({\nabla }_X\xi ,Y)$$

so ${\nabla }^{*}\eta =0$. With a simple calculation, we have:

$${\nabla }^{*}{\mathsf{\Gamma }}^{\infty }\left(ker\eta \right)\subset {\mathsf{\Gamma }}^{\infty }\left(ker\eta \right).$$

□

Corollary 4.

Let M be a manifold of dimension $2n+1$ and $W=M\times \mathbb{R}$. The following assertions are equivalent:

1. 

The gauge equation of dual connections on M admits a skew-symmetric solution of rank $2n$.

2. 

M admits an almost cosymplectic structure (almost contact structure).

3. 

W admits an almost symplectic structure.

4. 

The gauge equation of dual connections on W admits a skew-symmetric solution of rank $2n+2.$

Proof. 

(1)⟺(2) is exactly the assertion of the previous theorem.

Let us first prove that (2)⟺(3).

The necessary part $\left(2\right)⟹\left(3\right)$.

Starting with an almost cosymplectic structure $(\omega ,\eta )$, according to [16], there exists an almost contact metric $(\theta ,\xi ,\eta ,g)$ on M associated to the almost cosymplectic structure. From [31], $W=M\times \mathbb{R}$ admits a almost complex structure J defined by:

$$J(X,f\frac{\partial }{\partial s})=(\theta X-f\xi ,\eta \left(X\right)\frac{\partial }{\partial s}).$$

The claim follows, since the existence of an almost complex structures is equivalent to the one of an almost symplectic structure [32].

The sufficient part $\left(2\right)⟸\left(3\right)$.

Let us denote by $p:W=\mathbb{R}\times M\to M$ the canonical projection and by $l\left(a\right)=(0,a):M\to W=\mathbb{R}\times M$ a fixed section. Let $\mathsf{\Omega }$ be an almost symplectic 2-form on W ie (${\mathsf{\Omega }}^{n+1}\ne 0)$, s the coordinate in $\mathbb{R}$ and $\frac{\partial }{\partial s}$ the corresponding coordinate vector field. We define $(\eta ,\omega )$ by

$$\omega =l^{*}\mathsf{\Omega },\eta =l^{*}{i}_{\frac{\partial }{\partial s}}\mathsf{\Omega }.$$

On $W=M\times \mathbb{R}$

$$\mathsf{\Omega }=p^{*}\omega +p^{*}\eta \wedge ds.$$

Then from [24], ${\mathsf{\Omega }}^{n+1}=(n+1)p^{*}(\eta \wedge {\omega }^n)\wedge ds.$ The 2-form $\mathsf{\Omega }$ satisfies ${\mathsf{\Omega }}^{n+1}\ne 0$; thus, $\eta \wedge {\omega }^n$ is a volume form on M; consequently, the pair $(\eta ,\omega )$ is an almost cosymplectic structure on M.

Let us prove now that $\left(3\right)⟺\left(4\right).$

The necessary part $\left(3\right)⟹\left(4\right)$.

Let $\mathsf{\Omega }$ be an almost symplectic 2-form on W. From [33,34], there exist almost-symplectic connections ∇ defined by:

$${\nabla }_{X}Y={\nabla }_X^{0}Y+A(X,Y)$$

where A is a $(2,1)$-tensor, and ${\nabla }^0$ is such that

$${\nabla }_X^0\mathsf{\Omega }(Y,Z)=\mathsf{\Omega }(A(X,Y),Z).$$

The almost symplectic connections satisfy:

$$\nabla \mathsf{\Omega }=0.$$

There exists a skew symmetric $\theta \in \mathsf{\Gamma }(T{W}^{\star }\otimes TW)$ and a Riemannian metric g on W, such that the next identity holds:

$$\mathsf{\Omega }(X,Y)=g(\theta X,Y),{\theta }^2=-I{d}_{TW}.$$

Then:

$$X.\mathsf{\Omega }(Y,Z)-\mathsf{\Omega }({\nabla }_{X}Y,Z)-\mathsf{\Omega }(Y,{\nabla }_{X}Z)=0$$

$$X.g(\theta Y,Z)-g(\theta {\nabla }_{X}Y,Z)-g(\theta Y,{\nabla }_{X}Z)=0$$

$$g({\nabla }_X^{*}\theta Y-\theta {\nabla }_{X}Y,Z)=0$$

So, we have

$${\nabla }_X^{*}\theta Y=\theta {\nabla }_{X}Y\mathrm{and}rank\left(\theta \right)=rank\left(\mathsf{\Omega }\right)=2n+2.$$

The sufficient part (3)⟸(4).

Let $\theta$ be a skew-symmetric solution of the gauge equation of dual connections $(\nabla ,{\nabla }^{*})$ on $W=M\times \mathbb{R}$ of $rank\left(\theta \right)=2n+2.$ The 2-form $p_{\theta }$ is nondegenerate on W; thus, $p_{\theta }$ is an almost symplectic structure on W. □

Proceeding in the same way, we have the following corollary:

Corollary 5.

Let M be an even-dimensional manifold of dimension $2n$, and $W=M\times \mathbb{R}$. The following assertions are equivalent:

1. 

The gauge equation of dual connections on M admits a skew-symmetric solution θ such that $rank\theta =2n.$

2. 

M admits an almost symplectic structure (almost contact structure).

3. 

W admits an almost cosymplectic structure (almost contact structure).

4. 

The gauge equation of dual connections on W admits a skew-symmetric solution θ such that $rank\theta =2n.$

Proposition 8.

Let $(\theta ,\eta ,\xi )$ be an almost contact manifold. The following assertions are equivalent:

1. 

$\nabla \theta =0,\nabla \xi =0,$

2. 

$\nabla \theta =0,\nabla \eta =0.$

Proof. 

Let $(\theta ,\eta ,\xi )$ be an almost contact structure, i.e.,

$$\theta \circ \theta +I=\eta \otimes \xi ,\eta \left(\xi \right)=1.$$

By a simple calculation, it comes:

$$\left({\nabla }_X\theta \right)\left(\theta Y\right)+\theta \left(\left({\nabla }_X\theta \right)Y\right)=\eta \left(Y\right)\left({\nabla }_X\xi \right)+\left(\left({\nabla }_X\eta \right)Y\right)\xi$$

and the claim follows. □

Proposition 9.

Let $(\omega ,\eta )$ be an almost cosymplectic manifold with associated almost contact metric structure $(\theta ,\eta ,\xi ,g)$. If $\nabla \omega =0$; then, the next assertions are equivalent:

1. 

$\nabla \theta =0,$

2. 

g is ∇-paralell, i.e., $(\nabla g=0)$,

3. 

${\left({\nabla }_X\xi \right)}^{♭}={\nabla }_X\eta$ or ${\nabla }_X\xi ={\left({\nabla }_X\eta \right)}^{♯}.$

Proof. 

Let us prove that $\left(1\right)⟺\left(2\right)$.

$\left(2\right)⟹\left(1\right)$.

For any $X,Y,Z$, it follows:

$$\begin{array}{cc}& {\nabla }_Z\left(\omega \right)(X,Y)=Z\left(g(\theta X,Y)\right)-g(\theta {\nabla }_{Z}X,Y)-g(\theta X,{\nabla }_{Z}Y)\hfill \\ & =g({\nabla }_Z\theta X,Y)+g(\theta X,{\nabla }_{Z}Y)-g(\theta {\nabla }_{Z}X,Y)-g(\theta X,{\nabla }_{Z}Y)\hfill \\ & =g({\nabla }_Z\theta X,Y)-g(\theta {\nabla }_{Z}X,Y)\hfill \\ & =g(\left({\nabla }_Z\theta \right)X,Y)\hfill \end{array}$$

So, we deduce the necessary part.

$\left(1\right)⟹\left(2\right)$.

$\nabla \omega =0$ is equivalent to ${\nabla }_X^{*}\theta Y=\theta {\nabla }_{X}Y$, (1) implies ${\nabla }_X\theta Y={\nabla }_X^{*}\theta Y$, we deduce that $\nabla ={\nabla }^{*}$; then, $\nabla g=0.$ This proves the sufficient part. Let us prove that $\left(1\right)⟺\left(3\right)$.

$\left(1\right)⟹\left(3\right)$.

Assume that $\nabla \theta =0$, then $\nabla ={\nabla }^{*}$; by using the formula ${\left({\nabla }_Y^{*}X\right)}^{♭}={\nabla }_Y{X}^{♭}$ (resp,${\left({\nabla }_X\omega \right)}^{♯}={\nabla }_X^{*}{\omega }^{♯}$), we deduce that ${\left({\nabla }_X\xi \right)}^{♭}={\nabla }_X\eta$ (resp, ${\nabla }_X\xi ={\left({\nabla }_X\eta \right)}^{♯}$).

$\left(3\right)⟹\left(1\right)$.

${\left({\nabla }_X\xi \right)}^{♭}={\nabla }_X\eta ={\left({\nabla }_X^{*}\xi \right)}^{♭}$, so $\nabla ={\nabla }^{*}$. □

3.2. Gauge Equation of Self-Dual Connections

When $\nabla ={\nabla }^{*}$, the gauge equation is equivalent to

$$\left({\nabla }_X\theta \right)Y=0\forall X,Y\in \mathcal{X}\left(M\right).$$

Theorem 4.

The following assertions are equivalent:

1. 

M admits an almost contact metric structure;

2. 

there exists a metric on M, such that the gauge equation of self-dual connections with respect to it admits a skew-symmetric solution θ of rank $2n.$

Proof. 

This is essentially a corollary of Theorem 3. Let us prove that (1) implies (2). Assume that a $(\theta ,\xi ,\eta ,g)$-structure (almost contact metric structure) is given on M. From [31] (Theorem 11), [35] (Theorem 2) there exists a linear connection, such that

$$\nabla \xi =0,\nabla \theta =0,\nabla \eta =0,\nabla g=0.$$

We deduce that $\theta$ is a skew-symmetric solution of the gauge equation of rank $2n.$ Let us prove that (2) implies (1). Let $\theta$ be a skew-symmetric solution of the gauge equation of sel-fdual connections ∇, such that $rank\theta =2n.$ From Theorem 3, M admits an almost cosymplectic structure. From [16], there exists an almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on M.

□

Corollary 6.

Let M be a $2n+1$-dimensional manifold, Let $W=M\times \mathbb{R}$; the following assertions are equivalents:

1. 

The gauge equation of self-dual connections on M admits a skew-symmetric solution θ, such that $rank\theta =2n.$

2. 

M admits an almost contact metric structure.

3. 

$W=M\times \mathbb{R}$ has an almost Hermitian structure.

4. 

The gauge equation of self-dual connections on W admits a skew-symmetric solution θ, such that $rank\theta =2n+2.$

Proof. 

$\left(1\right)⟺\left(2\right)$ is exactly the assertion of the previous theorem.

Let us prove that $\left(2\right)⟺\left(3\right):$

$\left(2\right)⟹\left(3\right)$.

Let $(\theta ,\xi ,\eta ,g)$ be an almost contact metric structure on M. From [16], one can find a pair $(J,h)$ where J is an almost complex structure defined by:

$$J(X,f\frac{\partial }{\partial s})=(\theta X-f\xi ,\eta \left(X\right)\frac{\partial }{\partial s})$$

and $h=g+d{t}^2$ is a product metric on W. It comes:

$$h(J(X,f\frac{\partial }{\partial t}),J(Y,f\frac{\partial }{\partial t}))=h((X,f\frac{\partial }{\partial t}),(Y,f\frac{\partial }{\partial t}))$$

so pair $(J,h)$ is an almost Hermitian structure in W.

$\left(2\right)⟸\left(3\right)$.

Let $(J,h)$ be an almost Hermitian structure on W. The almost Hermitian form defined by $\mathsf{\Omega }(X,Y)=h(JX,Y)$ is a nondegenerate 2-form on W. Let s be the coordinate in $\mathbb{R}$, and $\frac{\partial }{\partial s}$ be its coordinate vector field. We define $(\eta ,\omega )$ with

$$\omega ={\pi }^{*}\mathsf{\Omega },\eta ={\pi }^{*}{i}_{\frac{\partial }{\partial s}}\mathsf{\Omega },$$

where $\pi$ is the canonical projection on the second factor. Pair $(\omega ,\eta )$ is an almost cosymplectic structure on M. From [16], there exists an almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on M.

$\left(3\right)⟺\left(4\right)$.

$\left(3\right)⟹\left(4\right)$.

Let $(J,h)$ be an almost Hermitian structure on W. From [36] (Theorem 15.1, Corollary 1), almost Hermitian connections exist, namely, linear connections ∇ defined by:

$$\nabla ={\nabla }^h-\frac{1}{2}J{\nabla }^{h}J$$

and satisfying:

$$\nabla J=0,\nabla h=0.$$

Then, the gauge equation of self-dual connections on M admits a skew-symmetric solution J such that $rankJ=2n+2.$

$\left(3\right)⟸\left(4\right)$.

Let $\theta$ be a skew-symmetric solution of the gauge equation of a self-dual connection ∇ on $W=M\times \mathbb{R}$ of rank $2n+2.$ The 2-form $p_{\theta }$ is non-degenerate on W, thus there exists on W an almost Hermitian structure $(J,h)$ such that $p_{\theta }(X,Y)=h(JX,Y).$ □

3.3. Gauge Equation of Torsionless Dual Connections, Modular Class, and Cosymplectic Manifold (Symplectic Mapping Torus)

Theorem 5.

The following assertions are equivalent:

1. 

M Admits a cosymplectic structure.

2. 

The gauge equation of dual torsionless connections admits a skew-symmetric solution θ, such that $rank\theta =2n$ and the modular class of the image of θ vanishes.

Proof. 

Let us prove that (1) implies (2).

Assume that M admits a cosymplectic structure $(\omega ,\eta )$, with $d\eta =0,d\omega =0$ and such that $\eta \wedge {\omega }^n\ne 0$ is a volume-form. From [16], there exists an almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on M, where $\xi$ is the Reeb vector field defined by $i_{\xi }\omega =0,\eta \left(\xi \right)=1)$. $(\theta ,g)$ can be obtained by polarizing $\omega$ onto the codimension one foliation $\mathrm{H}=ker\left(\eta \right)$. This satisfies the following identities:

$$\eta \left(\xi \right)=1,{\theta }^2=-Id+\eta \otimes \xi ,g(\theta X,\theta Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right)$$

The fundamental 2-form $\mathsf{\Omega }$ of the almost contact metric structure coincides with $\omega$, so we have:

$$\mathsf{\Omega }(X,Y)=\omega (X,Y)=g(\theta X,Y).$$

The $\eta \wedge {\omega }^n\ne 0$ condition implies that the restriction of $\omega$ to the leaves of the codimension one foliation $\mathrm{H}=ker\left(\eta \right)$ is a symplectic form. From [37], a symplectic connection on H can be obtained from an arbitrary torsionless linear connection ${\nabla }^0$ by first defining a tensor N using the relation:

$${\nabla }_X^0\omega (Y,Z)=\omega (N(X,Y),Z)$$

Connection ∇ defined as:

$${\nabla }_{X}Y={\nabla }_X^{0}Y+\frac{1}{3}N(X,Y)+\frac{1}{3}N(Y,X)\forall X,Y\in \mathsf{\Gamma }\left(\mathrm{H}\right).$$

is then symplectic. According to the decomposition of the tangent bundle as

$$TM=\mathbb{R}\xi \oplus \mathrm{H},$$

where $\pi :TM⟶\mathrm{H}$ denotes the projection on the second factor, symplectic connections ∇ admits a torsionless lift $\tilde{\nabla }$:

$$\nabla :=\pi \tilde{\nabla }{|}_H\mathrm{and}\tilde{\nabla }\xi =0.$$

t $\nabla \omega =0$ implies:

$$\tilde{\nabla }\omega =0\mathrm{and}\tilde{\nabla }\xi =0.$$

We have, using Blair’s definition,

$$\omega (X,Y)=\mathsf{\Omega }(X,Y)=g(\theta X,Y)\mathrm{and}g(\theta X,Y)=-g(X,\theta Y)$$

It follows:

$$\begin{array}{cc}\hfill & \tilde{\nabla }\omega =\tilde{\nabla }\mathsf{\Omega }=0\hfill \\ \hfill & X.\mathsf{\Omega }(Y,Z)-\mathsf{\Omega }({\tilde{\nabla }}_{X}Y,Z)-\mathsf{\Omega }(Y,{\tilde{\nabla }}_{X}Z)=0\hfill \\ \hfill & X.g(\theta Y,Z)-g(\theta {\tilde{\nabla }}_{X}Y,Z)-g(\theta Y,{\tilde{\nabla }}_{X}Z)=0\hfill \\ \hfill & g({\tilde{\nabla }}_X^{*}\theta Y,Z)-g(\theta {\tilde{\nabla }}_{X}Y,Z)=0\hfill \end{array}$$

We deduce that

$${\tilde{\nabla }}_X^{*}\theta Y=\theta {\tilde{\nabla }}_{X}Y\mathrm{and}g(\theta X,Y)=-g(X,\theta Y)$$

So, $\theta$ is a skew-symmetric solution of the gauge equation of torsionless dual connections $(\tilde{\nabla },{\tilde{\nabla }}^{*})$ such that $rank\left(\theta \right)=2n.$ We have: $ker\left(\theta \right)=ker\left(\omega \right)$, so $im\left(\theta \right)=ker{\left(\omega \right)}^{\perp }=ker\left(\eta \right).$ Then, [26], the modular class of the image of $\theta$ vanishes.

Let us now prove that (2) implies (1).

Let $\theta$ be a skew-symmetric solution of the gauge equation of torsionless dual connections $(\nabla ,{\nabla }^{*})$. From Corollary 2.2, $p_{\theta }$ is ∇-parallel; therefore, it is closed. By assuming the rank of $\theta$ is $2n$, 2-form $p_{\theta }$ has maximal rank, i.e., $p_{\theta }^n$ vanishes nowhere. We associate to $p_{\theta }$ a one-dimensional foliation $ker{p}_{\theta }=ker\left(\theta \right).$ From Proposition 7 $p_{\theta }$ is ∇-parallel, so the foliation $ker{p}_{\theta }$ is ∇-parallel, i.e., ($\nabla \mathsf{\Gamma }\left(ker{p}_{\theta }\right)\subset \mathsf{\Gamma }\left(ker{p}_{\theta }\right)).$ By using the duality of $(\nabla ,{\nabla }^{*})$:

$$X.g(v,v^{\perp })=g({\nabla }_{X}v,v^{\perp })+g(v,{\nabla }_X^{*}{v}^{\perp })$$

we deduce that $\mathrm{im}\left(\theta \right)$ is ${\nabla }^{*}$-parallel, i.e., ${\nabla }^{*}\mathsf{\Gamma }\left(im\theta \right)\subset \mathsf{\Gamma }\left(im\theta \right)$. By using the orientation on M in conjunction with $p_{\theta }^n$, we orient $ker{p}_{\theta }$, so $im\left(\theta \right)$ is transversally a codimension one foliation. By assumption, the modular class of the image of $\theta$ vanishes; thus, according to [26], there exists a closed one form ${\eta }_{\theta }$ on M, such that $im\theta =\mathit{ker}{\eta }_{\theta }$. We deduce that $(p_{\theta },{\eta }_{\theta })$ is a cosymplectic structure on M. □

Proceeding the same way as in Corollary 6, we have the next proposition.

Corollary 7.

In cosymplectic manifold $(M^{2n+1},\omega ,\eta )$, there are always dual torsionless connections $(\nabla ,{\nabla }^{*})$ adapted to the distributions $ker\left(\omega \right)$ and $ker\left(\eta \right)$, that is:

$$\nabla \left({\mathsf{\Gamma }}^{\infty }ker\left(\omega \right)\right)\subset {\mathsf{\Gamma }}^{\infty }ker\left(\omega \right),{\nabla }^{*}\left({\mathsf{\Gamma }}^{\infty }ker\left(\eta \right)\right)\subset {\mathsf{\Gamma }}^{\infty }ker\left(\eta \right).$$

Using the same technique as in the proof of Theorem 5, it follows:

Corollary 8.

Let $M^{2n+1}$ be an odd-dimensional manifold, Let $W=M^{2n+1}\times {\mathbb{S}}^1$; the following assertions are equivalents:

1. 

The gauge equation of dual torsionless connections on $M^{2n+1}$ admits a skew-symmetric solution θ, such that $rank\theta =2n$ and the modular class of image of θ vanish.

2. 

$M^{2n+1}$ admits a cosymplectic structure.

3. 

W admits a symplectic structure.

4. 

The gauge equation of dual torsionless connections on W admits a skew-symmetric solution θ, such that $rank\theta =2n+2.$

Corollary 9.

Let $M^{2n}$ be an even-dimensional manifold and $W=M^{2n}\times {\mathbb{S}}^1$. The following assertions are equivalent:

1. 

The gauge equation of dual torsion-less connections on $M^{2n}$ admits a skew-symmetric solution θ such that $rank\theta =2n.$

2. 

$M^{2n}$ admits a symplectic structure.

3. 

W admits an cosymplectic structure.

4. 

The gauge equation of dual torsionless connections on W admits a skew-symmetric solution θ, such that $rank\theta =2n$ and the modular class of image of θ vanish.

Proof. 

(1)⟺(2) is exactly the same computation as (3)⟺(4) in the previous corollary. (3)⟺(4) is also the same computation as (1)⟺(2). Let us proves that (2)⟺(3):

The necessary part (2)⟹(3).

Let $(M^{2n},\mathsf{\Omega })$ be a symplectic manifold. Symplectic mapping torus $W=M_{\phi }^{2n}=\frac{M^{2n}\times [0,1]}{(m,0)\sim (\phi m,1)}$, where $\phi$ is a symplectic diffeomorphism, admits an cosymplectic structure [17]. Letting $\phi =Id$ shows that $W=M^{2n}\times {\mathbb{S}}^1=\frac{M^{2n}\times [0,1]}{(m,0)\sim (m,1)}$ admits a cosymplectic structure.

The sufficient part $\left(2\right)⟸\left(3\right)$.

Let $(\mathsf{\Omega },\eta )$ be a cosymplectic structure on $W=M^{2n}\times {\mathbb{S}}^1$, and let us consider the fiber bundle $M^{2n}\to W=M_{Id}^{2n}\to {\mathbb{S}}^1$. Then, the 2-form $\omega =l^{*}\mathsf{\Omega }$, with $l:M^{2n}\to W$ as the inclusion map, provides $M^{2n}$ with a symplectic structure.

□

3.4. Gauge Equation of the Levi-Civita Connection and the Existence of Co-Khaler Structure in Dimension Three

3.4.1. Gauge Equation in the Levi-Civita Case

Proposition 10.

If θ is a solution of the gauge equation ${\nabla }^{lc}\theta =\theta {\nabla }^{lc}$, then the 2-form $p_{\theta }:(X,Y)\mapsto g\left(\theta X,Y\right)$ is harmonic, i.e., ${\Delta }^{\mathrm{lc}}{p}_{\theta }=0$.

Proof. 

For a torsionless connection ∇ and a k-form $\omega$:

$$d\omega \left(X_0,\dots ,X_k\right)=\sum _{i=0}^k{(-1)}^i\left({\nabla }_{X_i}\omega \right)\left(X_0,\dots ,{\widehat{X}}_i,\dots ,X_k\right)$$

Since $\nabla p_{\theta }=0$, the previous formula applied to $p_{\theta }$ shows that $d{p}_{\theta }=0$. Let $\theta$ be a skew-symmetric solution gauge equation. Then,

$$\begin{array}{cc}& {\delta }^{lc}{p}_{\theta }(Y_1,\dots ,Y_{r-1})=-\sum _{i=0}^{2n}\left({\nabla }_{E_i}{p}_{\theta }\right)(E_i,Y_1,\dots ,Y_{r-1})\hfill \\ & {\Delta }^{\mathrm{lc}}{p}_{\theta }=d\left({\delta }^{lc}{p}_{\theta }\right)+{\delta }^{lc}\left(d{p}_{\theta }\right)=0.\hfill \end{array}$$

□

3.4.2. Gauge Equation Solution and Pseudo-Kahler Structure

Pseudo-Kahler manifolds were introduced by André Lichnerowicz in [38].

Definition 3.

A $2n$-dimension manifold $(M,g,\mathsf{\Omega })$ is pseudo-Kahler if g is a Riemaniann metric and ${\nabla }^{lc}\mathsf{\Omega }=0$.

Proposition 11.

Let M be a $2n$-dimensional manifold. The following assertions are equivalent:

1. 

M admits a pseudo-Kahler structure.

2. 

There exists a metric g, such that the gauge equation of self-dual torsionless connections on M admits a skew-symmetric solution θ of rank $2n$.

Proof. 

(1)⟹(2).

Assuming that M admits a pseudo-Kahler structure $(\mathsf{\Omega },g)$. From Definition 3, we have ${\nabla }^{lc}\mathsf{\Omega }=0\mathrm{and}{\mathsf{\Omega }}^n\ne 0.$ There exists a skew-symmetric $\theta$ of rank $2n$, such that:

$$\mathsf{\Omega }(X,Y)=g(\theta X,Y)\forall X,Y\in \mathcal{X}\left(M\right).$$

From identity

$${\nabla }^{lc}\mathsf{\Omega }=g({\nabla }_Z^{\mathrm{lc}}\theta X,Y)-g(\theta {\nabla }_Z^{\mathrm{lc}}X,Y)$$

Then, ${\nabla }^{lc}\mathsf{\Omega }=0$ implies that ${\nabla }^{lc}\theta =0.$

(2)⟹(1).

Let g be a Riemaniann metric on M, and ${\nabla }^{lc}$ its levi-Civita connection. Let $\theta$ be a skew-symmetric solution of the linear equation ${\nabla }^{lc}\theta =0$ of rank $2n$. We have ${\nabla }^{lc}{p}_{\theta }=0\mathrm{and}{p}_{\theta }^n\ne 0.$ We deduce that $(g,p_{\theta })$ is a pseudo-Kahler structure on M.

□

3.5. Gauge Equation Solution and Curvature

For a fixed $p\in M$, the Riemaniann metric g admits an orthonormal basis $X_1,\dots ,X_n$ in $T_{p}M$. With respect to it, $\theta$ is represented by a skew-symmetric matrix $\mathsf{\Theta }$ with entries ${\mathsf{\Theta }}_{ij}=g(\theta X_j,X_i)$. It is well-known from elementary linear algebra that there exists a basis $Z_1,\dots ,Z_{2m},Z_{2m+1},\dots ,Z_n$ and real numbers ${\lambda }_1,\dots ,{\lambda }_m$, such that:

$$\begin{array}{cc}& \mathsf{\Theta }{Z}_{2k-1}={\lambda }_k{Z}_{2k},\mathsf{\Theta }{Z}_{2k}=-{\lambda }_k{Z}_{2k-1},k=1,\dots ,m\hfill \\ & \mathsf{\Theta }{Z}_{2m+k}=0,k=1,\dots ,n-2m\hfill \end{array}$$

Furthermore, the basis $Z_1,\dots ,Z_n$ can be chosen to be orthonormal. This is due to the fact that in any case: ${\mathsf{\Theta }}^2{Z}_i=-{\lambda }_{k\left(i\right)}^2{Z}_i$, where $\lambda$ is 0 if $i>2m$ and $k\left(i\right)=\lfloor (i+1)/2\rfloor$ otherwise. It thus comes:

$$g\left({\theta }^{{}^2}{Z}_i,Z_j\right)=-{\lambda }_{k\left(i\right)}^{2}g\left(Z_i,Z_j\right)=g\left(Z_i,{\theta }^{{}^2}{Z}_j\right)=-{\lambda }_{k\left(j\right)}g\left(Z_i,Z_j\right)$$

if ${\lambda }_{k\left(i\right)}\ne {\lambda }_{k\left(j\right)}$, then $g\left(Z_i,Z_j\right)=0$. Otherwise, $Z_i,Z_j$ belong to the same linear subspace of $T_{p}M$ and can thus be orthonormalized. In the $Z_i,i=1\dots n$ basis, matrix $\mathsf{\Theta }$ is block-diagonal with m blocks of the form:

$$\left(\begin{array}{cc}0& -\lambda \\ \lambda & 0\end{array}\right)$$

and the remaining entries all zero.

Remark 3.

As a complex matrix, Θ is diagonal in the base

$$X_{2k-1}-i{X}_{2k},X_{2k-1}+i{X}_{2k},k=0\dots m,Z_{2m+k},k=1\dots m-2N$$

with respective eigenvalues $i{\lambda }_k,,-i{\lambda }_k,0$.

Proposition 12.

For any $U,V\in T_{p}M$, curvature tensor $R(U,V)$ is block diagonal in the basis $Z_i=1\dots n$.

Proof. 

The complexification procedure used here is similar to the one in [39]. Let $U,V\in T_{p}M$ be fixed. In basis $X_1,\dots ,X_n$, $R(U,V)$ is represented by a skew-symmetric matrix still denoted by $R(U,V)$. Since ${\nabla }_X^{\mathrm{lc}}\circ \theta =\theta \circ {\nabla }_X^{\mathrm{lc}}$, $R(U,V)$ and $\mathsf{\Theta }$ commute and are both diagonalizable (as complex matrices), they must have the same eigenspaces. □

Remark 4.

Proposition 12 also shows that any gauge transformation ${\theta }^{\prime }$ satisfying ${\nabla }_X^{\mathrm{lc}}\circ {\theta }^{\prime }={\theta }^{\prime }\circ {\nabla }_X^{\mathrm{lc}}$ commutes with $R(U,V)$ and so is block diagonal in base $Z_1,\dots ,Z_n$. It must thus commute with θ.

Proposition 13.

Curvature tensor R is such that:

$$\left\{\begin{array}{cc}& R(Z_{2k},Z_{2k-1})Z_{2j}=-{\mu }_{kj}{Z}_{2j-1}\hfill \\ & R(Z_{2k-1},Z_{2k})Z_{2j}={\mu }_{kj}{Z}_{2j-1}\hfill \\ & R(Z_{2k},Z_{2k-1})Z_{2j-1}={\mu }_{kj}{Z}_{2j}\hfill \\ & R(Z_{2k-1},Z_{2k})Z_{2j-1}=-{\mu }_{kj}{Z}_{2j}\hfill \\ & 0otherwise.\hfill \end{array}\right.$$

Proof. 

Let us first recall that, for any $X,Y,U,V$,

$$g\left(R(U,V)X,Y\right)=g\left(R(X,Y)U,V\right)$$

Then, using the expression of R in basis $Z_i,i=1\dots n$, it follows that only terms

$$R(Z_{2k},Z_{2k-1})=-R(Z_{2k-1},Z_{2k})$$

can be nonzero. The claim follows by using the block diagonal expression of R. □

Remark 5.

A direct computation shows that the Ricci tensor is diagonal in the basis $Z_i,i=1,\dots ,n$ and:

$$Ric(Z_{2k},Z_{2k})=Ric(Z_{2k-1},Z_{2k-1})={\mu }_{kk}.$$

3.6. Gauge Equation Solution and K-Cosymplectic Structures

Definition 4

([18]). A $2n+1$-dimensional manifold M is K-cosymplectic if it is endowed with a cosymplectic structure, such that the Reeb vector field is Killing with respect to some Riemannian metric on M.

Remark 6.

By using Blair’s definition of cosymplectic manifold, Giovanni Bazzoni and Oliver Goertsches in [18] proved that the previous definition is equivalent to the existence of a $(\theta ,\xi ,\eta ,g)$-structure, such that the Reeb vector field ξ is Killing.

Proposition 14.

In a $2n+1$-dimensionally oriented Riemannian manifold $(M,g)$, if the gauge equation of the Levi-Civita connection admits a skew-symmetric solution of rank $2n$, then M admits a K-cosymplectic structure.

Proof. 

Let $\theta$ be a skew-symmetric solution of the gauge equation. By assuming that the rank of $\theta$ is 2n, 2-form $p_{\theta }$ has maximal rank, i.e., $p_{\theta }^n$ vanishes nowhere. Gauge equation ${\nabla }^{lc}\theta =0$ implies that ${\nabla }^{lc}{p}_{\theta }=0$ and $d{p}_{\theta }=0.$ Distribution $ker{p}_{\theta }$ is ${\nabla }^{lc}$-parallel and then associated to $p_{\theta }$ is a 1-dimensional foliation given by $ker{p}_{\theta }$. Using the orientation on M in conjunction with $p_{\theta }^n$, we orient $ker{p}_{\theta }.$ Let $\widehat{{\xi }_{\theta }}$ be a unit norm section in $\mathit{ker}{p}_{\theta }$. H denotes the mean curvature vector of foliation $ker{p}_{\theta }$

$$\mathrm{H}=({\nabla }_{\widehat{{\xi }_{\theta }}}^{lc}\widehat{{\xi }_{\theta }}){|}_{\mathit{ker}{p}_{\theta }^{\perp }}$$

and ${\eta }_{\theta }$ the volume form of $ker{p}_{\theta }$:

$${\eta }_{\theta }\left(X\right)=g(X,\widehat{{\xi }_{\theta }})\forall X\in \mathcal{X}\left(M\right)$$

It follows:

$$\begin{array}{cc}\hfill d{\eta }_{\theta }(\widehat{{\xi }_{\theta }},X)& ={\widehat{\xi }}_{\theta }.<\widehat{{\xi }_{\theta }},X>-X.{\left|\widehat{{\xi }_{\theta }}\right|}^2-<\widehat{{\xi }_{\theta }},[\widehat{{\xi }_{\theta }},X]>\hfill \\ & =<{{\nabla }^{\mathrm{lc}}}_{\widehat{{\xi }_{\theta }}}\widehat{{\xi }_{\theta }},X>-\frac{1}{2}X.{\left|\widehat{{\xi }_{\theta }}\right|}^2\hfill \\ & =<\mathrm{H},X>\hfill \end{array}$$

One-dimensional foliation $\mathit{ker}{p}_{\theta }$ is minimal:

$$d{\eta }_{\theta }(\widehat{{\xi }_{\theta }},X)=0\forall X\in \mathcal{X}\left(M\right).$$

(26)

Distribution $\mathit{ker}{\eta }_{\theta }$ is ${\nabla }^{lc}$-parallel; thus, $\mathit{ker}{\eta }_{\theta }$ is a codimension one, co-orientable foliation. Using integrability condition

$${\eta }_{\theta }\left([X,Y]\right)=0\forall X,Y\in \mathsf{\Gamma }\left(ker{\eta }_{\theta }\right),$$

we have:

$$d{\eta }_{\theta }(X,Y)=0\forall X,Y\in \mathsf{\Gamma }\left(ker{\eta }_{\theta }\right)$$

(27)

Then,

$$d{\eta }_{\theta }=0$$

$(p_{\theta },{\eta }_{\theta })$ is thus a cosymplectic structure on M, and $\widehat{{\xi }_{\theta }}$ is its Reeb vector field.

Since

$${{\nabla }^{\mathrm{lc}}}_{\widehat{{\xi }_{\theta }}}\widehat{{\xi }_{\theta }}=0,$$

the flow lines of $\widehat{{\xi }_{\theta }}$ are geodesics.

Now,

$$\left(L_{\widehat{{\xi }_{\theta }}}g\right)(X,Y)=g({\nabla }_X^{lc}\widehat{{\xi }_{\theta }},Y)+g(X,{\nabla }_Y^{lc}\widehat{{\xi }_{\theta }})=0\forall X,Y\in \mathit{ker}{\eta }_{\theta }.$$

Thus, $\widehat{{\xi }_{\theta }}$ is a Riemannian flow.

From [40] (Proposition 10.10), the Reeb vector field $\widehat{{\xi }_{\theta }}$ is Killing, i.e., $L_{{\widehat{\xi }}_{\theta }}g=0.$

□

Corollary 10.

Let M be a pseudo-Kahler manifold in the Lichnerowicz sense, and the manifold $W=M\times {\mathbb{S}}^1$ admits K-cosymplectic structures.

Proof. 

W is a fiber bundle over ${\mathbb{S}}^1$, let $\pi :W\to {\mathbb{S}}^1$ denote the natural projection on ${\mathbb{S}}^1$. Let $d\alpha$ be the angular form on ${\mathbb{S}}^1$, and $\frac{d}{d\alpha }$ its dual vector field. This satisfies $d\alpha \left(\frac{d}{d\alpha }\right)=1$ and so induces naturally on W, a nonvanishing closed 1-form ${\eta }_{\alpha }={\pi }^{\star }\left(d\alpha \right)$, and a nonvanishing vector field ${\xi }_{\alpha }$, such that:

$${\eta }_{\alpha }\left({\xi }_{\alpha }\right)=d\alpha \left(\frac{d}{d\alpha }\right)=1.$$

By assuming that M admits a pseudo-Kahler structure $(g,{\mathsf{\Omega }}_{\theta })$, then, on M, we have

$$\nabla {\mathsf{\Omega }}_{\theta }=0\mathrm{and}{\mathsf{\Omega }}_{\theta }^n\ne 0.$$

Let $p:W\to M$ denote the natural projection on the first factor. Let us denote by ${\overline{\mathsf{\Omega }}}_{\theta }$ the closed 2-form defined by:

$${\overline{\mathsf{\Omega }}}_{\theta }=p^{\star }{\mathsf{\Omega }}_{\theta }.$$

We have:

$${\overline{\mathsf{\Omega }}}_{\theta }^n\wedge {\eta }_{\alpha }\ne 0$$

$ker{\overline{\mathsf{\Omega }}}_{{\theta }_p}$ is one-dimensional for all $p\in W$, and ${\overline{\mathsf{\Omega }}}_{\theta }$ determines a line bundle with:

$$l_{{\overline{\mathsf{\Omega }}}_{\theta }}={\cup }_{p\in W}(p,ker{\overline{\mathsf{\Omega }}}_{{\theta }_p})$$

$ker\left({\eta }_{\alpha }\right)$ is an hyperplane distribution transverse to $l_{{\overline{\mathsf{\Omega }}}_{\theta }}$; hence, ${\overline{\mathsf{\Omega }}}_{\theta }$ restricts to a nondegenerate form on $ker\left({\eta }_{\alpha }\right)$. Let ${\xi }_{\alpha }$ be the unique section of $l_{{\overline{\mathsf{\Omega }}}_{\theta }}$ satisfying ${\eta }_{\alpha }\left({\xi }_{\alpha }\right)=1$. We see that ${\overline{\mathsf{\Omega }}}_{\theta }^n\wedge {\eta }_{\alpha }\ne 0$, so tangent bundle TM splits as the direct sum of a line bundle with a preferred nowhere vanishing section, and a symplectic vector bundle

$$TM=\mathbb{R}{\xi }_{\theta }\oplus (ker{\eta }_{\alpha },{\overline{\mathsf{\Omega }}}_{\theta }).$$

Let $h=g+{\left(d\alpha \right)}^2$ be the product metric of W. ${\xi }_{\theta }$ is Killing for metric h; thus, $({\overline{\mathsf{\Omega }}}_{\theta },{\eta }_{\alpha })$ is a K-cosymplectic structure on W.

□

3.6.1. Co-Kähler Structure in Dimension Three and Gauge Equation Solutions

Definition 5.

An almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on an odd-dimensional smooth manifold M is co-Kähler if it is cosymplectic and normal, that is, $N_{\theta }+d\eta \otimes \xi =0$, where $N_{\theta }$ is the Nijenhuis torsion of θ:

$$N_{\theta }(X,Y)={\theta }^2[X,Y]-\theta ([\theta X,Y]+[X,\theta Y])+[\theta X,\theta Y].$$

An almost contact metric structure is co-Kähler if and only if both ${\nabla }^{lc}\eta =0$ and ${\nabla }^{lc}\mathsf{\Omega }=0$, where ${\nabla }^{lc}$ is the covariant differentiation with respect to g, and $\mathsf{\Omega }$ is the fundamental 2-form of the almost contact metric structure. From [16] (Theorem 6.7), we have the following assertion:

Proposition 15.

An almost contact manifold $(M,\theta ,\xi ,\eta ,g)$ is co-Kähler if and only if ${\nabla }^{lc}\theta =0.$

From [17], co-Kähler manifolds are an odd-dimensional analog of Kähler manifolds:

Theorem 6

([17]). Any co-Kähler manifold is a Kähler mapping torus.

Co-Kähler manifolds coincide with cosymplectic manifolds in Blair’s sense.

Theorem 7.

Let M be a 3-dimensional manifold; the following assertions are equivalent:

1. 

M admits a co-Kähler structure(Kähler mapping torus).

2. 

There exists a metric on M, such that the gauge equation of the Levi-Civita connection admits a nonzero skew-symmetric solution.

Proof. 

The necessary part (1) implies (2).

Assuming that M admits a co-Kähler structure, there exists an almost contact metric structure $(\theta ,\xi ,\eta ,g)$ on M, where $\eta$ is a 1-form, $\theta$ is an endomorphism of $TM$, and $\xi$ is a nonvanishing vector field, such that:

$$\eta \left(\xi \right)=1\mathrm{and}{\theta }^2=-I+\eta \otimes \xi .$$

The compatible Riemannian metric g satisfies, for any two vector fields $X,Y\in \mathcal{X}\left(\mathrm{M}\right)$:

$$g(\theta X,\theta Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right)\mathrm{and}g(\theta X,Y)=-g(X,\theta Y).$$

From [16] (Theorem 6.7), the Levi-Civita connection ${\nabla }^{lc}$ of the compatible metric g satisfies ${\nabla }^{lc}\theta =0.$ Thus, there exists a metric g on M, such that gauge equation of the Levi-Civita connection admits a nonzero skew-symmetric solution.

Let us prove the sufficient part (2) implies (1).

Let $\theta$ be a skew-symmetric solution of the gauge equation (${\nabla }^{lc}\theta =0$). Assuming the rank of $\theta$ is 2, by Proposition 14, M admits a K-cosymplectic structures. From [18] (Proposition 2.8), M admits a co-Kähler structure. □