On Codazzi Couplings on the Metric (E⁴ = I)−Manifolds

Abstract

Let ${\mathcal{M}}^k$ be a metric $\left(E^4=I\right)-$manifold equipped with electromagnetic-type structure E, a pseudo-Riemannian metric g and a nondegenerate $2-$form $\widehat{\omega }$. The paper deals with Codazzi couplings of an affine connection ∇ with E, g and $\widehat{\omega }$. We present some results concerning the relationship of these Codazzi couplings. In addition, we construct the connection between Codazzi couplings and $e-(E^4=I)$ Kaehler manifolds.

1. Introduction

A metric $\left(E^4=I\right)-$manifold is a $k-$dimensional pseudo-Riemannian manifold ${\mathcal{M}}^k$ which consists of a $(1,1)$ tensor field E and a pseudo-Riemannian metric g satisfying the following:

(a)

$E^4=I$, whose characteristic polynomial is ${(\lambda -1)}^{{\beta }_1}{(\lambda +1)}^{{\beta }_2}{({\lambda }^2+1)}^s$ with ${\beta }_1+{\beta }_2+2s=k$;

(b)

$g\left(EK,L\right)+g\left(K,EL\right)=0$, then g is necessarily pseudo-Riemannian and ${\beta }_1={\beta }_2.$

A $\left(E^4=I\right)-$structure combines an almost-product structure and an almost-complex structure. In addition, it is a generalization of the electromagnetic tensor field. The condition (b) is the condition that the pseudo-Riemannian metric g is an aem (adapted in the electromagnetic sense metric). In addition, this condition generalizes in a sense of that of Mishra [1] and Hlavaty [2]. For g being an aem, a metric $\left(E^4=I\right)-$manifold will be called $e-(E^4=I)$ Kaehler manifold if E is parallel relative to the Levi–Civita connection ${\nabla }^g$ of g $({\nabla }^{g}E=0)$ [3].

Let the triple $({\mathcal{M}}^k,E,g)$ be a metric $\left(E^4=I\right)-$manifold. Then, fundamental $2-$form $\widehat{\omega }$ can be defined by the formula:

$$\widehat{\omega }\left(K,L\right)=g\left(EK,L\right)=-g\left(K,EL\right)=-\widehat{\omega }(L,K).$$

In the present paper, we will take manifold as a smooth $k-$manifold and use the character $E$, g and $\widehat{\omega }$ for the electromagnetic-type structure, the pseudo-Riemannian metric g and the $2-$form, respectively. Furthermore, the quadruple $({\mathcal{M}}^k,E,g,\widehat{\omega })$ is denoted as $e-(E^4=I)$ Kaehler manifold. It is easy to say that the following satisfy:

(i)

if E is an electromagnetic-type structure, $\widehat{E}=E^{-1}=E^3$ is an electromagnetic-type structure, which we will call an $\widehat{E}$ conjugate electromagnetic-type structure.

(ii)

if E is an electromagnetic-type structure, $P=E^2$ is an almost-product structure.

Note that substitution $K=\widehat{E}K$ and $L=\widehat{E}L$ in $g\left(EK,L\right)+g\left(K,EL\right)=0$ immediately gives that $g\left(\widehat{E}K,L\right)+g\left(K,\widehat{E}L\right)=0$. Moreover, ${\nabla }^{g}E=0$ if and only if ${\nabla }^g\widehat{E}=0$. Hence, $({\mathcal{M}}^k,\widehat{E},g)$ is another $e-(E^4=I)$ Kaehler manifold.

The $e-(E^4=I)$ Kaehler manifolds were firstly constructed by Gadea and Amilibia in [3]. In this paper, the authors showed many of the results obtained for Kaehler manifolds in this more general context. In particular, the Riemannian curvature tensor R satisfies an identity, involving E, in addition to the usual symmetries. As in the Kaehler case, this is used to show that R is determined by its values on quadruplets $(K,EK,K,EK),$$K\in$ Sec $T\left({\mathcal{M}}^k\right)$. This, in turn, leads to an analogue of the notion of constant holomorphic sectional curvature. The authors show that such manifolds are locally products of Kaehler manifolds and ones in which $E^2=I$. They give models for the latter in which ${\mathcal{M}}^k$ is the tangent bundle to a sphere.

In [4], Fei and Zhang studied the interaction of Codazzi couplings with para-Kaehler geometry. The authors obtain the structural results that the Kleinian group acts on an arbitrary affine connection by $g-$conjugation, $\widehat{\omega }-$conjugation, and $\mathfrak{L}-$gauge transformation, where g is the pseudo-Riemannian metric, $\widehat{\omega }$ is a non-degenerate $2-$form and $\mathfrak{L}$ is the tangent bundle isomorphism on smooth manifolds. They established the relationship of Codazzi couplings of a torsion-free connection with a compatible triple. They also showed the compatibility of a pair of connections with Kaehler and para-Kaehler structures, which generalizes special Kaehler geometry (where the connection is curvature-free) to Codazzi-Kaehler geometry (where the connection need not be curvature-free). Later, Gezer and Cakicioglu [5] obtained some new results concerning with Codazzi pairs on the anti-Hermitian context by using different arguments. The paper aims to study Codazzi couplings on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. The analogous case with almost Hermitian case was worked out earlier by Fei and Zhang [4].

2. Conjugate Connection and Codazzi Coupling

Let $({\mathcal{M}}^k,E,g,\widehat{\omega })$ be a metric $\left(E^4=I\right)-$manifold and ∇ be an affine connection. Next, we define, respectively, the conjugate connections of ∇ according to g, $\widehat{\omega }$ and E by the equations [6]:

$$Mg\left(K,L\right)=g\left({\nabla }_{M}K,L\right)+g\left(K,{\nabla }_M^{*}L\right),$$

$$M\widehat{\omega }\left(K,L\right)=\widehat{\omega }\left({\nabla }_{M}K,L\right)+\widehat{\omega }(K,{\nabla }_M^{†}L)$$

and

$${\nabla }^E(K,L)=\widehat{E}\left({\nabla }_{K}EL\right)$$

for all vector fields $K,L,M$ on ${\mathcal{M}}^k$. These connections are called a $g-$conjugate connection, $\widehat{\omega }-$conjugate connection and $E-$conjugate connection, respectively. Note that both $g-$conjugate connection and $\widehat{\omega }-$conjugate connection satisfy ${\left({\nabla }^{*}\right)}^{*}=\nabla$ and ${\left({\nabla }^{†}\right)}^{†}=\nabla$. It is clear that $\nabla g=0$ if and only if ${\nabla }^{*}$ (or ${\nabla }^{†})$ coincides with ∇. For conjugate connections, we also refer to [7,8,9].

Considering the pair $(\nabla ,g)$, the $(0,3)-$tensor fields $\mathfrak{C}$ and $\mathrm{\Gamma }$ are constructed, respectively, by

$$\mathfrak{C}(K,L,M):=\left({\nabla }_{M}g\right)(K,L)$$

and

$$\mathrm{\Gamma }(K,L,M):=-\left({\nabla }_M^{†}\widehat{\omega }\right)(K,L)=\left({\nabla }_M\widehat{\omega }\right)(K,L),$$

where the tensor field $\mathfrak{C}$ is referred to as the cubic form associated to the pair $(\nabla ,g)$ and $\mathrm{\Gamma }$ is in analogous to the cubic form $\mathfrak{C}$ [4].

The curvature tensor field R of an affine connection ∇ is defined by, for all vector fields $K,L,M$,

$$R(K,L)M={\nabla }_K{\nabla }_{L}M-{\nabla }_L{\nabla }_{K}M-{\nabla }_{[K,L]}M$$

and its $(0,4)-$curvature tensor field is as follows:

$$R(K,L,M,N)=g\left(R\right(K,L)M,N).$$

For the curvature tensor fields of ∇, ${\nabla }^{*}$ and ${\nabla }^E$, the theorem is given below.

Theorem 1.

Let$({\mathcal{M}}^k,E,g,\widehat{\omega })$be a metric $\left(E^4=I\right)-$manifold. ${\nabla }^{*}$ and ${\nabla }^E$ assign, respectively, $g-$conjugation and $E-$conjugation of an affine connection ∇ on ${\mathcal{M}}^k$. The relationship between the $(\mathit{0},\mathit{4})-$curvature tensor fields $R,R^{*}$ and $R^E$ of ∇, ${\nabla }^{*}$ and ${\nabla }^E$ is as follows:

$$R\left(K,L,EM,N\right)=-R^{*}\left(K,L,N,EM\right)=R^E(K,L,M,EN)$$

for all vector fields $K,L,M$ on ${\mathcal{M}}^k$.

Proof. 

It suffices to prove it only on one basis, because the relation is linear in the arguments $K,$ $L,N$ and M. Thus, we suppose that $K,L,N,M\in \{\frac{\partial }{\partial x^1},\dots ,\frac{\partial }{\partial x^k}\}$ and take computational advantage of the following vanishing Lie brackets

$$[K,L]=[L,N]=[N,M]=0.$$

From here, it is obtained that

$$\begin{array}{ccc}\hfill KL\widehat{\omega }\left(M,N\right)& =& K\left(Lg\left(EM,N\right)\right)\hfill \\ & =& K\left(g\left({\nabla }_L\left(EM\right),N\right)+g\left(EM,{\nabla }_L^{*}N\right)\right)\hfill \\ & =& K\left(g\left({\nabla }_L\left(EM\right),N\right)\right)+K\left(g\left(EM,{\nabla }_L^{*}N\right)\right)\hfill \\ & =& g\left({\nabla }_K{\nabla }_L\left(EM\right),N\right)+g\left({\nabla }_L\left(EM\right),{\nabla }_K^{*}N\right)\hfill \\ & & +g\left({\nabla }_K\left(EM\right),{\nabla }_L^{*}N)+g(EM,{\nabla }_K^{*}{\nabla }_L^{*}N\right)\hfill \end{array}$$

and similarly

$$\begin{array}{ccc}\hfill LK\widehat{\omega }\left(M,N\right)& =& L\left(Kg\left(EM,N\right)\right)\hfill \\ & =& g\left({\nabla }_L{\nabla }_K\left(EM\right),N\right)+g\left({\nabla }_K\left(EM\right),{\nabla }_L^{*}N\right)\hfill \\ & & +g\left({\nabla }_L\left(EM\right),{\nabla }_K^{*}N)+g(EN,{\nabla }_L^{*}{\nabla }_K^{*}N\right).\hfill \end{array}$$

When we subtract the above equations from each other, we find

$$\begin{array}{ccc}\hfill 0& =& \left[K,L\right]\widehat{\omega }\left(M,N\right)=KL\widehat{\omega }\left(M,N\right)-LK\widehat{\omega }\left(M,N\right)\hfill \\ \hfill 0& =& g\left({\nabla }_K{\nabla }_L\left(EM\right)-{\nabla }_L{\nabla }_K\left(EM\right),N\right)\hfill \\ & & +g\left(EM,{\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N\right)\hfill \\ & & -g\left({\nabla }_L{\nabla }_K\left(EM\right),N\right)-g\left({\nabla }_K\left(EM\right),{\nabla }_L^{*}N\right)\hfill \\ & & -g\left({\nabla }_L\left(EM\right),{\nabla }_K^{*}N)-g(EM,{\nabla }_L^{*}{\nabla }_K^{*}N\right)\hfill \\ \hfill 0& =& g\left(R\left(K,L\right)EM,N\right)+g\left(R^{*}\left(K,L\right)N,EM\right)\hfill \\ \hfill 0& =& R\left(K,L,EM,N\right)+R^{*}\left(K,L,N,EM\right)\hfill \end{array}$$

$$\Rightarrow R\left(K,L,EM,N\right)=-R^{*}\left(K,L,N,EM\right)$$

and similarly

$$\begin{array}{ccc}& & g\left({\nabla }_K{\nabla }_L\left(EM\right)-{\nabla }_L{\nabla }_K\left(EM\right),N\right)+g\left(EM,{\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N\right)=0\hfill \\ & & \widehat{\omega }\left(\widehat{E}\left({\nabla }_K{\nabla }_L\left(EM\right)-{\nabla }_L{\nabla }_K\left(EM\right)\right),N\right)+\widehat{\omega }\left(M,{\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N\right)=0\hfill \\ & & \widehat{\omega }\left(\widehat{E}{\nabla }_{K}E\left(\widehat{E}{\nabla }_{L}EM\right)-\widehat{E}{\nabla }_{L}E\left(\widehat{E}{\nabla }_{K}EM\right),N\right)+\widehat{\omega }\left(M,{\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N\right)=0\hfill \\ & & \widehat{\omega }\left(\widehat{E}{\nabla }_{K}E\left({\nabla }_L^{E}M\right)-\widehat{E}{\nabla }_{L}E\left({\nabla }_K^{E}M\right),N\right)+\widehat{\omega }\left(M,{\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N\right)=0\hfill \\ & & \widehat{\omega }\left({\nabla }_K^E{\nabla }_L^{E}M-{\nabla }_L^E{\nabla }_K^{E}M,N\right)+\widehat{\omega }\left(M,{\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N\right)=0\hfill \\ & & -g\left({\nabla }_K^E{\nabla }_L^{E}M-{\nabla }_L^E{\nabla }_K^{E}M,EN\right)+g\left({\nabla }_K^{*}{\nabla }_L^{*}N-{\nabla }_L^{*}{\nabla }_K^{*}N,EM\right)=0\hfill \\ & & -R^E\left(K,L,M,EN\right)+R^{*}\left(K,L,N,EM\right)=0\hfill \\ & & \Rightarrow R^E\left(K,L,M,EN\right)=R^{*}\left(K,L,N,EM\right).\hfill \end{array}$$

Thus, it is obtained that $R\left(K,L,EM,N\right)=-R^{*}\left(K,L,N,EM\right)=R^E(K,L,M,EN)$ which completes the proof. □

Given an arbitrary affine connection ∇ on a pseudo-Riemannian manifold $({\mathcal{M}}^k,g)$, for any $(1,1)-$tensor field E and a symmetric bilinear form $\rho$ on ${\mathcal{M}}^k$, we call $(\nabla ,E)$ and $(\nabla ,\rho )$, respectively, Codazzi-coupled, if their covariant derivative $(\nabla E)$ and $(\nabla \rho )$, respectively, is (totally) symmetric in $K,L,M$ [6]:

$$\left({\nabla }_{M}E\right)K=\left({\nabla }_{K}E\right)M,\left({\nabla }_M\rho \right)\left(K,L\right)=\left({\nabla }_K\rho \right)\left(M,L\right).$$

Next, we search characterization of Codazzi couplings of an affine connection ∇ on ${\mathcal{M}}^k$ with a pseudo-Riemannian metric g and an electromagnetic-type structure E. We give the following proposition, which is analogous to the result given in [4] for a Hermitian setting.

Proposition 1.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. If $(\nabla ,\widehat{\omega })$ is Codazzi-coupled, the following are provided:

(i) 

$\mathrm{\Gamma }\left(K,L,M\right)=\left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)$ is not totally symetric;

(ii) 

$\mathrm{\Gamma }\left(K,L,M\right)=\mathrm{\Gamma }\left(K,M,L\right)\iff {\nabla }^{†}$ and ∇ have equal torsions;

(iii) 

${\nabla }^{*}$ and ∇ have equal torsionsif and only if $({\nabla }^{*},E)$ is Codazzi-coupled.

Proof. (i) Since $\widehat{\omega }$ is skew-symmetric, $\mathrm{\Gamma }\left(K,L,M\right)=-\mathrm{\Gamma }\left(L,K,M\right)$. Therefore, $\mathrm{\Gamma }$ is not totally symetric.

(ii)

$$\begin{array}{ccc}& & \mathrm{\Gamma }\left(K,L,M\right)-\mathrm{\Gamma }\left(K,M,L\right)=\left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)-\left({\nabla }_L\widehat{\omega }\right)\left(K,M\right)\hfill \\ & & =Z\widehat{\omega }\left(K,L\right)-\widehat{\omega }\left({\nabla }_{M}K,L\right)-\widehat{\omega }\left(K,{\nabla }_{M}L\right)\hfill \\ & & -L\widehat{\omega }\left(K,M\right)+\widehat{\omega }\left({\nabla }_{L}K,M\right)+\widehat{\omega }(K,{\nabla }_{L}M)\hfill \\ & & =\widehat{\omega }\left({\nabla }_{M}K,L\right)+\widehat{\omega }\left(K,{\nabla }_M^{†}L\right)-\widehat{\omega }\left({\nabla }_{M}K,L\right)-\widehat{\omega }\left(K,{\nabla }_{M}L\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_{L}K,M\right)-\widehat{\omega }\left(K,{\nabla }_L^{†}M\right)+\widehat{\omega }\left({\nabla }_{L}K,M\right)+\widehat{\omega }(K,{\nabla }_{L}M)\hfill \\ & & =\widehat{\omega }\left(K,{\left({\nabla }^{†}-\nabla \right)}_{M}L\right)-\widehat{\omega }(K,{\left({\nabla }^{†}-\nabla \right)}_{L}M)\hfill \\ & & =\widehat{\omega }\left(K,{\left({\nabla }^{†}-\nabla \right)}_{M}L-{\left({\nabla }^{†}-\nabla \right)}_{L}M\right)\hfill \\ & & =\widehat{\omega }\left(K,T^{{\nabla }^{†}}\left(M,L\right)-T^{\nabla }\left(M,L\right)\right)=0\iff T^{{\nabla }^{†}}=T^{\nabla },\hfill \end{array}$$

where

$$\begin{array}{ccc}& & T^{{\nabla }^{†}}\left(M,L\right)={\nabla }_M^{†}L-{\nabla }_L^{†}M-[M,L]\hfill \\ & & T^{\nabla }\left(M,L\right)={\nabla }_{M}L-{\nabla }_{L}M-[M,L].\hfill \end{array}$$

(iii) From the covariant derivative, we have

$$\begin{array}{ccc}& & \left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)=\left({\nabla }_K\widehat{\omega }\right)\left(M,L\right)\hfill \\ & & \widehat{\omega }\left(K,L\right)-\widehat{\omega }\left({\nabla }_{M}K,L\right)-\widehat{\omega }\left(K,{\nabla }_{M}L\right)=K\widehat{\omega }\left(M,L\right)-\widehat{\omega }\left({\nabla }_{K}M,L\right)-\widehat{\omega }(M,{\nabla }_{K}L)\hfill \\ & & Mg\left(EK,L\right)-g\left(E{\nabla }_{M}K,L\right)-g\left(EK,{\nabla }_{M}L\right)=Kg\left(EM,L\right)-g\left(E{\nabla }_{K}M,L\right)-g(EM,{\nabla }_{K}L)\hfill \\ & & g\left({\nabla }_M^{*}\left(EK\right),L\right)-g\left(E{\nabla }_{M}K,L\right)=g\left({\nabla }_K^{*}\left(EM\right),L\right)-g(E{\nabla }_{K}M,L)\hfill \\ & & \widehat{\omega }\left(\widehat{E}{\nabla }_M^{*}\left(EK\right),L\right)-\widehat{\omega }\left({\nabla }_{M}K,L\right)=\widehat{\omega }\left(\widehat{E}{\nabla }_K^{*}\left(EM\right),L\right)-\widehat{\omega }\left({\nabla }_{K}M,L\right)\hfill \\ & & \widehat{\omega }\left(\widehat{E}\left\{{\nabla }_M^{*}\left(EK\right)-{\nabla }_K^{*}\left(EM\right)\right\},L\right)=\widehat{\omega }\left({\nabla }_{M}K-{\nabla }_{K}M,L\right)\hfill \\ & & E^{-1}\left\{{\nabla }_M^{*}\left(EK\right)-{\nabla }_K^{*}\left(EM\right)\right\}={\nabla }_{M}K-{\nabla }_{K}M\hfill \\ & & \widehat{E}\left\{\left({\nabla }_M^{*}E\right)K+E\left({\nabla }_M^{*}K\right)-\left({\nabla }_K^{*}E\right)M-E\left({\nabla }_K^{*}M\right)\right\}={\nabla }_{M}K-{\nabla }_{K}M\hfill \\ & & \widehat{E}\{\left({\nabla }_M^{*}E\right)K-\left({\nabla }_K^{*}E\right)M\}+\widehat{E}\left\{E({\nabla }_M^{*}K-{\nabla }_K^{*}M)\right\}={\nabla }_{M}K-{\nabla }_{K}M\hfill \\ & & \widehat{E}\{\left({\nabla }_M^{*}E\right)K-\left({\nabla }_K^{*}E\right)M\}+(({\nabla }_M^{*}K-{\nabla }_K^{*}M)-[M,K])={\nabla }_{M}K-{\nabla }_{K}M-[M,K]\hfill \\ & & \widehat{E}\{\left({\nabla }_M^{*}E\right)K-\left({\nabla }_K^{*}E\right)M\}=T^{\nabla }(M,K)-T^{{\nabla }^{*}}(M,K),\hfill \end{array}$$

where

$$T^{{\nabla }^{*}}\left(M,L\right)={\nabla }_M^{*}L-{\nabla }_L^{*}M-[M,L].$$

Therefore, $T^{{\nabla }^{*}}(M,K)=T^{\nabla }\left(M,K\right)\iff \left({\nabla }_M^{*}E\right)K=\left({\nabla }_K^{*}E\right)M$, that is,$({\nabla }^{*},E)$ is Codazzi-coupled. □

Proposition 2.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. The followings are equivalent:

(i) 

$(\nabla ,E)$ is Codazzi-coupled;

(ii) 

∇ and ${\nabla }^E$ have equal torsions;

(iii) 

$({\nabla }^E,\widehat{E})$ is Codazzi-coupled,

where $\widehat{E}$ is a conjugate electromagnetic-type structure on $({\mathcal{M}}^k,E,g,\widehat{\omega })$.

Proof. 

Let $(\nabla ,E)$ be Codazzi-coupled. Using $T^{\nabla }\left(K,L\right)={\nabla }_{K}L-{\nabla }_{L}K-[K,L]$ and $T^{{\nabla }^E}\left(K,L\right)={\nabla }_K^{E}L-{\nabla }_L^{E}K-[K,L]$, we yield

$$\begin{array}{ccc}& & T^{{\nabla }^E}\left(K,L\right)-T^{\nabla }\left(K,L\right)={\nabla }_K^{E}L-{\nabla }_{K}L-{\nabla }_L^{E}K+{\nabla }_{L}K\hfill \\ & & =\widehat{E}\left({\nabla }_{K}EL\right)-{\nabla }_{K}L-\widehat{E}\left({\nabla }_{L}EK\right)+{\nabla }_{L}K\hfill \\ & & =\widehat{E}\left[\left({\nabla }_{K}E\right)L+E\left({\nabla }_{K}L\right)\right]-{\nabla }_{K}L-\widehat{E}\left[\left({\nabla }_{L}E\right)K+E\left({\nabla }_{L}K\right)\right]+{\nabla }_{L}K\hfill \\ & & =\widehat{E}\left({\nabla }_{K}E\right)L+{\nabla }_{K}L-{\nabla }_{K}L-\widehat{E}\left({\nabla }_{L}E\right)K-{\nabla }_{L}K+{\nabla }_{L}K\hfill \\ & & =\widehat{E}\left[\left({\nabla }_{K}E\right)L-\left(\nabla LE\right)K\right]=0.\hfill \end{array}$$

On the other hand, it is straightforward to obtain

$$\begin{array}{ccc}& & \left({\nabla }_K^E\widehat{E}\right)L-\left({\nabla }_L^E\widehat{E}\right)K={\nabla }_K^E\left(\widehat{E}L\right)-\widehat{E}\left({\nabla }_K^{E}L\right)-{\nabla }_L^E\left(\widehat{E}K\right)+\widehat{E}\left({\nabla }_L^{E}K\right)\hfill \\ & & =\widehat{E}\left({\nabla }_{K}L\right)-\widehat{E}\left(\widehat{E}{\nabla }_{K}EL\right)-\widehat{E}\left({\nabla }_{L}K\right)+\widehat{E}\left(\widehat{E}{\nabla }_{L}EK\right)\hfill \\ & & =\widehat{E}\left({\nabla }_{K}L\right)-{\widehat{E}}^2\left[\left({\nabla }_{K}E\right)L+E\left({\nabla }_{K}L\right)\right]-\widehat{E}\left({\nabla }_{L}K\right)+{\widehat{E}}^2\left[\left({\nabla }_{L}E\right)K+E\left({\nabla }_{L}K\right)\right]\hfill \\ & & =\widehat{E}\left({\nabla }_{K}L\right)-P\left({\nabla }_{K}E\right)L-\widehat{E}\left({\nabla }_{K}L\right)-\widehat{E}\left({\nabla }_{L}K\right)+P\left({\nabla }_{L}E\right)K+\widehat{E}\left({\nabla }_{L}K\right)\hfill \\ & & =P\left[\left({\nabla }_{L}E\right)K-\left({\nabla }_{K}E\right)L\right],\hfill \end{array}$$

which gives to us: $\left({\nabla }_K^E\widehat{E}\right)L-\left({\nabla }_L^E\widehat{E}\right)K=0$$\iff \left({\nabla }_{L}E\right)K-\left({\nabla }_{K}E\right)L=0$. Hence, the proof is completed. □

Proposition 3.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. If $(\nabla ,E)$ and $(\nabla ,P)$ are Codazzi-coupled, $({\nabla }^E,E)$ is Codazzi-coupled, where $P=E^2$.

Proof. 

Standard calculations give

$$\begin{array}{ccc}& & \left({\nabla }_K^{E}E\right)K-\left({\nabla }_L^{E}E\right)K=\left({\nabla }_K^{E}EL\right)-E\left({\nabla }_K^{E}L\right)-\left({\nabla }_L^{E}EK\right)+E\left({\nabla }_L^{E}K\right)\hfill \\ & & =E^{-1}\left({\nabla }_{K}PL\right)-E\left(\widehat{E}{\nabla }_{K}EL\right)-\widehat{E}\left({\nabla }_{L}PK\right)+E\left(\widehat{E}{\nabla }_{L}EK\right)\hfill \\ & & =\widehat{E}\left({\nabla }_{K}PL\right)-{\nabla }_{K}EL-\widehat{E}\left({\nabla }_{L}PK\right)+{\nabla }_{L}EK\hfill \\ & & =\widehat{E}\left[\left({\nabla }_{K}P\right)L+P{\nabla }_{K}L\right]-\left({\nabla }_{K}E\right)L-E{\nabla }_{K}L\hfill \\ & & -\widehat{E}\left[\left({\nabla }_{L}P\right)K+P{\nabla }_{L}K\right]+\left({\nabla }_{L}E\right)K+E{\nabla }_{L}K\hfill \\ & & =\widehat{E}\left({\nabla }_{K}P\right)L+E{\nabla }_{K}L-\left({\nabla }_{K}E\right)L-E{\nabla }_{K}L\hfill \\ & & -\widehat{E}\left({\nabla }_{L}P\right)K-E{\nabla }_{L}K+\left({\nabla }_{L}E\right)K+E{\nabla }_{L}K\hfill \\ & & =\widehat{E}\left[\left({\nabla }_{K}P\right)L-\left({\nabla }_{L}P\right)K\right]+\left({\nabla }_{L}E\right)K-\left({\nabla }_{K}E\right)L.\hfill \end{array}$$

Thus, the result is given. □

Proposition 4.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. In that case, the following are equivalent:

(i) 

$\widehat{\omega }\left(T^{\nabla }\left(K,L\right),M\right)=\widehat{\omega }\left(T^{{\nabla }^{†}}\left(K,L\right),M\right)+\left({\nabla }^{†}\widehat{\omega }\right)\left(K,L,M\right)-\left({\nabla }^{†}\widehat{\omega }\right)\left(L,K,M\right)$;

(ii) 

$\nabla \widehat{\omega }$ is symmetric if and only if ${\nabla }^{†}\widehat{\omega }$ is symmetric.

Proof. (i) From the definition of $\widehat{\omega }-$conjugation, it follows that

$$\begin{array}{ccc}& & \widehat{\omega }\left(T^{\nabla }\left(K,L\right),M\right)=\widehat{\omega }\left({\nabla }_{K}L-{\nabla }_{L}K-\left[K,L\right],M\right)\hfill \\ & & =\widehat{\omega }\left({\nabla }_{K}L,M\right)-\widehat{\omega }\left({\nabla }_{L}K,M\right)-\widehat{\omega }\left(\left[K,L\right],M\right)\hfill \\ & & =K\widehat{\omega }\left(L,M\right)-\widehat{\omega }\left(L,{\nabla }_K^{†}M\right)-L\widehat{\omega }\left(K,M\right)+\widehat{\omega }\left(K,{\nabla }_L^{†}M\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_K^{†}L-{\nabla }_L^{†}K-T^{{\nabla }^{†}}\left(K,L\right),M\right)\hfill \\ & & =K\widehat{\omega }\left(L,M\right)-\widehat{\omega }\left(L,{\nabla }_K^{†}M\right)-L\widehat{\omega }\left(K,M\right)+\widehat{\omega }\left(K,{\nabla }_L^{†}M\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_K^{†}L,M\right)+\widehat{\omega }\left({\nabla }_L^{†}K,M\right)+\widehat{\omega }\left(T^{{\nabla }^{†}}\left(K,L\right),M\right)\hfill \\ & & =\left({\nabla }_K^{†}\widehat{\omega }\right)\left(L,M\right)-\left({\nabla }_L^{†}\widehat{\omega }\right)\left(K,M\right)+\widehat{\omega }\left(T^{{\nabla }^{†}}\left(K,L\right),M\right).\hfill \end{array}$$

(ii) Similarly, we obtain

$$\begin{array}{ccc}& & \left({\nabla }^{†}\widehat{\omega }\right)\left(K,L,M\right)=\left({\nabla }_K^{†}\widehat{\omega }\right)\left(L,M\right)\hfill \\ & & =K\widehat{\omega }\left(L,M\right)-\widehat{\omega }\left({\nabla }_K^{†}L,M\right)-\widehat{\omega }\left(L,{\nabla }_K^{†}M\right)\hfill \\ & & =K\widehat{\omega }\left(L,M\right)-K\widehat{\omega }\left(L,M\right)+\widehat{\omega }\left(L,{\nabla }_{K}M\right)-K\widehat{\omega }\left(L,M\right)+\widehat{\omega }\left({\nabla }_{K}L,M\right)\hfill \\ & & =-K\widehat{\omega }\left(L,M\right)+\widehat{\omega }\left(L,{\nabla }_{K}M\right)+\widehat{\omega }\left({\nabla }_{K}L,M\right)\hfill \\ & & =-\left({\nabla }_K\widehat{\omega }\right)\left(L,M\right)=-\left(\nabla \widehat{\omega }\right)\left(K,L,M\right).\hfill \end{array}$$

□

Proposition 5.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. In that case,

(i) 

$\widehat{\omega }\left(R\left(K,L,M\right),N\right)=-\widehat{\omega }\left(M,R^{†}\left(K,L,N\right)\right)=\widehat{\omega }\left(R^{†}\left(K,L,N\right),M\right)$;

(ii) 

If ${\nabla }^{†}$ is flat, ∇ is flat, too,

where ${\nabla }^{†}$ denotes $\widehat{\omega }-$conjugation of ∇ on $M^k$ and R and $R^{†}$ are, respectively, the curvature tensor fields of ∇ and ${\nabla }^{†}$.

Proof. 

$$\begin{array}{ccc}& & \widehat{\omega }\left(R^{\nabla }\left(K,L,M\right),N\right)=\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M-{\nabla }_L{\nabla }_{K}M-{\nabla }_{\left[K,L\right]}M,N\right)\hfill \\ & & =\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M,N\right)-\widehat{\omega }\left({\nabla }_L{\nabla }_{K}M,N\right)-\widehat{\omega }\left({\nabla }_{\left[K,L\right]}M,N\right)\hfill \\ & & =K\widehat{\omega }\left({\nabla }_{L}M,N\right)-\widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)-L\widehat{\omega }\left({\nabla }_{K}M,N\right)+\widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)\hfill \\ & & -\left[K,L\right]\widehat{\omega }\left(M,N\right)+\widehat{\omega }\left(M,{\nabla }_{\left[K,L\right]}^{†}N\right).\hfill \end{array}$$

On the other hand, using $\left[K,L\right]\widehat{\omega }\left(M,N\right)=KL\widehat{\omega }\left(M,N\right)-LK\widehat{\omega }\left(M,N\right)$, we have

$$\begin{array}{ccc}\hfill KL\widehat{\omega }\left(M,N\right)& =& K\left(L\widehat{\omega }\left(M,N\right)\right)\hfill \\ & =& K(\widehat{\omega }\left({\nabla }_{L}M,N\right)+\widehat{\omega }\left(M,{\nabla }_L^{†}N\right)\hfill \\ & =& K\left(\widehat{\omega }\left({\nabla }_{L}M,N\right)\right)+K\left(\widehat{\omega }\left(M,{\nabla }_L^{†}N\right)\right)\hfill \\ & =& \widehat{\omega }\left({\nabla }_K{\nabla }_{L}M,N\right)+\widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)\hfill \\ & +& \widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)+\widehat{\omega }\left(M,{\nabla }_K^{†}{\nabla }_L^{†}N\right)\hfill \end{array}$$

and similarly

$$\begin{array}{ccc}\hfill LK\widehat{\omega }\left(M,N\right)& =& \widehat{\omega }\left({\nabla }_L{\nabla }_{K}M,N\right)+\widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)\hfill \\ & +& \widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)+\widehat{\omega }\left(M,{\nabla }_L^{†}{\nabla }_K^{†}N\right).\hfill \end{array}$$

Thus

$$\begin{array}{ccc}& & LK\widehat{\omega }\left(M,N\right)-KL\widehat{\omega }\left(M,N\right)=\widehat{\omega }\left({\nabla }_L{\nabla }_{K}M,N\right)+\widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)\hfill \\ & & +\widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)+\widehat{\omega }\left(M,{\nabla }_L^{†}{\nabla }_K^{†}N\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M,N\right)-\widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)-\widehat{\omega }\left(M,{\nabla }_K^{†}{\nabla }_L^{†}N\right)\hfill \\ & & =-\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M-{\nabla }_L{\nabla }_{K}M,N\right)-\widehat{\omega }\left(M,{\nabla }_K^{†}{\nabla }_L^{†}N-{\nabla }_L^{†}{\nabla }_K^{†}N\right).\hfill \end{array}$$

From the last equation, we obtain

$$\begin{array}{ccc}& & \widehat{\omega }\left(R^{\nabla }\left(K,L,M\right),N\right)=\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M,N\right)+\widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_{L}M,{\nabla }_K^{†}N\right)-\widehat{\omega }\left({\nabla }_L{\nabla }_{K}M,N\right)-\widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)+\widehat{\omega }\left({\nabla }_{K}M,{\nabla }_L^{†}N\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M-{\nabla }_L{\nabla }_{K}M,N\right)\hfill \\ & & -\widehat{\omega }\left(M,{\nabla }_K^{†}{\nabla }_L^{†}N-{\nabla }_L^{†}{\nabla }_K^{†}N\right)+\widehat{\omega }\left(M,{\nabla }_{\left[K,L\right]}^{†}N\right)\hfill \\ & & =\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M-{\nabla }_L{\nabla }_{K}M,N\right)-\widehat{\omega }\left({\nabla }_K{\nabla }_{L}M-{\nabla }_L{\nabla }_{K}M,N\right)\hfill \\ & & -\widehat{\omega }\left(M,{{\nabla }_K^{†}\nabla }_L^{†}N-{\nabla }_L^{†}{\nabla }_K^{†}N-{\nabla }_{\left[K,L\right]}^{†}N\right)\hfill \\ & & =-\widehat{\omega }\left(M,R^{{\nabla }^{†}}\left(K,L,N\right)\right)=\widehat{\omega }\left(R^{{\nabla }^{†}}\left(K,L,N\right),M\right).\hfill \end{array}$$

□

Proposition 6.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $(M^k,E,g,\widehat{\omega })$. If $\nabla \widehat{\omega }$ is symmetric and $(\nabla ,E)$ is Codazzi-coupled, $({\nabla }^{†},E)$ is so.

Proof. 

$$\begin{array}{ccc}& & \widehat{\omega }\left(\left({\nabla }_K^{†}E\right)L-\left({\nabla }_L^{†}E\right)K,M\right)=\widehat{\omega }\left(\left({\nabla }_K^{†}E\right)L,M\right)-\widehat{\omega }\left(\left({\nabla }_L^{†}E\right)K,M\right)\hfill \\ & & =\widehat{\omega }\left({\nabla }_K^{†}EL-E{\nabla }_K^{†}L,M\right)-\widehat{\omega }\left({\nabla }_L^{†}EK-E{\nabla }_L^{†}K,M\right)\hfill \\ & & =\widehat{\omega }\left({\nabla }_K^{†}EL,M\right)-\widehat{\omega }\left(E{\nabla }_K^{†}L,M\right)-\widehat{\omega }\left({\nabla }_L^{†}EK,M\right)+\widehat{\omega }\left(E{\nabla }_L^{†}K,M\right)\hfill \\ & & =-\widehat{\omega }\left(M,{\nabla }_K^{†}EL\right)-\widehat{\omega }\left(E{\nabla }_K^{†}L,M\right)+\widehat{\omega }\left(M,{\nabla }_L^{†}EK\right)+\widehat{\omega }\left(E{\nabla }_L^{†}K,M\right)\hfill \\ & & =-K\widehat{\omega }\left(M,EL\right)+\widehat{\omega }\left({\nabla }_{K}M,EL\right)+L\widehat{\omega }\left(M,EK\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_{L}M,EK\right)-\widehat{\omega }\left(E{\nabla }_K^{†}L,M\right)+\widehat{\omega }\left(E{\nabla }_L^{†}K,M\right)\hfill \\ & & =K\widehat{\omega }\left(EL,M\right)-\widehat{\omega }\left(EL,{\nabla }_{K}M\right)-L\widehat{\omega }\left(EK,M\right)\hfill \\ & & +\widehat{\omega }\left(EK,{\nabla }_{L}M\right)-\widehat{\omega }\left(E\left({\nabla }_K^{†}L-{\nabla }_L^{†}K\right),M\right)\hfill \\ & & =K\widehat{\omega }\left(EL,M\right)-\widehat{\omega }\left(EL,{\nabla }_{K}M\right)-L\widehat{\omega }\left(EK,M\right)\hfill \\ & & +\widehat{\omega }\left(EK,{\nabla }_{L}M\right)-\widehat{\omega }\left(E\left(T^{{\nabla }^{†}}\left(K,L\right)+\left[K,L\right]\right),M\right).\hfill \end{array}$$

From $T^{{\nabla }^{†}}=T^{\nabla }$, we obtain

$$\begin{array}{ccc}& & =K\widehat{\omega }\left(EL,M\right)-\widehat{\omega }\left(EL,{\nabla }_{K}M\right)-L\widehat{\omega }\left(EK,M\right)\hfill \\ & & +\widehat{\omega }\left(EK,{\nabla }_{L}M\right)-\widehat{\omega }\left(E\left({\nabla }_{K}L-{\nabla }_{L}K\right),M\right)\hfill \\ & & =\left({\nabla }_K\widehat{\omega }\right)\left(EL,M\right)+\widehat{\omega }\left({\nabla }_{K}EL,M\right)+\widehat{\omega }\left(EL,{\nabla }_{K}M\right)\hfill \\ & & -\widehat{\omega }\left(EL,{\nabla }_{K}M\right)-\left({\nabla }_L\widehat{\omega }\right)\left(EK,M\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_{L}EK,M\right)-\widehat{\omega }\left(EK,{\nabla }_{L}M\right)+\widehat{\omega }\left(EK,{\nabla }_{L}M\right)\hfill \\ & & -\widehat{\omega }\left(E{\nabla }_{K}L,M\right)+\widehat{\omega }\left(E{\nabla }_{L}K,M\right)\hfill \\ & & =\left({\nabla }_K\widehat{\omega }\right)\left(EL,M\right)+\widehat{\omega }\left({\nabla }_{K}EL,M\right)-\left({\nabla }_L\widehat{\omega }\right)\left(EK,M\right)\hfill \\ & & -\widehat{\omega }\left({\nabla }_{L}EK,M\right)-\widehat{\omega }\left(E{\nabla }_{K}L,M\right)+\widehat{\omega }\left(E{\nabla }_{L}K,M\right)\hfill \\ & & =\left({\nabla }_K\widehat{\omega }\right)\left(EL,M\right)+\widehat{\omega }\left(\left({\nabla }_{K}E\right)L,M\right)+\widehat{\omega }\left(E{\nabla }_{K}L,M\right)-\left({\nabla }_L\widehat{\omega }\right)\left(EK,M\right)\hfill \\ & & -\widehat{\omega }\left(\left({\nabla }_{L}E\right)K,M\right)-\widehat{\omega }\left(E{\nabla }_{L}K,M\right)-\widehat{\omega }\left(E{\nabla }_{K}L,M\right)+\widehat{\omega }\left(E{\nabla }_{L}K,M\right)\hfill \\ & & =\left({\nabla }_K\widehat{\omega }\right)\left(EL,M\right)-\left({\nabla }_L\widehat{\omega }\right)\left(EK,M\right)+\widehat{\omega }\left(({\nabla }_{K}E\right)L-\left({\nabla }_{L}E\right)K,M).\hfill \end{array}$$

Hence, the proof is completed. □

Recall that a structure is integrable if $N_E=0$, where $N_E$ is Nijenhuis tensor. In that case, the integrability of the electromagnetic-type structure E is equivalent to $N_E=0$:

$$N_E(K,L)=\left[EK,EL\right]-E\left[EK,L\right]-E\left[K,EL\right]+E^2\left[K,L\right].$$

Proposition 7.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. In the case that $(\nabla ,E)$ is Codazzi-coupled, $\left(\nabla ,P\right)$ is Codazzi-coupled if and only if E is integrable, where $P=E^2$.

Proof. 

From the condition that $(\nabla ,E)$ being Codazzi-coupled, we have

$$\begin{array}{ccc}\hfill N_E\left(K,L\right)& =& \left[EK,EL\right]-E\left[K,EL\right]-E\left[EK,L\right]+E^2\left[K,L\right]\hfill \\ & =& {\nabla }_{EK}EL-{\nabla }_{EL}EK-E\left({\nabla }_{K}EL-{\nabla }_{EL}K\right)\hfill \\ & & -E\left({\nabla }_{EK}L-{\nabla }_{L}EK\right)+P\left({\nabla }_{K}L-{\nabla }_{L}K\right)\hfill \\ & =& \left(\left({\nabla }_{EK}E\right)L+E\left({\nabla }_{EK}\right)L\right)-\left(\left({\nabla }_{EL}E\right)K+E\left({\nabla }_{EL}\right)K\right)\hfill \\ & & -E\left(\left({\nabla }_{K}E\right)L+E{\nabla }_{K}L-{\nabla }_{EL}K\right)\hfill \\ & & -E\left({\nabla }_{EK}L-\left({\nabla }_{L}E\right)K-E\left({\nabla }_{L}K\right)\right)+P\left({\nabla }_{K}L-{\nabla }_{L}K\right)\hfill \\ \hfill & =& \left({\nabla }_{EK}E\right)L+E\left({\nabla }_{EK}\right)L-\left({\nabla }_{EL}E\right)K-E\left({\nabla }_{EL}\right)K-E\left({\nabla }_{K}E\right)L\hfill \\ & & -P{\nabla }_{K}L+E{\nabla }_{EL}K-E{\nabla }_{EK}L+E\left({\nabla }_{L}E\right)K\hfill \\ & & +P{\nabla }_{L}K+P{\nabla }_{K}L-P{\nabla }_{L}K\hfill \\ \hfill & =& \left({\nabla }_{EK}E\right)L-\left({\nabla }_{EL}E\right)K-E\left({\nabla }_{K}E\right)L+E\left({\nabla }_{L}E\right)K\hfill \\ & =& \left({\nabla }_{L}E\right)EK-\left({\nabla }_{K}E\right)EL+E\left(\left({\nabla }_{L}E\right)K-\left({\nabla }_{K}E\right)L\right)\hfill \\ & =& \left({\nabla }_{L}P\right)K-E\left({\nabla }_{L}E\right)K-\left({\nabla }_{K}P\right)L+E\left({\nabla }_{K}E\right)L\hfill \\ & & +E\left(\left({\nabla }_{L}E\right)K-\left({\nabla }_{K}E\right)L\right)\hfill \\ & =& \left({\nabla }_{L}P\right)K-\left({\nabla }_{K}P\right)L.\hfill \end{array}$$

From this, we can say that $\left(\nabla ,P\right)$ is Codazzi-coupled if and only if E is integrable. □

3. (Codazzi) $\mathit{e}-({\mathit{E}}^{\mathbf{4}}=\mathit{I})$ Kaehler Manifold

Next, we search the Codazzi couplings with respect to the torsion-free connection ∇: Codazzi coupling of ∇ with E, Codazzi coupling of ∇ with g, and $\nabla \widehat{\omega }=0$ (that is, ∇ is an almost-symplectic connection). By means of these Codazzi couplings, we plan to approach (Codazzi) $e-(E^4=I)$ Kaehler manifold.

Let $({\mathcal{M}}^k,g)$ be a (pseudo-)Riemannian manifold with the torsion-free connection ∇. If $(\nabla ,g)$ is Codazzi-coupled, the manifold ${\mathcal{M}}^k$ with a statistical structure $(\nabla ,g)$ is named a statistical manifold. This type of manifold was first described by Lauritzen [10]. Statistical manifolds have been extensively researched in affine differential geometry [8,10] and have an important role plays in information geometry. The following theorem is analogue to the theorem given by Fei and Zhang [4] for a Hermitian setting.

Theorem 2.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $({\mathcal{M}}^k,E,g,\widehat{\omega })$. Assuming that

(i) 

$(\nabla ,g)$ is Codazzi-coupled;

(ii) 

$(\nabla ,J)$ and $(\nabla ,P)$ are Codazzi-coupled, where $P=J^2$.

Then, $(M^k,g,E)$ is a (Codazzi) $e-(E^4=I)$ Kaehler manifold.

Proof. 

We shall prove that $E$is integrable and $\widehat{\omega }$ is closed.

From Proposition 7, we have that E is integrable if $(\nabla ,E)$ and $(\nabla ,P)$ are Codazzi-coupled to a torsion-free connection $\nabla .$ Therefore, we only shall prove that $d\widehat{\omega }=0$. We obtain

$$\begin{array}{c}\left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)=M\left(\widehat{\omega }\left(K,L\right)\right)-\widehat{\omega }\left({\nabla }_{M}K,L\right)-\widehat{\omega }\left(K,{\nabla }_{M}L\right)\hfill \\ =M\left(g\left(EK,L\right)\right)-g\left(E{\nabla }_{M}K,L\right)-g\left(EK,{\nabla }_{M}L\right)\hfill \\ =\left({\nabla }_{M}g\right)\left(EK,L\right)+g\left(\left({\nabla }_{M}E\right)K,L\right)\hfill \\ =\mathfrak{C}\left(EK,L,M\right)+g\left(\left({\nabla }_{M}E\right)K,L\right).\hfill \end{array}$$

(1)

Using $\left({\nabla }_{M}g\right)\left(EK,L\right)=Mg\left(EK,L\right)-g\left({\nabla }_{M}EK,L\right)-g\left(EK,{\nabla }_{M}L\right)$, we obtain

$$=Mg\left(EK,L\right)-g\left(\left({\nabla }_{M}E\right)K,L\right)-g\left(E{\nabla }_{M}K,L\right)-g\left(EK,{\nabla }_{M}L\right)).$$

Similarly

$$\begin{array}{c}\left({\nabla }_K\widehat{\omega }\right)\left(L,M\right)=K\left(\widehat{\omega }\left(L,M\right)\right)-\widehat{\omega }\left({\nabla }_{K}L,M\right)-\widehat{\omega }\left(L,{\nabla }_{K}M\right)\hfill \\ =\mathfrak{C}\left(EL,M,K\right)+g\left(\left({\nabla }_{K}E\right)L,M\right)\hfill \\ =Kg\left(EL,M\right)-g\left(\left({\nabla }_{K}E\right)L,M\right)-g\left(E{\nabla }_{K}L,M\right)-g\left(EL,{\nabla }_{K}M\right))\hfill \end{array}$$

(2)

and

$$\begin{array}{c}\left({\nabla }_L\widehat{\omega }\right)\left(M,K\right)=L\left(\widehat{\omega }\left(M,K\right)\right)-\widehat{\omega }\left({\nabla }_{L}M,K\right)-\widehat{\omega }(M,{\nabla }_{L}K)\hfill \\ =\mathfrak{C}\left(EM,K,L\right)+g\left(\left({\nabla }_{L}E\right)M,K\right)\hfill \\ =Lg\left(EM,K\right)-g\left(\left({\nabla }_{L}E\right)M,K\right)-g\left(E{\nabla }_{L}M,K\right)-g\left(EM,{\nabla }_{L}K\right)).\hfill \end{array}$$

(3)

We use (1), (2) and (3) in the following equation

$$d\widehat{\omega }\left(K,L,M\right)=\left({\nabla }_K\widehat{\omega }\right)\left(L,M\right)+\left({\nabla }_L\widehat{\omega }\right)\left(M,K\right)+\left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)$$

(see also [4]). In addition, we find

$$\begin{array}{ccc}\hfill d\widehat{\omega }\left(K,L,M\right)& =& \left({\nabla }_K\widehat{\omega }\right)\left(L,M\right)+\left({\nabla }_L\widehat{\omega }\right)\left(M,K\right)+\left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)\hfill \\ & =& \mathfrak{C}\left(EK,L,M\right)+\mathfrak{C}\left(EL,M,K\right)+\mathfrak{C}\left(EM,K,L\right)\hfill \\ & & +g\left(\left({\nabla }_{M}E\right)K,L\right)+g\left(\left({\nabla }_{K}E\right)L,M\right)+g\left(\left({\nabla }_{L}E\right)M,K\right)\hfill \end{array}$$

and similarly

$$\begin{array}{ccc}\hfill d\widehat{\omega }\left(M,L,K\right)& =& \left({\nabla }_M\widehat{\omega }\right)\left(L,K\right)+\left({\nabla }_L\widehat{\omega }\right)\left(K,M\right)+\left({\nabla }_K\widehat{\omega }\right)\left(M,L\right)\hfill \\ & =& \mathfrak{C}\left(EM,L,K\right)+\mathfrak{C}\left(EL,K,M\right)+\mathfrak{C}\left(EK,M,L\right)\hfill \\ & & +g\left(\left({\nabla }_{K}E\right)M,L\right)+g\left(\left({\nabla }_{M}E\right)L,K\right)+g\left(\left({\nabla }_{L}E\right)K,M\right).\hfill \end{array}$$

Moreover, we know that $(\nabla ,g)$ is Codazzi-coupled, then $\mathfrak{C}$ is totally symetric [4]. Using $(\nabla ,E)$ being Codazzi-coupled and $d\widehat{\omega }$ being totally skew-symetric, we obtain

$$\begin{array}{ccc}\hfill d\widehat{\omega }\left(K,L,M\right)-d\widehat{\omega }\left(M,L,K\right)& =& 0\hfill \\ \hfill -2d\widehat{\omega }\left(K,L,M\right)=0\Rightarrow d\widehat{\omega }\left(K,L,M\right)& =& 0.\hfill \end{array}$$

This gives a result. □

4. $\mathit{E}-$Parallel Affine Connections

Let ∇ be an affine connection and E be an electromagnetic-type structure. If

$${\nabla }_{K}EL=E{\nabla }_{K}L$$

is satisfied for any vector fields $K,L$ on ${\mathcal{M}}^k$, ∇ is named an $E-$parallel affine connection on ${\mathcal{M}}^k$.

Proposition 8.

Let ∇ be an affine connection on the metric $\left(E^4=I\right)-$manifold $\left({\mathcal{M}}^k,E,g,\widehat{\omega }\right)$. ${\nabla }^{*}$ and ${\nabla }^{†}$ assign, respectively, $g-$conjugation and $\widehat{\omega }-$conjugation of ∇ on ${\mathcal{M}}^k$. In that case,

(i) 

${\nabla }^{*}$ is $E-$parallel if and only if ∇ is so.

(ii) 

${\nabla }^{†}$ is $E-$parallel if and only if ∇ is so.

Proof. (i) From the definitions of $g-$conjugation and $E-$parallel, it is obtained that

$$\begin{array}{ccc}& & \widehat{\omega }\left({\nabla }_K^{*}EL-E{\nabla }_K^{*}L,M\right)=\widehat{\omega }\left({\nabla }_K^{*}EL,M\right)-\widehat{\omega }\left(E{\nabla }_K^{*}L,M\right)\hfill \\ & & =g\left(E{\nabla }_K^{*}EL,M\right)-g\left(E\left(E{\nabla }_K^{*}L\right),M\right)=-g\left({\nabla }_K^{*}EL,EM\right)+g\left(E{\nabla }_K^{*}L,EM\right)\hfill \\ & & =-g\left(EM,{\nabla }_K^{*}EL\right)-g\left(E^{2}M,{\nabla }_K^{*}L\right)\hfill \\ & & =-Kg\left(EM,EL\right)+g\left({\nabla }_{K}EM,EL\right)-Kg\left(E^{2}M,L\right)+g\left({\nabla }_K{E}^{2}M,L\right)\hfill \\ & & =-Kg\left(EM,EL\right)+g\left({\nabla }_{K}EM,EL\right)+Kg\left(EM,EL\right)+g\left({\nabla }_K{E}^{2}M,L\right)\hfill \\ & & =g\left(EL,{\nabla }_{K}EM\right)+g\left(\left({\nabla }_{K}E\right)EM,L\right)+g\left(E{\nabla }_{K}EM,L\right)\hfill \\ & & =g\left(EL,{\nabla }_{K}EM\right)+g\left(\left({\nabla }_{K}E\right)EM,L\right)-g\left({\nabla }_{K}EM,EL\right)\hfill \\ & & =g\left(\left({\nabla }_{K}E\right)EM,L\right).\hfill \end{array}$$

Thus, ${\nabla }_K^{*}EL=E{\nabla }_K^{*}L$ if and only if ${\nabla }_{K}EM=E{\nabla }_{K}M$.

(ii) Firstly, we have

$$\widehat{\omega }\left(EK,L\right)=g\left(E\left(EK\right),L\right)=-g\left(EK,EL\right)=-\widehat{\omega }\left(K,EL\right).$$

From the definitions of $\widehat{\omega }-$conjugation and the above equation, we obtain

$$\begin{array}{ccc}& & \widehat{\omega }\left({\nabla }_K^{†}EL-E{\nabla }_K^{†}L,M\right)=\widehat{\omega }\left({\nabla }_K^{†}EL,M\right)-\widehat{\omega }\left(E{\nabla }_K^{†}L,M\right)\hfill \\ & & =-\widehat{\omega }\left(M,{\nabla }_K^{†}EL\right)+\widehat{\omega }\left(M,E{\nabla }_K^{†}L\right)=-\widehat{\omega }\left(M,{\nabla }_K^{†}EL\right)-\widehat{\omega }\left(EM,{\nabla }_K^{†}L\right)\hfill \\ & & =-K\widehat{\omega }\left(M,EL\right)+\widehat{\omega }\left({\nabla }_{K}M,EL\right)-K\widehat{\omega }\left(EM,L\right)+\widehat{\omega }\left({\nabla }_{K}EM,L\right)\hfill \\ & & =-K\widehat{\omega }\left(M,EL\right)+\widehat{\omega }\left({\nabla }_{K}M,EL\right)+K\widehat{\omega }\left(M,EL\right)+\widehat{\omega }\left({\nabla }_{K}EM,L\right)\hfill \\ & & =-\widehat{\omega }\left(EL,{\nabla }_{K}M\right)-\widehat{\omega }\left(L,{\nabla }_{K}EM\right)=\widehat{\omega }\left(L,E{\nabla }_{K}M\right)-\widehat{\omega }\left(L,{\nabla }_{K}EM\right),\hfill \end{array}$$

which completes the proof. □

Proposition 9.

Let ∇ be an $E-$parallel affine connection on the metric $\left(E^4=I\right)-$manifold $\left({\mathcal{M}}^k,E,g,\widehat{\omega }\right)$. ${\nabla }^{*}$and ${\nabla }^{†}$assign respectively $g-$conjugation and $\widehat{\omega }-$conjugation of ∇ on ${\mathcal{M}}^k$. The followings are provided:

(i) 

${\nabla }^{†}={\nabla }^{*}$;

(ii) 

$\left(\nabla ,\widehat{\omega }\right)$ is a Codazzi-coupled if and only if $\left(\nabla ,g\right)$ is so.

Proof. (i) From the definitions of $g-$conjugation, $\widehat{\omega }-$conjugation and $E-$parallel, we obtain

$$\begin{array}{ccc}& & M\widehat{\omega }\left(K,L\right)=\widehat{\omega }\left({\nabla }_{M}K,L\right)+\widehat{\omega }\left(K,{\nabla }_M^{†}L\right)\hfill \\ & & Mg\left(EK,L\right)=g\left(E{\nabla }_{M}K,L\right)+g\left(EK,{\nabla }_M^{†}L\right)\hfill \\ & & Mg\left(EK,L\right)-g\left(E{\nabla }_{M}K,L\right)=g\left(EK,{\nabla }_M^{†}L\right)\hfill \\ & & Mg\left(EK,L\right)-g\left({\nabla }_{M}EK,L\right)=g\left(EK,{\nabla }_M^{†}L\right)\hfill \\ & & g\left(EK,{\nabla }_M^{*}L\right)=g\left(EK,{\nabla }_M^{†}L\right)\iff {\nabla }_M^{*}L={\nabla }_M^{†}L\iff {\nabla }^{*}={\nabla }^{†}.\hfill \end{array}$$

(ii) From the Codazzi equation, we have

$$\begin{array}{ccc}\hfill \left({\nabla }_M\widehat{\omega }\right)\left(K,L\right)& =& \left({\nabla }_K\widehat{\omega }\right)\left(M,L\right)\hfill \\ \hfill M\widehat{\omega }\left(K,L\right)-\widehat{\omega }\left({\nabla }_{M}K,L\right)-\widehat{\omega }\left(K,{\nabla }_{M}L\right)& =& K\widehat{\omega }\left(M,L\right)-\widehat{\omega }\left({\nabla }_{K}M,L\right)-\widehat{\omega }\left(M,{\nabla }_{K}L\right)\hfill \\ \hfill Mg\left(EK,L\right)-g\left(E{\nabla }_{M}K,L\right)-g\left(EK,{\nabla }_{M}L\right)& =& Kg\left(EM,L\right)-g\left(E{\nabla }_{K}M,L\right)-g\left(EM,{\nabla }_{K}L\right)\hfill \end{array}$$

$$\begin{array}{ccc}& & -Mg\left(K,EL\right)+g\left({\nabla }_{M}K,EL\right)+g\left(M,E{\nabla }_{M}L\right)\hfill \\ & =& -Kg\left(M,EL\right)+g\left({\nabla }_{K}M,EL\right)+g\left(M,E{\nabla }_{K}L\right)-Mg\left(K,EL\right)\hfill \\ & & +g\left({\nabla }_{M}K,EL\right)+g\left(K,{\nabla }_{M}EL\right)\hfill \\ & =& -Kg\left(M,EL\right)+g\left({\nabla }_{K}M,EL\right)+g\left(M,{\nabla }_{K}EL\right)\hfill \\ & & \left({\nabla }_{M}g\right)\left(K,EL\right)=\left({\nabla }_{K}g\right)\left(M,EL\right),\hfill \end{array}$$

which gives the proof. □

To give results about statistical structures, for a moment, we shall assume a torsion-free $E-$parallel affine connection ∇.

Theorem 3.

Let ∇ be a torsion-free $E-$parallel affine connection on the metric $\left(E^4=I\right)-$manifold $\left({\mathcal{M}}^k,E,g,\widehat{\omega }\right)$, by ${\nabla }^{*}$and ${\nabla }^{†}$we assign respectively $g-$conjugation and $\widehat{\omega }-$conjugation of ∇ on ${\mathcal{M}}^k$. If $(\nabla ,\widehat{\omega })$is a Codazzi-coupled, the followings are provided:

(i) 

$({\nabla }^{*},g)$is a statistical structure;

(ii) 

$\left({\nabla }^{†},g\right)$ is a statistical structure.

Proof. 

The proof is obtained from Proposition 9 and Proposition 2.10 in [4]. □

Theorem 4.

Let ∇ be a torsion-free $E-$parallel affine connection on the metric $\left(E^4=I\right)-$manifold $\left({\mathcal{M}}^k,E,g,\widehat{\omega }\right)$.${\nabla }^{*}$assigns the $g-$conjugation of ∇ on ${\mathcal{M}}^k$. In the case that $(\nabla ,\widehat{\omega })$ being a Codazzi coupled, $\left({\nabla }^{*},\widehat{\omega }\right)$ is a Codazzi-coupled if and only if $\left({\nabla }_{EM}g\right)\left(K,L\right)=\left({\nabla }_{M}g\right)\left(EK,L\right)$.

Proof. 

Let ∇ be a torsion-free $E-$parallel affine connection and $(\nabla ,\widehat{\omega })$ be a Codazzi coupled. Then, from Proposition 9, we also have that $(\nabla ,g)$ is Codazzi-coupled. Moreover, from Proposition 8, we can say that ${\nabla }^{*}$ is $E-$parallel. Using all the above, we obtain

$$\begin{array}{ccc}& & \left({\nabla }_M^{*}\widehat{\omega }\right)\left(K,L\right)-\left({\nabla }_K^{*}\widehat{\omega }\right)\left(M,L\right)=M\widehat{\omega }\left(K,L\right)-\widehat{\omega }\left({\nabla }_M^{*}K,L\right)-\widehat{\omega }\left(K,{\nabla }_M^{*}L\right)\hfill \\ & & -K\widehat{\omega }\left(M,L\right)+\widehat{\omega }\left({\nabla }_K^{*}M,L\right)+\widehat{\omega }\left(M,{\nabla }_K^{*}L\right)\hfill \\ & & =Mg\left(EK,L\right)-g\left(E{\nabla }_M^{*}K,L\right)-g\left(EK,{\nabla }_M^{*}L\right)\hfill \\ & & -Kg\left(EM,L\right)+g\left(E{\nabla }_K^{*}M,L\right)+g\left(EM,{\nabla }_K^{*}L\right)\hfill \\ & & =g\left({\nabla }_M^{*}EK,L\right)+g\left(EK,{\nabla }_{M}L\right)-g\left(E{\nabla }_M^{*}K,L\right)-g\left(EK,{\nabla }_M^{*}L\right)\hfill \\ & & -g\left({\nabla }_K^{*}EM,L\right)-g\left(EM,{\nabla }_{K}L\right)+g\left(E{\nabla }_K^{*}M,L\right)+g\left(EM,{\nabla }_K^{*}L\right)\hfill \\ & & =g\left(\left({\nabla }_M^{*}E\right)K,L\right)+g\left(E{\nabla }_M^{*}K,L\right)+g\left(EK,{\nabla }_{M}L\right)-g\left(E{\nabla }_M^{*}K,L\right)-g\left(EK,{\nabla }_M^{*}L\right)\hfill \\ & & -g\left(\left({\nabla }_K^{*}E\right)M,L\right)-g\left(E{\nabla }_K^{*}M,L\right)-g\left(EM,{\nabla }_{K}L\right)+g\left(E{\nabla }_K^{*}M,L\right)+g\left(EM,{\nabla }_K^{*}L\right)\hfill \\ & & =g\left(EK,{\nabla }_{M}L\right)-g\left(EK,{\nabla }_M^{*}L\right)-g\left(EM,{\nabla }_{K}L\right)+g\left(EM,{\nabla }_K^{*}L\right)\hfill \\ & & =g\left(EK,{\nabla }_{M}L\right)-Mg\left(EK,L\right)+g\left({\nabla }_{M}EK,L\right)-g\left(EM,{\nabla }_{K}L\right)\hfill \\ & & +Kg\left(EM,L\right)-g\left({\nabla }_{K}EM,L\right)=-\left({\nabla }_{M}g\right)\left(EK,L\right)+\left({\nabla }_{K}g\right)\left(EM,L\right)\hfill \\ & & =-\left({\nabla }_{M}g\right)\left(EK,L\right)+\left({\nabla }_{EM}g\right)\left(K,L\right).\hfill \end{array}$$

□