Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds

Abstract

In this work, we have characterized the frame bundle $FM$ admitting metallic structures on almost quadratic $\varphi$-manifolds ${\varphi }^2=p\varphi +qI-q\eta \otimes \zeta$, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic $\varphi$-structure to the metallic structure on $FM$ are constructed. We also prove the existence of a metallic structure on $FM$ with the aid of the $\tilde{J}$ tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field $\tilde{J}$ on $FM$. Finally, we construct an example of it to finish.

1. Introduction

Numerous types of f-structures on a differentiable manifold M have been studied by Yano [1], Ishihara and Yano [2], Blair [3], Nakagawa [4] and others. Yano proposed the notion of an f-structure obeying $f^3+f=0,f$ is a tensor field of type (1,1), which is the generalization of an almost complex structure and an almost contact structure [5] and investigated some basic results of it. Later, Goldberg and Yano [6] and Goldberg and Perridis [7] defined a polynomial structure $P\left(J\right)=J^n+a_n{J}^{n-1}+...+a_{2}J+a_{1}I,$ where $a_1,a_2,\cdots ,a_n$ are real numbers, J is a tensor field of type (1,1) and I is an identity tensor field of type (1,1) on M. Moreover, some important polynomial structures such as an $f(3,\epsilon )$-structure [8], a general quadratic structure [9], an almost complex structure and an almost product structure [1], $\varphi (4,\pm 2)$-structures [10] and an almost r-contact structure [11] are studied and the fundamental results are established in these papers.

Recently, the polynomial structure $J^2=pJ+qI,p,q\in \mathbb{N},$ where $\mathbb{N}$ is the set of natural numbers, of degree 2 is known as a metallic structure on M [12,13,14]. For specific values of p and q, metallic structures become prominent structures given below:

p	q	Structure
0	1	an almost product structure [15]
0	−1	an almost complex structure [16,17]
1	1	a golden structure [18,19]
2	1	a silver structure [20]

Hretceanu and Crasmareanu [21] initiated the study of golden and metallic structures on a Riemannian manifold and interpreted the geometry of submanifolds admitting both structures on M. The various geometric properties of such structures in a metallic (and golden) Riemannian manifold and a metallic (and golden) warped product Riemannian manifold were studied in [22,23,24,25,26]. Debnath and Konar [27] defined a new type of structure named as an almost quadratic $\varphi$-structure $(\varphi ,\zeta ,\eta )$ on M and studied some geometric properties of such structures. Next, Gonul et al. [28] established the relationship between an almost quadratic metric $\varphi$-structure and a metallic structure on M. Most recently, Gok et al. [29] defined a generalized structure namely $f_{(a,b)}(3,2,1)$-structures on manifolds and construct a framed $f_{(a,b)}(3,2,1)$-structures on M.

On the other hand, let M be an m-dimensional differentiable manifold, $TM$ its tangent bundle and $FM$ its frame bundle. The notion of the mappings, namely vertical, complete and horzontal lifts from the manifold M to its tangent bundle $TM$ were introduced by Sasaki [30], Yano and Ishihara [31] and Yano and Davis [32]. Kabayashi and Nomizu [33], Mok [34] and Okubo [35] have studied the complete lift of a vector field $\mathcal{A}$ to $FM$. The geometric structures such as an almost contact metric structure $(\varphi ,\zeta ,\eta ,g)$, and almost complex structures J on $FM$ have been studied by Bonome et al. [16], who established the integrability and normality of such structures on $FM$.

In [36], Khan has introduced a tensor field $\tilde{J}$ on $FM$ and proved that $\tilde{J}$ is a metallic structure on $FM$. The integrability condition for the diagonal and horizontal lifts of the metallic structure $\tilde{J}$ on $FM$ is established. The geometric structures on $FM$ have been studied by Cordero et al. [37], Kowalski [38], Sekizawa [39], Kowalski and Sekizawa [40], Niedzialomski [41], Lachieze-Rey [42], Khan [43,44,45] and many more.

The main objective of this paper can be summarized as follows:

• We study the complete lifts of an almost quadratic $\varphi$-structure to the metallic structure on $FM$.
• We establish the existence of a metallic structure on $FM$ in the tensor field $\tilde{J}$, which we define.
• We obtain results on the 2-Form and its derivative on $FM$.
• We derive the expressions of the Nijenhuis tensor of a tensor field $\tilde{J}$ on $FM$.
• We construct an example related to it.

Remark: ${\Im }_a^b\left(M\right)$ and ${\Im }_a^b\left(FM\right)$ are symbolized as the set of all $(a,b)$-type tensor fields in M and $FM$ respectively [17].

2. Preliminaries

Let $F,\mathcal{A},f$ and $\eta$ be a tensor field of type (1,1), a vector field, a function and a 1-form, respectively, on M. The horizontal, vertical and $\alpha$-vertical lifts of $F,\mathcal{A},f$ and $\eta$ are represented by $F^H,{\mathcal{A}}^H,{\mathcal{A}}^{\left(\alpha \right)},f^H,{\eta }^V$ and ${\eta }^{H_{\alpha }}$ on $FM$ and they are expressed in terms of partial differential equations as [16,17]

$$\begin{array}{ccc}\hfill {\mathcal{A}}^H& =& {\mathcal{A}}^i\frac{\partial }{\partial {\mathcal{A}}^i}-{\mathcal{A}}^i{\Gamma }_{ik}^h{\mathcal{A}}_{\alpha }^k\frac{\partial }{\partial {\mathcal{A}}^h},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\mathcal{A}}^{\left(\alpha \right)}& =& {\mathcal{A}}^i\frac{\partial }{\partial {\mathcal{A}}_{\alpha }^i},\hfill \\ \hfill F^H& =& F_j^h\frac{\partial }{\partial {\mathcal{A}}^h}\otimes d{x}^j+{\mathcal{A}}_{\alpha }^k({\Gamma }_{jk}^i{F}_i^h-{\Gamma }_{ik}^h{F}_j^i)\frac{\partial }{\partial {\mathcal{A}}_{\alpha }^h},\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & \otimes d{x}^j+{\delta }_{\alpha }^{\beta }{F}_j^h\frac{\partial }{\partial {\mathcal{A}}_{\alpha }^h}\otimes d{X}_{\beta }^j,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\eta }^V& =& {\eta }_{i}d{x}^i,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\eta }^{H_{\alpha }}& =& {\mathcal{A}}_{\alpha }^j{\Gamma }_{ij}^h{\eta }_{h}d{x}^i+{\eta }_{i}d{X}_{\alpha }^i,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\mathcal{A}}^H& =& \sum _{\alpha =1}^m({\mathcal{A}}_{\alpha }^j{\Gamma }_{ij}^h{\eta }_{h}d{x}^i+{\eta }_{i}d{X}_{\alpha }^i),\hfill \end{array}$$

(6)

where ${\Gamma }_{ij}^h,{\mathcal{A}}^i,F_j^h$ and ${\eta }_i$ are the local components of a linear connection ∇, $\mathcal{A}$, F and $\eta$, respectively on M.

Proposition 1. 

$\forall \mathcal{A},\mathcal{B}\in {\Im }_0^1\left(M\right)$, by using mathematical operators, we have the following

$$\begin{array}{ccc}\hfill {\mathcal{A}}^H\left(f^V\right)& =& {\left(\mathcal{A}\left(f\right)\right)}^V),\hfill \\ \hfill {\mathcal{A}}^{\left(\alpha \right)}\left(f^V\right)& =& 0,\hfill \\ \hfill F^H\left({\mathcal{A}}^{\left(\alpha \right)}\right)& =& {\left(F\left(\mathcal{A}\right)\right)}^{\alpha },\hfill \\ \hfill F^H\left({\mathcal{A}}^H\right)& =& {\left(F\left(\mathcal{A}\right)\right)}^H,\hfill \\ \hfill {\eta }^V\left({\mathcal{A}}^H\right)& =& {\left(F\left(\mathcal{A}\right)\right)}^V,\hfill \\ \hfill {\eta }^V\left({\mathcal{A}}^{\left(\alpha \right)}\right)& =& 0,\hfill \\ \hfill {\eta }^{H_{\alpha }}\left({\mathcal{A}}^H\right)& =& 0,\hfill \\ \hfill {\eta }^{H_{\alpha }}\left({\mathcal{A}}^{\left(\beta \right)}\right)& =& {\delta }_{\alpha }^{\beta }{\left(\eta \left(\mathcal{A}\right)\right)}^V,\hfill \end{array}$$

(7)

where $\alpha ,\beta =1,\cdots ,m$ and ${\delta }_{\beta }^{\alpha }$ denotes the Kronecker delta.

Proposition 2. 

Let $\forall \mathcal{A},\mathcal{B}\in {\Im }_0^1\left(M\right)$. Then, we have the following

$$\begin{array}{ccc}\hfill [{\mathcal{A}}^{\left(\alpha \right)},{\mathcal{B}}^{\left(\beta \right)}]& =& 0,\hfill \\ \hfill [{\mathcal{A}}^H,{\mathcal{B}}^{\left(\alpha \right)}]& =& {\left({\nabla }_{X}Y\right)}^{\left(\alpha \right)},\hfill \\ \hfill [{\mathcal{A}}^H,{\mathcal{B}}^H]& =& {[\mathcal{A},\mathcal{B}]}^H-\gamma R(\mathcal{A},\mathcal{B}),\hfill \end{array}$$

(8)

where $R(\mathcal{A},\mathcal{B})=[{\nabla }_{\mathcal{A}},{\nabla }_{\mathcal{B}}]-{\nabla }_{[\mathcal{A},\mathcal{B}]}$, R is the curvature tensor of ∇.

Let g be a Riemannian metric on a Riemannian manifold M and $g^D$ its diagonal metric on $FM$, then

$$\begin{array}{ccc}\hfill g^D({\mathcal{A}}^H,{\mathcal{B}}^H)& =& {\left\{g(\mathcal{A},\mathcal{B})\right\}}^V,\hfill \\ \hfill g^D({\mathcal{A}}^H,{\mathcal{B}}^{\left(\alpha \right)})& =& 0,\hfill \\ \hfill g^D({\mathcal{A}}^{\left(\alpha \right)},{\mathcal{B}}^{\left(\beta \right)})& =& {\delta }^{\alpha \beta }{\left\{g(\mathcal{A},\mathcal{B})\right\}}^V,\forall \alpha ,\beta =1,\cdots ,m\hfill \end{array}$$

(9)

and

$$\begin{array}{ccc}\hfill 2g^D({\tilde{\nabla }}_{\tilde{\mathcal{A}}}\tilde{\mathcal{B}},\tilde{\mathcal{C}})& =& \tilde{\mathcal{A}}\left(g^D(\tilde{\mathcal{B}},\tilde{\mathcal{C}})\right)+\tilde{\mathcal{B}}\left(g^D(\tilde{\mathcal{C}},\tilde{\mathcal{A}})\right)-\tilde{\mathcal{C}}\left(g^D(\tilde{\mathcal{A}},\tilde{\mathcal{B}})\right)\hfill \\ & +& g^D([\tilde{\mathcal{A}},\tilde{\mathcal{B}}],\tilde{\mathcal{C}})+g^D([\tilde{\mathcal{C}},\tilde{\mathcal{A}}],\tilde{\mathcal{B}})+g^D(\tilde{\mathcal{A}},[\tilde{\mathcal{C}},\tilde{\mathcal{B}}]),\hfill \end{array}$$

(10)

$\forall \tilde{\mathcal{A}},\tilde{\mathcal{B}}\in {\Im }_0^1\left(FM\right)$, where ∇ and $\tilde{\nabla }$ represent the Levi-Civita connection of $(M,g)$ and $(FM,g^D)$, respectively.

Proposition 3. 

$\forall \mathcal{A},\mathcal{B}\in {\Im }_0^1\left(M\right)$, by using mathematical operators, we have the following

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{{\mathcal{A}}^{\left(\alpha \right)}}{\mathcal{B}}^{\left(\beta \right)}& =& 0,\hfill \\ \hfill g^D({\tilde{\nabla }}_{{\mathcal{A}}^{\left(\alpha \right)}}{\mathcal{B}}^H,{\mathcal{C}}^{\left(\beta \right)})& =& 0,\hfill \\ \hfill g^D({\tilde{\nabla }}_{{\mathcal{A}}^{\left(\alpha \right)}}{\mathcal{B}}^H,{\mathcal{C}}^H)& =& -\frac{1}{2}{g}^D(\gamma R(\mathcal{C},\mathcal{B}),{\mathcal{A}}^{\left(\alpha \right)}),\hfill \\ \hfill g^D({\tilde{\nabla }}_{{\mathcal{A}}^H}{\mathcal{B}}^{\left(\alpha \right)},{\mathcal{C}}^{\left(\beta \right)})& =& {\delta }^{\alpha \beta }{\left\{g({\nabla }_{\mathcal{A}}\mathcal{B},\mathcal{C})\right\}}^V,\hfill \\ \hfill g^D({\tilde{\nabla }}_{{\mathcal{A}}^H}{\mathcal{B}}^{\left(\alpha \right)},{\mathcal{C}}^H)& =& -\frac{1}{2}{g}^D(\gamma R(\mathcal{C},\mathcal{A}),{\mathcal{B}}^{\left(\alpha \right)}),\hfill \\ \hfill g^D({\tilde{\nabla }}_{{\mathcal{A}}^H}{\mathcal{B}}^H,{\mathcal{C}}^{\left(\alpha \right)})& =& -\frac{1}{2}{g}^D(\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^{\left(\alpha \right)}),\hfill \\ \hfill g^D({\tilde{\nabla }}_{{\mathcal{A}}^H}{\mathcal{B}}^H,{\mathcal{C}}^H)& =& {\left\{g({\nabla }_{\mathcal{A}}\mathcal{B},\mathcal{C})\right\}}^V.\hfill \end{array}$$

(11)

2.1. Metallic Structure

If a (1, 1) tensor field J obeying

$$J^2=pJ+qI,p,q\in \mathbb{N},$$

(12)

where $\mathbb{N}$ is the set of natural numbers and I is an identity operator, determines a polynomial structure on a manifold M, the structure is referred to as metallic. A metallic manifold is defined as $(M,J)$ when a manifold M possesses a metallic structure (MS) J.

The Nijenhuis tensor $N_J$ of J is expressed as

$$N_J(\mathcal{A},\mathcal{B})=[J\mathcal{A},J\mathcal{B}]-J[J\mathcal{A},\mathcal{B}]-J[\mathcal{A},J\mathcal{B}]+J^2[\mathcal{A},\mathcal{B}],$$

(13)

$\forall \mathcal{A},\mathcal{B}\in {\Im }_0^1\left(M\right)$.

2.2. Almost Quadratic $\varphi$-Structure

An $m(=2n+1)$-dimensional differentiable manifold M with a non-null tensor field $\varphi$ of type (1,1), a 1-form $\eta$ and a vector field $\zeta$ on M satisfies

$$\begin{array}{ccc}\hfill {\varphi }^2& =& p\varphi +qI-q\eta \otimes \zeta ,p^2+4q\ne 0,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \eta \left(\zeta \right)& =& 1,\eta \circ \varphi =0,\varphi \left(\zeta \right)=0,\hfill \end{array}$$

(15)

where p is an arbitrary constant and $q\ne 0$. The structure $(\varphi ,\zeta ,\eta )$ is called an almost quadratic $\varphi$-structure on M and the manifold $(M,\varphi ,\zeta ,\eta )$ is called an almost quadratic $\varphi$-manifold [27,28].

Furthermore,

$$g(\varphi \mathcal{A},\mathcal{B})=g(\mathcal{A},\varphi \mathcal{B})$$

(16)

and

$$g(\varphi \mathcal{A},\varphi \mathcal{B})=pg(\varphi \mathcal{A},\mathcal{B})+qg(\mathcal{A},\mathcal{B})-q\eta \left(\mathcal{A}\right)\eta \left(\mathcal{B}\right)).$$

(17)

The structure $(\varphi ,\zeta ,\eta ,g)$ is referred to as an almost quadratic metric $\varphi$-structure and $(M,\varphi ,\zeta ,\eta ,g)$ is called an almost quadratic metric $\varphi$-manifold.

In addition, the 1-form $\eta$ is associated with g such that

$$g(\mathcal{A},\zeta )=\eta \left(\mathcal{A}\right)$$

and the fundamental 2-Form $\Phi$ is given by [3]

$$\Phi (\mathcal{A},\mathcal{B})=g(\mathcal{A},\varphi \mathcal{B}).$$

(18)

The Nijenhuis tensor of $(\varphi ,\zeta ,\eta )$ is denoted by $N_{\varphi }$ and is given by

$$N_{\varphi }(\mathcal{A},\mathcal{B})=[\varphi \mathcal{A},\varphi \mathcal{B}]-\varphi [\varphi \mathcal{A},\mathcal{B}]-\varphi [\mathcal{A},\varphi \mathcal{B}]+{\varphi }^2[\mathcal{A},\mathcal{B}],$$

(19)

$\forall \mathcal{A},\mathcal{B}\in {\Im }_0^1\left(M\right)$.

3. Proposed Theorems on FM Admitting Metallic Structures on Almost Quadratic $\varphi$-Manifolds

In this section, we construct the complete lifts of an almost quadratic $\varphi$-structure to the metallic structure on $FM$.

Next, we obtain the results on the 2-Form and its derivative on $FM$.

Boname et al. [16] proposed and gave the definition of $\tilde{J}$ on $FM$ as

$$\begin{array}{ccc}\hfill \tilde{J}& =& {\varphi }^H+\sum _{\alpha =1}^n{\eta }^{H_{\alpha }}\otimes {\zeta }^{(\alpha +n)}-\sum _{\alpha =1}^n{\eta }^{H_{\alpha +n}}\otimes {\zeta }^{\left(\alpha \right)}\hfill \\ & +& {\eta }^V\otimes {\zeta }^{(2n+1)}-{\eta }^{H_{2n+1}}\otimes {\zeta }^H.\hfill \end{array}$$

(20)

Recently, Khan [36] proposed and gave the definition of the tensor field $\tilde{J}$ on $FM$ as

$$\begin{array}{ccc}\hfill \tilde{J}& =& \frac{p}{2}I-\left(\frac{2{\sigma }_p^q-p}{2}\right)[{\varphi }^H+\sum _{\alpha =1}^n{\eta }^{H_{\alpha }}\otimes {\zeta }^{(\alpha +n)}\hfill \\ & -& \sum _{\alpha =1}^n{\eta }^{H_{\alpha +n}}\otimes {\zeta }^{\left(\alpha \right)}+{\eta }^V\otimes {\zeta }^{(2n+1)}-{\eta }^{H_{2n+1}}\otimes {\zeta }^H],\hfill \end{array}$$

(21)

where $\eta ={\eta }_{i}d{x}^i$, ${\eta }^V={\eta }_{i}d{x}^i$ and ${\eta }^{H_{\alpha }}={\mathcal{A}}_{\alpha }^j{\Gamma }_{ij}^h{\eta }_{h}d{x}^i+{\eta }_{i}d{x}_{\alpha }^i$.

Motivated by the above definitions, let us introduce a tensor field $\tilde{J}$ of type (1,1) on $FM$ as

$$\begin{array}{ccc}\hfill \tilde{J}& =& \frac{p}{2}I-A[{\varphi }^H+\sqrt{q}\{\sum _{\alpha =1}^n{\eta }^{H_{\alpha }}\otimes {\zeta }^{(\alpha +n)}\hfill \\ & -& \sum _{\alpha =1}^n{\eta }^{H_{\alpha +n}}\otimes {\zeta }^{\left(\alpha \right)}+{\eta }^V\otimes {\zeta }^{(2n+1)}-{\eta }^{H_{2n+1}}\otimes {\zeta }^H\left\}\right],\hfill \end{array}$$

(22)

where $A=\frac{2{\sigma }_p^q-p}{2\sqrt{p{\varphi }^H+q}}$, $\eta ={\eta }_{i}d{x}^i$,

${\eta }^V={\eta }_{i}d{x}^i$ and ${\eta }^{H_{\alpha }}={\mathcal{A}}_{\alpha }^j{\Gamma }_{ij}^h{\eta }_{h}d{x}^i+{\eta }_{i}d{x}_{\alpha }^i$.

Theorem 1. 

Let $\tilde{\mathcal{A}}$ be a vector field on $FM$. Then $\tilde{J}$ given by (22) is a metallic structure on $FM$.

Proof. 

To prove that $\tilde{J}$ defined in (22) is a metallic structure, we have to prove that

$${\tilde{J}}^2\tilde{\mathcal{A}}=p\tilde{J}\left(\tilde{\mathcal{A}}\right)+qI;p,q\in \mathbb{N}.$$

(23)

□

Taking the horizontal lift ${\mathcal{A}}^H$ and ${\beta }^{th}$-vertical lift ${\mathcal{A}}^{\left(\beta \right)}$ for each $\beta =1,\cdots 2n+1$ on both sides of (22), we infer

$$\begin{array}{ccc}\hfill \tilde{J}\left({\mathcal{A}}^{\left(\beta \right)}\right)& =& \frac{p}{2}{\mathcal{A}}^{\left(\beta \right)}-A[{\left(\varphi \mathcal{A}\right)}^{\left(\beta \right)}+\sqrt{q}\{\epsilon \left(\beta \right){\zeta }^{(\beta +\epsilon (\beta \left)n\right)}\hfill \\ & -& {\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^V{\xi }^H\left\}\right],\hfill \end{array}$$

(24)

where

$$\epsilon \left(\beta \right)=\left\{\begin{array}{cc}1,\hfill & \beta \le n,\hfill \\ -1,\hfill & n<\beta \le 2n,\hfill \\ 0,\hfill & \beta =2n+1,\hfill \end{array}\right.$$

(25)

and

$$\tilde{J}\left({\mathcal{A}}^H\right)=\frac{p}{2}{\mathcal{A}}^H-A[{\left(\varphi \mathcal{A}\right)}^H+\sqrt{q}\left\{\eta {\left(\mathcal{A}\right)}^V{\zeta }^{(2n+1)}\right\}].$$

(26)

In view of (22), we provide

$$\begin{array}{ccc}\hfill \tilde{J}\left({\varphi }^H\tilde{\mathcal{A}}\right)& =& \frac{p}{2}{\varphi }^H\tilde{\mathcal{A}}-A[-\tilde{\mathcal{A}}+\sqrt{q}\{\sum _{\alpha =1}^n{\eta }^{H_{\alpha }}\left(\tilde{\mathcal{A}}\right){\zeta }^{(\alpha +n)}\hfill \\ & -& \sum _{\alpha =1}^n{\eta }^{H_{\alpha +n}}\left(\tilde{\mathcal{A}}\right){\zeta }^{\left(\alpha \right)}+{\eta }^V\left(\tilde{\mathcal{A}}\right){\zeta }^{(2n+1)}-{\eta }^{H_{2n+1}}\left(\tilde{\mathcal{A}}\right){\zeta }^H\left\}\right],\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \tilde{J}\left({\zeta }^{\left(\alpha \right)}\right)& =& \frac{p}{2}{\zeta }^{\left(\alpha \right)}-A\sqrt{q}({\zeta }^{(\alpha +n)}-{\zeta }^H),\hfill \end{array}$$

$$\tilde{J}\left({\zeta }^H\right)=\frac{p}{2}{\zeta }^H-A\sqrt{q}{\zeta }^{(2n+1)},$$

and

$$\begin{array}{ccc}\hfill {\tilde{J}}^2\left(\tilde{\mathcal{A}}\right)& =& \frac{p}{2}J\tilde{\mathcal{A}}-A[\tilde{J}\left({\varphi }^H\tilde{\mathcal{A}}\right)+\sqrt{q}\{\sum _{\alpha =1}^n{\eta }^{H_{\alpha }}\left(\tilde{\mathcal{A}}\right)\tilde{J}\left({\zeta }^{(\alpha +n)}\right)\hfill \\ & -& \sum _{\alpha =1}^n{\eta }^{H_{\alpha +n}}\left(\tilde{\mathcal{A}}\right)\tilde{J}\left({\zeta }^{\left(\alpha \right)}\right)+{\eta }^V\left(\tilde{\mathcal{A}}\right)\tilde{J}\left({\zeta }^{(2n+1)}\right)-{\eta }^{H_{2n+1}}\left(\tilde{\mathcal{A}}\right)\tilde{J}\left({\zeta }^H\right)\left\}\right],\hfill \\ \hfill {\tilde{J}}^2\left(\tilde{\mathcal{A}}\right)& =& p\tilde{J}\left(\tilde{\mathcal{A}}\right)+q\tilde{\mathcal{A}}.\hfill \end{array}$$

(28)

Definition 1. 

The 2-Form Ω of $\tilde{J}$ is given by

$$\Omega (\tilde{\mathcal{A}},\tilde{\mathcal{B}})=g^D(\tilde{\mathcal{A}},\tilde{J}\tilde{\mathcal{B}}),$$

(29)

$\forall \tilde{\mathcal{A}},\tilde{\mathcal{B}}\in {\Im }_0^1\left(FM\right)$.

Theorem 2. 

The 2-Form Ω of $(g^D,\tilde{J})$ on $FM$ is given by

$$\begin{array}{ccc}\hfill \left(i\right)\Omega ({\mathcal{A}}^H,{\mathcal{B}}^H)& =& \frac{p}{2}g{(\mathcal{A},\mathcal{B})}^V-A\Phi {(\mathcal{A},\mathcal{B})}^V,\hfill \\ \hfill \left(ii\right)\Omega ({\mathcal{A}}^H,{\mathcal{B}}^{\left(\beta \right)})& =& A\sqrt{q}{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V,\hfill \\ \hfill \left(iii\right)\Omega ({\mathcal{A}}^{\left(\beta \right)},{\mathcal{B}}^{\left(\mu \right)})& =& \frac{p}{2}{\delta }_{\mu }^{\beta }{\left(g(\mathcal{A},\mathcal{B})\right)}^V-A[{\delta }_{\mu }^{\beta }\Phi {(\mathcal{A},\mathcal{B})}^V\hfill \\ & +& \sqrt{q}\epsilon \left(\mu \right){\delta }_{\mu +\epsilon \left(\mu \right)n}^{\beta +\epsilon \left(\beta \right)n}\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V],\hfill \end{array}$$

where $\alpha ,\beta ,\mu =1,\cdots ,2n+1$ and $\forall \mathcal{A},\mathcal{B}\in {\Im }_0^1\left(M\right)$.

Proof. 

Using (9) and (29), we infer

$$\begin{array}{ccc}\hfill \left(i\right)\Omega ({\mathcal{A}}^H,{\mathcal{B}}^H)& =& g^D({\mathcal{A}}^H,\frac{p}{2}{\mathcal{B}}^H-A[{\left(\varphi \mathcal{B}\right)}^H+\sqrt{q}\eta {\left(\mathcal{B}\right)}^V{\zeta }^{(2n+1)}]),\hfill \\ & =& \frac{p}{2}g{(\mathcal{A},\mathcal{B})}^V-A\Phi {(\mathcal{A},\mathcal{B})}^V,\hfill \\ \hfill \left(ii\right)\Omega ({\mathcal{A}}^H,{\mathcal{B}}^{\left(\beta \right)})& =& g^D({\mathcal{A}}^H,\frac{p}{2}{\mathcal{B}}^{\left(\beta \right)}-A[{\left(\varphi \mathcal{B}\right)}^{\left(\beta \right)}\hfill \\ & +& \sqrt{q}\{\epsilon \left(\beta \right)\eta {\left(\mathcal{B}\right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}-{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{B}\right)}^V{\zeta }^H]\left)\right\}.\hfill \\ & =& A\sqrt{q}{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V,\hfill \\ \hfill \left(iii\right)\Omega ({\mathcal{A}}^{\left(\beta \right)},{\mathcal{B}}^{\left(\mu \right)})& =& g^D({\mathcal{A}}^{\left(\beta \right)},\frac{p}{2}{\mathcal{B}}^{\left(\mu \right)}-A[{\left(\varphi \mathcal{B}\right)}^{\left(\mu \right)}\hfill \\ & +& \sqrt{q}\{\epsilon \left(\beta \right)\eta {\left(\mathcal{B}\right)}^V{\zeta }^{(\mu +\epsilon (\mu \left)n\right)}-{\delta }_{2n+1}^{\mu }\eta {\left(\mathcal{B}\right)}^V{\zeta }^H]\left)\right\}\hfill \\ & =& \frac{p}{2}{\delta }_{\mu }^{\beta }{\left(g(\mathcal{A},\mathcal{B})\right)}^V-A[{\delta }_{\mu }^{\beta }\Phi {(\mathcal{A},\mathcal{B})}^V\hfill \\ & +& \sqrt{q}\epsilon \left(\mu \right){\delta }_{\mu +\epsilon \left(\mu \right)n}^{\beta +\epsilon \left(\beta \right)n}\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V].\hfill \end{array}$$

(30)

□

Theorem 3. 

The differential $d\Omega$ on $FM$ is expressed as

$$\begin{array}{ccc}\hfill \left(i\right)d\Omega ({\mathcal{A}}^H,{\mathcal{B}}^H,{\mathcal{C}}^H)& =& \frac{1}{3}\left\{\frac{p}{2}\right[{\left(Xg(\mathcal{B},\mathcal{C})\right)}^V-g{([\mathcal{A},\mathcal{B}],\mathcal{C})}^V-{\left(Yg(\mathcal{B},\mathcal{C})\right)}^V\hfill \\ & +& g{([\mathcal{A},\mathcal{C}],\mathcal{B})}^V+{\left(Zg(\mathcal{A},\mathcal{B})\right)}^V-g{([\mathcal{B},\mathcal{C}],\mathcal{A})}^V]\hfill \\ & -& {A\left[\right(\mathcal{A}(\Phi (\mathcal{B},\mathcal{C}))}^V{-(\mathcal{B}(\Phi (\mathcal{A},\mathcal{C}))}^V\hfill \\ & +& (\mathcal{C}{(\Phi (\mathcal{A},\mathcal{B}))}^V-(\Phi {([\mathcal{A},\mathcal{B}],\mathcal{C})}^V)+(\Phi {([\mathcal{A},\mathcal{C}],\mathcal{B})}^V)\hfill \\ & -& (\Phi {([\mathcal{B},\mathcal{C}],\mathcal{A})}^V)+\Omega (\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^H)\hfill \\ & -& \Omega (\gamma R(\mathcal{A},\mathcal{C}),{\mathcal{B}}^H)+\Omega (\gamma R(\mathcal{B},\mathcal{C}),{\mathcal{A}}^H)\},\hfill \\ \hfill \left(ii\right)d\Omega ({\mathcal{A}}^H,{\mathcal{B}}^H,{\mathcal{C}}^{\left(\beta \right)})& =& \frac{1}{3}\left\{A\sqrt{q}\right[{\delta }_{2n+1}^{\beta }{\left(\mathcal{A}\eta \left(\mathcal{C}\right)\eta \left(\mathcal{B}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\mathcal{B}\eta \left(\mathcal{C}\right)\eta \left(\mathcal{A}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\eta \left([\mathcal{A},\mathcal{B}]\right)\eta \left(\mathcal{C}\right)\right)}^V+\Omega (\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^{\left(\beta \right)})\hfill \\ & +& {\delta }_{2n+1}^{\beta }{\left(\eta \left({\nabla }_{X}Z\right)\eta \left(\mathcal{B}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\eta \left({\nabla }_{Y}Z\right)\eta \left(\mathcal{A}\right)\right)}^V\left]\right\},\hfill \\ \hfill \left(iii\right)d\Omega ({\mathcal{A}}^H,{\mathcal{B}}^{\left(\beta \right)},{\mathcal{C}}^{\left(\mu \right)})& =& \frac{1}{3}\{\frac{p}{2}{\delta }_{\alpha }^{\beta }\left({\nabla }_{X}g\right){(\mathcal{B},\mathcal{C})}^V-A{\delta }_{\alpha }^{\beta }({\nabla }_{\mathcal{A}}\Phi ){(\mathcal{B},\mathcal{C})}^V\hfill \\ & +& \sqrt{q}\epsilon \left(\alpha \right){\delta }_{\alpha +\sqrt{q}\epsilon \left(\alpha \right)n}^{\beta }\eta {\left(\mathcal{B}\right)}^V\left({\nabla }_{\mathcal{A}}\eta \right)\mathcal{C}{{)}^V+\eta {\left(\mathcal{C}\right)}^V\left({\nabla }_{\mathcal{A}}\eta \right)\mathcal{B})}^V\},\hfill \\ \hfill \left(iv\right)d\Omega ({\mathcal{A}}^{\left(\alpha \right)},{\mathcal{B}}^{\left(\beta \right)},{\mathcal{C}}^{\left(\mu \right)})& =& 0,\hfill \end{array}$$

$\forall \mathcal{A},\mathcal{B},\mathcal{C}\in {\Im }_0^1\left(M\right)$.

Proof. 

The differential $d\Omega$ is given by

$$\begin{array}{ccc}\hfill 3d\Omega (\tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{\mathcal{C}})& =& \{\tilde{\mathcal{A}}(\Omega (\tilde{\mathcal{B}},\tilde{\mathcal{C}}))-\tilde{\mathcal{B}}(\Omega (\tilde{\mathcal{A}},\tilde{\mathcal{C}}))+\tilde{\mathcal{C}}(\Omega (\tilde{\mathcal{A}},\tilde{\mathcal{B}}))\hfill \\ & -& \Omega ([\tilde{\mathcal{A}},\tilde{\mathcal{B}}],\tilde{\mathcal{C}})+\Omega ([\tilde{\mathcal{A}},\tilde{\mathcal{C}}],\tilde{\mathcal{B}})-\Omega ([\tilde{\mathcal{B}},\tilde{\mathcal{C}}],\tilde{\mathcal{A}})\},\hfill \end{array}$$

$\forall \tilde{\mathcal{A}},\tilde{\mathcal{B}},\tilde{\mathcal{C}}\in {\Im }_0^1\left(FM\right)$.

$$\begin{array}{ccc}\hfill \left(i\right)3d\Omega ({\mathcal{A}}^H,{\mathcal{B}}^H,{\mathcal{C}}^H)& =& \frac{p}{2}[{\mathcal{A}}^H\left(g{(\mathcal{B},\mathcal{C})}^V\right)-{\mathcal{B}}^H\left(g{(\mathcal{A},\mathcal{C})}^V\right)\hfill \\ & +& {\mathcal{C}}^H\left(g{(\mathcal{A},\mathcal{B})}^V\right)]-A[{\mathcal{A}}^H(\Phi {(\mathcal{B},\mathcal{C})}^V)\hfill \\ & -& {\mathcal{B}}^H(\Phi {(\mathcal{A},\mathcal{C})}^V)+{\mathcal{C}}^H(\Phi {(\mathcal{A},\mathcal{B})}^V)]\hfill \\ & -& \frac{p}{2}g{([\mathcal{A},\mathcal{B}],\mathcal{C})}^V+A(\Phi {([\mathcal{A},\mathcal{B}],\mathcal{C})}^V)\hfill \\ & +& \Omega (\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^H)+\frac{p}{2}g{([\mathcal{A},\mathcal{C}],\mathcal{B})}^V\hfill \\ & +& A(\Phi {([\mathcal{A},\mathcal{C}],\mathcal{B})}^V)-\Omega (\gamma R(\mathcal{A},\mathcal{C}),{\mathcal{B}}^H)\hfill \\ & -& \frac{p}{2}g{([\mathcal{B},\mathcal{C}],\mathcal{A})}^V+A(\Phi {([\mathcal{B},\mathcal{C}],\mathcal{A})}^V)\hfill \\ & +& \Omega (\gamma R(\mathcal{B},\mathcal{C}),{\mathcal{A}}^H)\hfill \\ & =& \frac{p}{2}{[\left(Xg(\mathcal{B},\mathcal{C})\right)}^V-g{([\mathcal{A},\mathcal{B}],\mathcal{C})}^V-{\left(Yg(\mathcal{B},\mathcal{C})\right)}^V\hfill \\ & +& g{([\mathcal{A},\mathcal{C}],\mathcal{B})}^V+{\left(Zg(\mathcal{A},\mathcal{B})\right)}^V-g{([\mathcal{B},\mathcal{C}],\mathcal{A})}^V]\hfill \\ & -& {A\left[\right(\mathcal{A}(\Phi (\mathcal{B},\mathcal{C}))}^V{-(\mathcal{B}(\Phi (\mathcal{A},\mathcal{C}))}^V\hfill \\ & +& (\mathcal{C}{(\Phi (\mathcal{A},\mathcal{B}))}^V-(\Phi {([\mathcal{A},\mathcal{B}],\mathcal{C})}^V)+(\Phi {([\mathcal{A},\mathcal{C}],\mathcal{B})}^V)\hfill \\ & -& (\Phi {([\mathcal{B},\mathcal{C}],\mathcal{A})}^V)+\Omega (\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^H)\hfill \\ & -& \Omega (\gamma R(\mathcal{A},\mathcal{C}),{\mathcal{B}}^H)+\Omega (\gamma R(\mathcal{B},\mathcal{C}),{\mathcal{A}}^H),\hfill \\ \hfill \left(ii\right)3d\Omega ({\mathcal{A}}^H,{\mathcal{B}}^H,{\mathcal{C}}^{\left(\beta \right)})& =& A\sqrt{q}[{\mathcal{A}}^H{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{C}\right)}^V\eta {\left(\mathcal{B}\right)}^V\hfill \\ & -& {\mathcal{B}}^H{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{C}\right)}^V\eta {\left(\mathcal{A}\right)}^V\hfill \\ & +& {\mathcal{C}}^{\left(\beta \right)}\{\frac{p}{2}g{(\mathcal{A},\mathcal{B})}^V-\Phi {(\mathcal{A},\mathcal{B})}^V\}\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\eta \left([\mathcal{A},\mathcal{B}]\right)\eta \left(\mathcal{C}\right)\right)}^V+\Omega (\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^{\left(\beta \right)})\hfill \\ & +& {\delta }_{2n+1}^{\beta }{\left(\eta \left({\nabla }_{X}Z\right)\eta \left(\mathcal{B}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\eta \left({\nabla }_{Y}Z\right)\eta \left(\mathcal{A}\right)\right)}^V]\hfill \\ & =& A\sqrt{q}[{\delta }_{2n+1}^{\beta }{\left(\mathcal{A}\eta \left(\mathcal{C}\right)\eta \left(\mathcal{B}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\mathcal{B}\eta \left(\mathcal{C}\right)\eta \left(\mathcal{A}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\eta \left([\mathcal{A},\mathcal{B}]\right)\eta \left(\mathcal{C}\right)\right)}^V+\Omega (\gamma R(\mathcal{A},\mathcal{B}),{\mathcal{C}}^{\left(\beta \right)})\hfill \\ & +& {\delta }_{2n+1}^{\beta }{\left(\eta \left({\nabla }_{X}Z\right)\eta \left(\mathcal{B}\right)\right)}^V\hfill \\ & -& {\delta }_{2n+1}^{\beta }{\left(\eta \left({\nabla }_{Y}Z\right)\eta \left(\mathcal{A}\right)\right)}^V].\hfill \end{array}$$

Formulas $\left(iii\right)$ and $\left(iv\right)$ can be easily obtained. □

4. Behavior of the Nijehuis Tensor on FM

The Nijenhuis tensor of $\tilde{J}$ is expressed by

$$N(\tilde{\mathcal{A}},\tilde{\mathcal{B}})=[\tilde{J}\tilde{\mathcal{A}},\tilde{J}\tilde{\mathcal{B}}]-\tilde{J}[\tilde{J}\tilde{\mathcal{A}},\tilde{\mathcal{B}}]-\tilde{J}[\tilde{\mathcal{A}},\tilde{J}\tilde{\mathcal{B}}]+{\tilde{J}}^2[\tilde{\mathcal{A}},\tilde{\mathcal{B}}].$$

Theorem 4. 

$\forall \tilde{\mathcal{A}},\tilde{\mathcal{B}}\in {\Im }_0^1\left(FM\right)$, then

$$\begin{array}{ccc}\hfill \left(i\right)N({\mathcal{A}}^H,{\mathcal{B}}^H)& =& \frac{pA}{2}\{{\left({\nabla }_{\varphi \mathcal{B}}\mathcal{A}\right)}^{\left(\beta \right)}-{\left({\nabla }_{\varphi \mathcal{A}}\mathcal{B}\right)}^{\left(\beta \right)}\}\hfill \\ & +& A^2{[\varphi \mathcal{A},\varphi \mathcal{B}]}^H-A\tilde{J}{[\varphi \mathcal{A},\mathcal{B}]}^H\hfill \\ & -& A\tilde{J}{[\mathcal{A},\varphi \mathcal{B}]}^H+{\tilde{J}}^2{[\mathcal{A},\mathcal{B}]}^H\hfill \\ & +& A^2{(\eta \left(\mathcal{B}\right)}^V({\left({\nabla }_{\varphi \mathcal{A}}\zeta \right)}^{(2n+1)}-{\left({\nabla }_{\varphi \mathcal{B}}\zeta \right)}^{(2n+1)})\hfill \\ & +& A^2({\left({\nabla }_{\varphi \mathcal{A}}\zeta \right)}^{(2n+1)}+{\left(\varphi {\nabla }_{\mathcal{A}}\zeta \right)}^{(2n+1)})(\eta {\left(\mathcal{B}\right)}^V\hfill \\ & -& A^2({\left({\nabla }_{\varphi \mathcal{B}}\zeta \right)}^{(2n+1)}+{\left(\varphi {\nabla }_{\mathcal{B}}\zeta \right)}^{(2n+1)})(\eta {\left(\mathcal{A}\right)}^V\hfill \\ & +& A^2(\eta {\left({\nabla }_{\mathcal{B}}\zeta \right)}^V\eta {\left(\mathcal{A}\right)}^V-\eta {\left({\nabla }_{\mathcal{A}}\zeta \right)}^V\eta {\left(\mathcal{B}\right)}^V){\zeta }^H\hfill \\ & +& \frac{pA}{2}\{{\left({\nabla }_{\mathcal{B}}\mathcal{A}\right)}^{(2n+1)}-{\left({\nabla }_{\mathcal{A}}\mathcal{B}\right)}^{(2n+1)}\}\hfill \\ & -& A^2\gamma R(\varphi \mathcal{A},\varphi \mathcal{B})+A\tilde{J}\gamma R(\varphi \mathcal{A},\mathcal{B})\hfill \\ & +& A\tilde{J}\gamma R(\varphi \mathcal{A},\mathcal{B})-{\tilde{J}}^2\gamma R(\mathcal{A},\mathcal{B}),\hfill \\ \hfill \left(ii\right)N({\mathcal{A}}^{\left(\alpha \right)},{\mathcal{B}}^{\left(\beta \right)})& =& \sqrt{q}\{A^2\left[\right({\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{B}\right)}^V{\left({\nabla }_{\zeta }\left(\varphi \mathcal{A}\right)\right)}^{\alpha }\hfill \\ & +& \epsilon \left(\alpha \right)\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V{\delta }_{2n+1}^{\beta }{\left({\nabla }_{\zeta }\zeta \right)}^{(\alpha +\epsilon (\alpha \left)n\right)}\hfill \\ & -& {\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^V{\left({\nabla }_{\zeta }\left(\varphi \mathcal{B}\right)\right)}^{\alpha }\hfill \\ & -& \epsilon \left(\beta \right)\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V{\delta }_{2n+1}^{\alpha }{\left({\nabla }_{\zeta }\zeta \right)}^{(\beta +\epsilon (\beta \left)n\right)}\hfill \\ & +& {\delta }_{2n+1}^{\alpha }{\delta }_{2n+1}^{\beta }({[\zeta ,\zeta ]}^H-\gamma R(\zeta ,\zeta ))]\hfill \\ & -& \frac{pA}{2}{\left({\nabla }_{\zeta }\mathcal{B}\right)}^{\left(\beta \right)}-\frac{p}{2}{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{B}\right)}^V{\left({\nabla }_{\mathcal{A}}\zeta \right)}^{\left(\alpha \right)}\hfill \\ & -& \frac{pA}{2}{\mathcal{A}}^{\left(\alpha \right)}{\delta }_{2n+1}^{\alpha }\eta {\left(\mathcal{A}\right)}^V\hfill \\ & +& A^2{X}^{\left(\alpha \right)}{\delta }_{2n+1}^{\alpha }\eta {\left(\mathcal{A}\right)}^V({\left(\varphi {\nabla }_{\zeta }\mathcal{B}\right)}^{\left(\beta \right)}\hfill \\ & +& \epsilon \left(\beta \right)\eta {\left({\nabla }_{\zeta }\mathcal{B}\right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}-{\delta }_{2n+1}^{\beta }\eta {\left({\nabla }_{\zeta }\mathcal{B}\right)}^V{\zeta }^H)\hfill \\ & -& \frac{pA}{2}{\mathcal{B}}^{\left(\beta \right)}{\delta }_{2n+1}^{\alpha }\eta {\left(\mathcal{B}\right)}^V\hfill \\ & +& A^2{Y}^{\left(\alpha \right)}{\delta }_{2n+1}^{\alpha }\eta {\left(\mathcal{B}\right)}^V({\left(\varphi {\nabla }_{\zeta }\mathcal{A}\right)}^{\left(\beta \right)}\hfill \\ & +& \epsilon \left(\beta \right)\eta {\left({\nabla }_{\zeta }\mathcal{A}\right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}-{\delta }_{2n+1}^{\beta }\eta {\left({\nabla }_{\zeta }\mathcal{A}\right)}^V{\zeta }^H\left)\right\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \left(iii\right)N({\mathcal{A}}^H,{\mathcal{B}}^{\left(\beta \right)})& =& -\frac{pA}{2}\sqrt{q}{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{B}\right)}^V{\left({\nabla }_{\zeta }\mathcal{A}\right)}^{\left(\beta \right)}-\frac{pA}{2}{\left({\nabla }_{\varphi \mathcal{A}}\mathcal{B}\right)}^{\left(\beta \right)}\hfill \\ & +& A^2{\left({\nabla }_{\varphi \mathcal{A}}\varphi \mathcal{B}\right)}^{\left(\beta \right)}+A^2\sqrt{q}\{\epsilon \left(\beta \right)\eta {\left(\mathcal{B}\right)}^V{\left({\nabla }_{\varphi \mathcal{A}}\zeta \right)}^{(\beta +\epsilon (\beta \left)n\right)}\hfill \\ & -& {\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{B}\right)}^V([\varphi \mathcal{A},\zeta ]-\gamma R(\varphi \mathcal{A},\zeta ))\hfill \\ & +& {\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^V\eta {\left(\mathcal{B}\right)}^V{\left({\nabla }_{\zeta }\zeta \right)}^{(2n+1)}-\varphi {\nabla }_{\varphi \mathcal{A}}{\mathcal{B})}^{\left(\beta \right)}\hfill \\ & +& \epsilon \left(\beta \right)\eta {\left({\nabla }_{\varphi \mathcal{A}}\mathcal{B}\right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}-{\delta }_{2n+1}^{\beta }\eta {\left({\nabla }_{\varphi \mathcal{A}}\mathcal{B}\right)}^V{\zeta }^H\left)\right\}\hfill \\ & +& \frac{pA}{2}{\left({\nabla }_{\varphi \mathcal{A}}\mathcal{B}\right)}^V-pA({\left(\varphi {\nabla }_{X}Y\right)}^{\left(\beta \right)}\hfill \\ & +& pA({\left(\varphi {\nabla }_{\mathcal{A}}\varphi \mathcal{B}\right)}^{\left(\beta \right)}\hfill \\ & +& \sqrt{q}\{\epsilon \left(\beta \right)\eta {\left({\nabla }_{X}Y\right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}-{\delta }_{2n+1}^{\beta }\eta {\left({\nabla }_{X}Y\right)}^V{\zeta }^H)\hfill \\ & +& +\epsilon \left(\beta \right)\eta {\left({\nabla }_{\mathcal{A}}\varphi \mathcal{B}\right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}\hfill \\ & -& {\delta }_{2n+1}^{\beta }\eta {\left({\nabla }_{\mathcal{A}}\varphi \mathcal{B}\right)}^V{\zeta }^H)+\epsilon \left(\beta \right)\eta {\left(\mathcal{B}\right)}^V{\left(\varphi {\nabla }_{\mathcal{A}}\zeta \right)}^{(\beta +\epsilon (\beta \left)n\right)}\hfill \\ & +& {\epsilon }^2\left(\beta \right)\eta {\left(\mathcal{B}\right)}^V\eta {\left(\varphi {\nabla }_{\mathcal{A}}\zeta \right)}^V{\zeta }^{(\beta +\epsilon (\beta \left)n\right)}-{\delta }_{2n+1}^{\beta }\epsilon \left(\beta \right)\eta {\left(\mathcal{B}\right)}^V\eta {\left(\varphi {\nabla }_{\mathcal{A}}\zeta \right)}^V{\zeta }^H)\hfill \\ & -& {\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{B}\right)}^V({\left(\varphi [\mathcal{A},\zeta ]\right)}^H+\eta {[\mathcal{A},\zeta ]}^V{\zeta }^{(2n+1)},-\gamma \tilde{J}R(\mathcal{A},\zeta ))\left)\right\}\hfill \end{array}$$

where $\alpha ,\beta =1,\cdots ,2n+1$.

Proof. 

Using (22) and Theorem (1), Theorem (4) is proven. □

5. Example

Let $\{e_i,\varphi e_i,\zeta \}$ be a basis in $(M,\varphi ,\zeta ,\eta ,g)$ where i denotes 1 to n. The coderivative $\delta \Omega$ with basis $\{e_i^H,{\left(\varphi e_i\right)}^H,{\zeta }^H,e_i^{\left(\alpha \right)},{\left(\varphi e_i\right)}^{\left(\alpha \right)},{\zeta }^{\left(\alpha \right)}\}$ can be expressed as [16]

$$\begin{array}{ccc}\hfill \delta \Omega \left(\tilde{\mathcal{A}}\right)& =& -\sum _{i=1}^n\{({\tilde{\nabla }}_{e_i^H}\Omega )(e_i^H,\tilde{\mathcal{A}})+({\tilde{\nabla }}_{{\left(\varphi e_i\right)}^H}\Omega )({\left(\varphi e_i\right)}^H,\tilde{\mathcal{A}})\}\hfill \\ & +& \sum _{j=1}^n({\tilde{\nabla }}_{{\zeta }^{\left(j\right)}}\Omega )({\zeta }^{\left(j\right)},\tilde{\mathcal{A}})-({\tilde{\nabla }}_{{\zeta }^{(2n+1)}}\Omega )({\zeta }^{(2n+1)},\tilde{\mathcal{A}})\hfill \\ & -& ({\tilde{\nabla }}_{{\zeta }^H}\Omega )({\zeta }^H,\tilde{\mathcal{A}})-\sum _{\alpha =1}^{2n+1}\sum _{i=1}^n\{{\tilde{\nabla }}_{e_i^{\left(\alpha \right)}}\Omega )(e_i^{\left(\alpha \right)},\tilde{\mathcal{A}})\hfill \\ & +& \left({\tilde{\nabla }}_{{\left(\varphi e_i\right)}^{\left(\alpha \right)}}F\right)({\left(\varphi e_i\right)}^{\left(\alpha \right)},\tilde{\mathcal{A}})\}.\hfill \end{array}$$

(31)

Taking $\tilde{\mathcal{A}}={\mathcal{A}}^{\left(\beta \right)}$ in (31), using (11) and (29), we acquire

$$\begin{array}{ccc}\hfill \delta \Omega \left({\mathcal{A}}^{\left(\beta \right)}\right)& =& -\sum _{i=1}^n\{g^D({\nabla }_{e_i^H}{e}_i^H,\tilde{J}{\mathcal{A}}^{\left(\beta \right)})+g^D({\nabla }_{{(\varphi E}_i{)}^H}{\left(\varphi e_i\right)}^H,\tilde{J}{\mathcal{A}}^{\left(\beta \right)})\}\hfill \\ & -& g^D({\nabla }_{{\zeta }^H}{\zeta }^H,\tilde{J}{\mathcal{A}}^{\left(\beta \right)})\hfill \\ & =& -\sum _{i=1}^n\{-g^D(\gamma R(e_i,e_i),\frac{p}{2}{\mathcal{A}}^{\left(\beta \right)})-A[-g^D(\gamma R(\varphi e_i,e_i),{\mathcal{A}}^{\left(\beta \right)})\hfill \\ & -& \sqrt{q}{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^{V}g{({\nabla }_{e_i}\zeta ,e_i)}^V-\sqrt{q}{\delta }_{2n+1}^{\beta }\eta {\left(\mathcal{A}\right)}^{V}g{({\nabla }_{\varphi e_i}\zeta ,\varphi e_i)}^V\}\hfill \\ & +& \sqrt{q}{\delta }_{2n+1}^{\beta }g({\nabla }_{{\zeta }^H}{\zeta }^H,{\mathcal{A}}^V)]\hfill \\ & =& \frac{p}{2}\sum _{i=1}^n\{g^D(\gamma R(e_i,e_i),{\mathcal{A}}^{\left(\beta \right)})-A[-g^D(\gamma R(e_i,\varphi e_i),{\mathcal{A}}^{\left(\beta \right)})\hfill \\ & +& \sqrt{q}{\delta }_{2n+1}^{\beta }\{\eta {\left(\mathcal{A}\right)}^V{\left(\delta \eta \right)}^V,\left({\nabla }_{\zeta }\eta \right){\mathcal{A}}^V\}],\hfill \end{array}$$

where

$$\begin{array}{c}\hfill \delta \eta =-\sum _{i=1}^n\{\left({\nabla }_{e_i}\eta \right){\zeta }_i+\left({\nabla }_{\varphi e_i}\eta \right)\varphi {\zeta }_i\}\end{array}$$

and

$$\begin{array}{c}\hfill \left({\nabla }_{\zeta }\eta \right)\mathcal{A}=g(\mathcal{A},{\nabla }_{\zeta }\zeta ).\end{array}$$

Taking $\tilde{\mathcal{A}}={\mathcal{A}}^H$ in (31), using (11) and (29), we acquire

$$\begin{array}{ccc}\hfill \delta \Omega \left({\mathcal{A}}^H\right)& =& -\sum _{i=1}^n\{g^D({\nabla }_{e_i^H}{e}_i^H,\tilde{J}{\mathcal{A}}^H)+g^D({\nabla }_{{(\varphi E}_i{)}^H}{\left(\varphi e_i\right)}^H,\tilde{J}{\mathcal{A}}^H\}\hfill \\ & -& g^D({\nabla }_{{\zeta }^{(2n+1)}}{\zeta }^{(2n+1)},\tilde{J}{\mathcal{A}}^H)-g^D({\nabla }_{{\zeta }^H}{\zeta }^H,\tilde{J}{\mathcal{A}}^H)\hfill \\ & -& \sum _{\alpha =1}^{2n+1}\sum _{i=1}^n(g^D({\nabla }_{e_i^{\left(\alpha \right)}}{e}_i^{\left(\alpha \right)},\tilde{J}{\mathcal{A}}^H)+g^D({\nabla }_{{(\varphi E}_i{)}^{\left(\alpha \right)}}{\left(\varphi e_i\right)}^{\left(\alpha \right)},\tilde{J}{\mathcal{A}}^H).\hfill \\ & =& -\frac{p}{2}\sum _{i=1}^n[{\left(g({\nabla }_{e_i}{e}_i,\mathcal{A})\right)}^V+{\left(g({\nabla }_{\varphi e_i}\varphi e_i,\mathcal{A})\right)}^V+{\left(g({\nabla }_{\zeta }\zeta ,\mathcal{A})\right)}^V]\hfill \\ & -& A[-\sum _{i=1}^n(-g{(\left({\nabla }_{e_i}\varphi \right)e_i,\mathcal{A})}^V-g{(\left({\nabla }_{\varphi e_i}\varphi \right)\varphi e_i,\mathcal{A})}^V)\hfill \\ & +& g{(\left({\nabla }_{\zeta }\varphi \right)\zeta ,\mathcal{A})}^V].\hfill \\ & =& -\frac{p}{2}\sum _{i=1}^n[{\left(g({\nabla }_{e_i}{e}_i,\mathcal{A})\right)}^V+{\left(g({\nabla }_{\varphi e_i}\varphi e_i,\mathcal{A})\right)}^V+{\left(g({\nabla }_{\zeta }\zeta ,\mathcal{A})\right)}^V]\hfill \\ & -& A{(\delta \Phi \left(\mathcal{A}\right))}^V,\hfill \end{array}$$

where

$$\delta \Phi \left(\mathcal{A}\right)=-\sum _{i=1}^n({\nabla }_{e_i}\Phi )(e_i,\mathcal{A})+({\nabla }_{\varphi e_i}\Phi )(\varphi e_i,\mathcal{A}))-({\nabla }_{\zeta }\Phi )(\zeta ,\mathcal{A}).$$

and

$$({\nabla }_{\mathcal{A}}\Phi )(\mathcal{B},\mathcal{C})=-g(\left({\nabla }_{\mathcal{A}}\varphi \right)(\mathcal{B},\mathcal{C}).$$