Metallic Structures for Tangent Bundles over Almost Quadratic ϕ-Manifolds

Abstract

This paper aims to explore the metallic structure $J^2=pJ+qI,$ where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle $TM$ over almost quadratic $\varphi$-structures (briefly, $(\varphi ,\xi ,\eta )$). Tensor fields $\tilde{F}$ and $F^{*}$ are defined on $TM$, and it is shown that they are metallic structures over $(\varphi ,\xi ,\eta )$. Next, the fundamental 2-form $\Omega$ and its derivative $d\Omega$, with the help of complete lift on $TM$ over $(\varphi ,\xi ,\eta )$, are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures $\tilde{F}$ and $F^{*}$ are determined using complete and horizontal lifts on $TM$ over $(\varphi ,\xi ,\eta )$, respectively. Finally, we prove the existence of almost quadratic $\varphi$-structures on $TM$ with non-trivial examples.

1. Introduction

Many ancient societies have made extensive use of the golden mean as a foundation for proportions, whether for creating music, sculptures, paintings, or buildings, such as temples and palaces [1]. Fractal geometry has been explained using the silver mean [2]. Some uses of a class of polynomial structures have been constructed on Riemannian manifolds for the metallic means family (a generalization of the golden mean) and generalized Fibonacci sequences in differential geometry. The geometric properties (such as totally geodesic, totally umbilical hypersurfaces, etc.) in metallic Riemannian manifolds have been explored in [3]. This manuscript is focused on studying the properties of metallic structures for tangent bundles over a class of metallic Riemannian manifolds.

A quadratic equation of type

$$x^2=px+q,$$

where p and q are natural numbers, whose positive solutions are given by

$${\sigma }_p^q=\frac{p+\sqrt{p^2+4q}}{2}$$

is known as a metallic means family [4]. The most notable member is the well-known “Golden Mean” for $p=q=1$. The metallic means family includes the silver mean for $p=2,q=1$, the bronze mean for $p=3,q=1$, the copper mean for $p=1,q=2$, and many others.

Let M be an n-dimensional differentiable manifold and $TM$ be its tangent bundle. Let $\Im \left(M\right)$ and $\Im \left(TM\right)$ be the algebra of tensor fields of M and $TM$, respectively. The differential geometry of tangent bundle has been broadly studied by Davis [5], Sasaki [6], Tachibana and Okumura [7], Yano and Ishihara [8], and others. Yano and Kabayashi [9] defined the natural mapping (say complete lift) of $\Im \left(M\right)$ into $\Im \left(TM\right)$ and studied complete lifts of an almost complex structure and the symplectic structure on $TM$. Tanno [10] studied complete and vertical lifts of an almost contact structure on $TM$ and defined a tensor field $\tilde{J}$ of type (1,1) and proved that it is an almost complex structure on $TM$. Numerous investigators have studied various geometric structures on $TM$—an almost complex structure by Yano [11], paracomplex structures by Tekkoyun [12], almost r-contact structures by Das and Khan [13], and many others [14,15,16,17,18,19].

In [20], Azami explored complete and horizontal lifts of metallic structures and analyzed the geometric properties of these structures. Salimov et al. [19] studied complete lifts of symplectic vector fields on tangent and cotangent bundles. Recently, Khan [21] introduced a new tensor field J of type (1,1) and demonstrated that J is a metallic structure ($MS$) on the frame bundle $FM$. Furthermore, the derivative and the coderivative of fundamental 2-form and the Nijenhuis tensor of J on $FM$ are discussed.

On the other hand, Sasaki [6] defined a structure named as an almost contact structure and demonstrated its basic algebraic properties such as a Riemannian metric, the fundamental 2-form, etc., on M. Later on, Sato [22] defined the notion of an almost paracontact structure and analyzed its geometrical properties.

Debnath et al. [23] defined the notion of a $(\varphi ,\xi ,\eta )$ on a differentiable manifold M and established its existence. Later on, Gonul et al. [24] developed a relation between $MS$ and $(\varphi ,\xi ,\eta )$. They proved that the warped product manifold has structure $(\varphi ,\xi ,\eta )$. Most recently, Gök et al. [25] introduced the notion of $f_{(a,b)}(3,2,1)$-structures and investigated a necessary condition for these structures to be a $(\varphi ,\xi ,\eta )$.

The main aim of this paper is summarized as:

• Tensor fields $\tilde{F}$ and $F^{*}$ are defined on $TM$ over the structure $(\varphi ,\xi ,\eta )$ and we prove that they are metallic structures, which generalize the notion of almost complex structure $\tilde{J}$ introduced by Tanno [10].
• The basic geometrical properties of fundamental 2-Form and its derivative on $TM$ over the structure $(\varphi ,\xi ,\eta )$ are studied.
• The integrability conditions and expressions of the Lie derivative of metallic structures $\tilde{F}$ and $F^{*}$ with the help of complete and horizontal lifts, respectively, on $TM$ over the structure $(\varphi ,\xi ,\eta )$ are investigated.
• The existence of almost quadratic $\varphi$-manifolds on $TM$ with non-trivial examples are proved.

2. Preliminaries

Let M be an n-dimensional differentiable manifold of class $C^{\infty }$ and $TM$ be the tangent bundle over a manifold M such that $TM={\bigcup }_{x\in M}{T}_{x}M$ with the projection map $\pi :TM\to M$, where $T_{x}M$ represents the tangent space at a point x of M. Let $(U,x^h)$ be a local chart in M and $(x^h,y^h)$ be a local coordinate in ${\pi }^{-1}\left(U\right)\subset TM$ and be called the induced coordinate in ${\pi }^{-1}\left(U\right)$.

Let $f,\eta ,{{\rm Y}}_1$, and F be a function, a 1-form, a vector field, and a tensor field of type (1,1) of M, respectively. The vertical lifts $f^V,{\eta }^V,{{{\rm Y}}_1}^V$, and $F^V$ on $TM$ in terms of partial differential equations are given by [8,25]

$$\begin{array}{ccc}\hfill f^V& =& f\circ \pi ,\hfill \\ \hfill {\eta }^V& =& {\left({\eta }_i\right)}^V{\left(d{x}^i\right)}^V,\hfill \\ \hfill {{{\rm Y}}_1}^V& =& x^h\frac{\partial }{\partial y^h},\hfill \\ \hfill F^V& =& F_i^h\frac{\partial }{\partial y^h}\otimes d{x}^i,\hfill \end{array}$$

where ${\eta }_i,x^h$, and $F_i^h,i,h=1,2,\dots ,n$ are local components of $\eta ,{{\rm Y}}_1$, and F on M, respectively.

The complete lifts $f^C,{\eta }^C,{{{\rm Y}}_1}^C$, and $F^C$ on $TM$ in the term of partial differential equations are given by

$$\begin{array}{ccc}\hfill f^C& =& y^i{\partial }_{i}f=\partial f,\hfill \\ \hfill {\eta }^C& =& y^i{\partial }_i\eta ,\hfill \\ \hfill {{{\rm Y}}_1}^C& =& x^h\frac{\partial }{\partial x^h}+\frac{\partial x^h}{\partial x^i}{y}^i\frac{\partial }{\partial y^h},\hfill \\ \hfill F^C& =& {\left(F_i^h\right)}^C\frac{\partial }{\partial y^h}\otimes d{x}^i+{\left(F_i^h\right)}^V\frac{\partial }{\partial x^h}\otimes d{x}^i+{\left(F_i^h\right)}^V\frac{\partial }{\partial y^h}\otimes d{y}^h.\hfill \end{array}$$

By the definition of the lift, we have

$$\begin{array}{ccc}\hfill \left(i\right){{{\rm Y}}_1}^V{f}^V& =& 0,{{{\rm Y}}_1}^V{f}^C={\left({{\rm Y}}_{1}f\right)}^V,\hfill \\ \hfill \left(ii\right){{{\rm Y}}_1}^C{f}^V& =& {\left({{\rm Y}}_{1}f\right)}^V,{{{\rm Y}}_1}^C{f}^C={\left({{\rm Y}}_{1}f\right)}^C,\hfill \\ \hfill \left(iii\right){\eta }^V\left({{{\rm Y}}_1}^V\right)& =& 0,{\eta }^V\left({{{\rm Y}}_1}^C\right)={\left(\eta {{\rm Y}}_1\right)}^V,\hfill \\ \hfill \left(iv\right){\eta }^C\left({{{\rm Y}}_1}^V\right)& =& {\left(\eta {{\rm Y}}_1\right)}^V,{\eta }^C\left({{{\rm Y}}_1}^C\right)={\left(\eta {{\rm Y}}_1\right)}^C,\hfill \\ \hfill \left(v\right)F^V{{{\rm Y}}_1}^V& =& 0,F^V{{{\rm Y}}_1}^C={\left(F{{\rm Y}}_1\right)}^V,\hfill \\ \hfill \left(vi\right)F^C{{{\rm Y}}_1}^V& =& {\left(F{{\rm Y}}_1\right)}^V,F^C{{{\rm Y}}_1}^C={\left(F{{\rm Y}}_1\right)}^C.\hfill \end{array}$$

(1)

By the definition of the Lie product of the lift, we have

$$[{{{\rm Y}}_1}^V,{{{\rm Y}}_2}^V]=0,[{{{\rm Y}}_1}^V,{{{\rm Y}}_2}^C]={[{{\rm Y}}_1,{{\rm Y}}_2]}^V,[{{{\rm Y}}_1}^C,{{{\rm Y}}_2}^C]={[{{\rm Y}}_1,{{\rm Y}}_2]}^C.$$

(2)

Let f be a function and ∇ is an affine connection on M. The horizontal lift is

$$f^H=f^C-{\nabla }_{\gamma }f,$$

where $\nabla f$ is a gradient of f on M, $\gamma$ is an operator, and ${\nabla }_{\gamma }f=\gamma (\nabla f)$ is in ${\pi }^{-1}\left(U\right)$ (see [8], p. 86).

Let ${{\rm Y}}_1,\eta$, and S be a vector field, a 1-form, and a tensor field of arbitrary type on M, respectively. The horizontal lifts ${{{\rm Y}}_1}^H,{\eta }^H$, and $S^H$ on $TM$ are given by

$$\begin{array}{ccc}\hfill {{{\rm Y}}_1}^H& =& {{{\rm Y}}_1}^C-{\nabla }_{\gamma }{{\rm Y}}_1,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\eta }^H& =& {\eta }^C-{\nabla }_{\gamma }\eta ,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill S^H& =& S^C-{\nabla }_{\gamma }S.\hfill \end{array}$$

(5)

By the definitions of the lifts, we have

$${{{\rm Y}}_1}^H{f}^V={\left({{\rm Y}}_{1}f\right)}^V,F^V{{{\rm Y}}_1}^H={\left(F{{\rm Y}}_1\right)}^V,{\eta }^V\left({{{\rm Y}}_1}^H\right)={\left(\eta \left({{\rm Y}}_1\right)\right)}^V.$$

(6)

By the definitions of the Lie product of the lifts, we have

$$\begin{array}{ccc}\hfill [{{{\rm Y}}_1}^V,{{{\rm Y}}_2}^H]& =& {[{{\rm Y}}_1,{{\rm Y}}_2]}^V-{\left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V=-{\left({\widehat{\nabla }}_{{{\rm Y}}_2}{{\rm Y}}_1\right)}^V,\hfill \\ \hfill [{{{\rm Y}}_1}^C,{{{\rm Y}}_2}^H]& =& {[{{\rm Y}}_1,{{\rm Y}}_2]}^H-\gamma {\pounds }_{{{\rm Y}}_1}{{\rm Y}}_2,\hfill \\ \hfill [{{{\rm Y}}_1}^H,{{{\rm Y}}_2}^H]& =& {[{{\rm Y}}_1,{{\rm Y}}_2]}^H-\gamma \widehat{R}({{\rm Y}}_1,{{\rm Y}}_2),\hfill \end{array}$$

(7)

where ${\pounds }_{{{\rm Y}}_1}$ represents the Lie derivative with respect to ${{\rm Y}}_1$ and $\widehat{R}$ represents the curvature tensor of $\widehat{\nabla }$ given by ${\widehat{\nabla }}_{{{\rm Y}}_1}{{\rm Y}}_2={\nabla }_{{{\rm Y}}_2}{{\rm Y}}_1+[{{\rm Y}}_1,{{\rm Y}}_2]$.

In addition, let P and Q be arbitrary elements of $\Im \left(M\right)$, then

$$\begin{array}{ccc}\hfill {(P\otimes Q)}^V& =& P^V\otimes Q^V,\hfill \\ \hfill {(P\otimes Q)}^C& =& P^C\otimes Q^V+P^V\otimes Q^C,\hfill \\ \hfill {(P\otimes Q)}^H& =& P^H\otimes Q^V+P^V\otimes Q^H.\hfill \end{array}$$

Let $g^C$ be the complete lift on $TM$ of a Riemannian metric g on M. Then [20]

$$\begin{array}{ccc}\hfill g^C({{{\rm Y}}_1}^C,{{{\rm Y}}_2}^V)& =& g^C({{{\rm Y}}_1}^V,{{{\rm Y}}_2}^C)={\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V,\hfill \\ \hfill g^C({{{\rm Y}}_1}^V,{{{\rm Y}}_2}^V)& =& 0,\hfill \\ \hfill g^C({{{\rm Y}}_1}^C,{{{\rm Y}}_2}^C)& =& {\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^C,\hfill \end{array}$$

where ${{\rm Y}}_1$ and ${{\rm Y}}_2$ are vector fields on M.

2.1. Metallic Structure

The quadratic structure J on M satisfying

$$J^2=pJ+qI,$$

(8)

where J denotes a tensor field of type (1,1), I is the identity vector field, and $p,q$ are natural numbers, named as a metallic structure. The structure $(M,J)$ is called a metallic manifold [26,27,28,29,30,31].

Let g be a Riemannian metric on M such that

$$g(J{{\rm Y}}_1,{{\rm Y}}_2)=g({{\rm Y}}_1,J{{\rm Y}}_2)$$

or equally,

$$g(J{{\rm Y}}_1,J{{\rm Y}}_2)=pg(J{{\rm Y}}_1,{{\rm Y}}_2)+qg({{\rm Y}}_1,{{\rm Y}}_2),$$

where ${{\rm Y}}_1$ and ${{\rm Y}}_2$ are vector fields on M. The structure $(M,J,g)$ is said to be a metallic Riemannian manifold [32,33].

The Nijenhuis tensor of J is denoted by $N_J$ and given by

$$N_J({{\rm Y}}_1,{{\rm Y}}_2)=[J{{\rm Y}}_1,J{{\rm Y}}_2]-J[J{{\rm Y}}_1,{{\rm Y}}_2]-J[{{\rm Y}}_1,J{{\rm Y}}_2]+J^2[{{\rm Y}}_1,{{\rm Y}}_2],$$

J is integrable if $N_J({{\rm Y}}_1,{{\rm Y}}_2)=0.$

2.2. Almost Quadratic $\varphi$-Structure

Debnath et al. [23] introduced the notion of structure $(\varphi ,\xi ,\eta )$ and discussed some geometric properties of such structures. Next, Gonul et al. [24] investigated the connection between $MS$ and almost quadratic $\varphi$-structures. Consider a non-null tensor fields $\varphi$ of type (1,1), a 1-form $\eta$ and a vector field $\xi$ on M satisfying

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

(9)

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

(10)

where p and q are constants and I is the identity vector field. The structure $(\varphi ,\xi ,\eta )$ is called an almost quadratic $\varphi$-structure on M and the manifold $(M,\varphi ,\xi ,\eta )$ is called an almost quadratic $\varphi$-manifold [23,24,34].

Furthermore,

$$g(\varphi {{\rm Y}}_1,{{\rm Y}}_2)=g({{\rm Y}}_1,\varphi {{\rm Y}}_2)$$

or equally,

$$g(\varphi {{\rm Y}}_1,\varphi {{\rm Y}}_2)=pg(\varphi {{\rm Y}}_1,{{\rm Y}}_2)+qg({{\rm Y}}_1,{{\rm Y}}_2)-q\eta \left({{\rm Y}}_1\right)\eta \left({{\rm Y}}_2\right).$$

The structure $(\varphi ,\xi ,\eta ,g)$ is termed as an almost quadratic metric $\varphi$-structure and the manifold $(M,\varphi ,\xi ,\eta ,g)$ is called an almost quadratic metric $\varphi$-manifold.

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

$$g({{\rm Y}}_1,\xi )=\eta \left({{\rm Y}}_1\right)$$

and the 2-Form $\Phi$ is given by [35]

$$\Phi ({{\rm Y}}_1,{{\rm Y}}_2)=g({{\rm Y}}_1,\varphi {{\rm Y}}_2)$$

(11)

is said to be the fundamental form of an almost quadratic metric $\varphi$-manifold.

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

$$N_{\varphi }({{\rm Y}}_1,{{\rm Y}}_2)=[\varphi {{\rm Y}}_1,\varphi {{\rm Y}}_2]-\varphi [\varphi {{\rm Y}}_1,{{\rm Y}}_2]-\varphi [{{\rm Y}}_1,\varphi {{\rm Y}}_2]+{\varphi }^2[{{\rm Y}}_1,{{\rm Y}}_2],$$

where ${{\rm Y}}_1$ and ${{\rm Y}}_2$ are vector fields on M.

Proposition 1

([24]). Let $(M,\varphi ,\xi ,\eta ,g,\nabla )$ be a $(\beta ,\varphi )$-Kenmotsu quadratic metric manifold such that $\left({\nabla }_{{{\rm Y}}_1}\varphi \right){{\rm Y}}_2=\beta g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\xi +\beta \eta \left({{\rm Y}}_2\right)\varphi {{\rm Y}}_1.$ Then the structure $(\varphi ,\xi ,\eta )$ is integrable; that is, the Nijenhuis tensor $N_{\varphi }=0$, where ∇ is the Levi-Civita connection.

3. Proposed Theorems for the Complete Lifts of Metallic Structures on the Tangent Bundle Over $(\mathit{\varphi },\mathit{\xi },\mathit{\eta })$

In this section, we study the structure $(\varphi ,\xi ,\eta )$ geometrically using complete lift on $TM$. A tensor field $\tilde{F}$ on the tangent bundle is defined and show that it is an $MS$ by using the complete lift on $TM$ over $(\varphi ,\xi ,\eta )$. Next, mathematical operators, namely fundamental 2-Form $\Omega$ and the derivative $d\Omega$ using the complete lift on $TM$ over $(\varphi ,\xi ,\eta )$, are calculated. Furthermore, the integrability condition and the Lie derivative of an $MS\left(\tilde{F}\right)$ by using the complete lift on $TM$ over $(\varphi ,\xi ,\eta )$ are established.

Let M be an n dimensional differentiable manifold and $\varphi$, $\eta$, and $\xi$ be a tensor field of type (1,1), a 1-form and a vector field on M, respectively.

Applying complete lifts on (9), (10) and using (1), we obtain

$$\begin{array}{ccc}\hfill {\left({\varphi }^C\right)}^2& =& p{\varphi }^C+qI-q({\eta }^V\otimes {\xi }^C+{\eta }^C\otimes {\xi }^V),\hfill \\ \hfill {\eta }^C\left({\xi }^C\right)& =& {\eta }^V\left({\xi }^V\right)=0,{\eta }^V\left({\xi }^C\right)={\eta }^C\left({\xi }^V\right)=1,\hfill \\ \hfill {\eta }^C\circ {\varphi }^C& =& {\eta }^V\circ {\varphi }^C={\eta }^C\circ {\varphi }^V={\eta }^V\circ {\varphi }^V=0,\hfill \\ \hfill {\varphi }^C\left({\xi }^V\right)& =& {\varphi }^V\left({\xi }^C\right)={\varphi }^C\left({\xi }^C\right)={\varphi }^V\left({\xi }^V\right)=0,\hfill \end{array}$$

where ${\varphi }^C,{\eta }^C,{\xi }^C,{\varphi }^V,{\eta }^V$, and ${\xi }^V$ are complete and vertical lifts of $\varphi$, $\eta$, and $\xi$, respectively, on $TM$. Azami [20] defined a tensor field J of type (1,1) on $TM$ with an almost paracontact structure $(\varphi ,\eta ,\xi ,g)$ as

$$J=\frac{p}{2}I-\left(\frac{2{\sigma }_p^q-p}{2}\right)\left({\varphi }^C+{\eta }^V\otimes {\xi }^V+{\eta }^C\otimes {\xi }^C\right)$$

and proved that it is an $MS$ on $TM$.

Recently, Khan [21] introduced a tensor $\tilde{J}$ on $FM$ immersed with an almost contact structure $(\varphi ,\eta ,\xi ,g)$ as

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

where ${\varphi }^H,{\eta }^{H_{\alpha }},\alpha =1,2,\dots ,n$ and ${\xi }^H$ are horizontal lifts of a tensor field $\varphi$ of type (1,1), a 1-form $\eta$ and a vector field $\xi$, respectively, and ${\xi }^{\left(\alpha \right)}$ is $\alpha$-vertical lift of $\xi$ on $FM$.

From Azami [20] and Khan [21], let us introduce a new $(1,1)$-type tensor field $\tilde{F}$ on $TM$ as

$$\tilde{F}=\frac{p}{2}I-A\left[{\varphi }^C+\sqrt{q}\left({\eta }^V\otimes {\xi }^V+{\eta }^C\otimes {\xi }^C\right)\right],$$

(12)

where $A=\frac{2{\sigma }_p^q-p}{2\sqrt{p{\varphi }^C+q}}$. Since $p,q$ are natural numbers and $\varphi$ is non-singular, therefore $p{\varphi }^C+q>0$ and $A\ne 0$.

Theorem 1.

Let $TM$ be a tangent bundle of M immersed with structure $(\varphi ,\xi ,\eta )$. Then $\tilde{F}$ given by (12) is a metallic structure on $TM$.

Proof. 

Let ${{\rm Y}}_1$ be a vector field on M and ${{{\rm Y}}_1}^C$ and ${{{\rm Y}}_1}^V$ be complete and vertical lifts of ${{\rm Y}}_1$, respectively, on $TM$. Applying ${\xi }^V$, ${\xi }^C$, and ${\varphi }^C$ on (12), we obtain

$$\begin{array}{ccc}& & \tilde{F}\left({\xi }^V\right)=\frac{p}{2}{\xi }^V-A\sqrt{q}{\xi }^C,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & \tilde{F}\left({\xi }^C\right)=\frac{p}{2}{\xi }^C-A\sqrt{q}{\xi }^V,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & \tilde{F}\left({\varphi }^C{\widetilde{{\rm Y}}}_1\right)=\frac{p}{2}{\varphi }^C{\widetilde{{\rm Y}}}_1-A[p{\varphi }^C{\widetilde{{\rm Y}}}_1+q{\widetilde{{\rm Y}}}_1-q({\eta }^V\left({\widetilde{{\rm Y}}}_1\right){\xi }^C+{\eta }^C\left({\widetilde{{\rm Y}}}_1\right){\xi }^V)],\hfill \end{array}$$

(15)

where ${\widetilde{{\rm Y}}}_1$ is a vector field on $TM$.

In the view of (12)–(15), we obtain

$$\begin{array}{ccc}\hfill {\tilde{F}}^2\left({\widetilde{{\rm Y}}}_1\right)& =& \frac{p}{2}\tilde{F}\left({\widetilde{{\rm Y}}}_1\right)-A\left[{\phi }^C{\widetilde{{\rm Y}}}_1+\sqrt{q}({\eta }^V\left({\widetilde{{\rm Y}}}_1\right)\tilde{F}\left({\xi }^V\right)+{\eta }^C\left({\widetilde{{\rm Y}}}_1\right)\tilde{F}\left({\xi }^C\right))\right],\hfill \\ & =& p\tilde{F}\left({\widetilde{{\rm Y}}}_1\right)+q\left({\widetilde{{\rm Y}}}_1\right).\hfill \end{array}$$

This shows that $\tilde{F}$ is an $MS$ on $TM$. □

Corollary 1.

Let ${{\rm Y}}_1$ and ${{\rm Y}}_2$ be vector fields on M and $\tilde{F}$ be an $MS$ on $TM$ given by (12) such that $\eta \left({{\rm Y}}_1\right)=0$, then

$$\begin{array}{ccc}\hfill \tilde{F}{{{\rm Y}}_1}^V& =& \frac{p}{2}{{{\rm Y}}_1}^V-A\left[{\left(\varphi {{\rm Y}}_1\right)}^V+\sqrt{q}{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^C\right],\hfill \\ \hfill \tilde{F}{{\rm Y}}_1^C& =& \frac{p}{2}{{\rm Y}}_1^C-A\left[{\left(\varphi {{\rm Y}}_1\right)}^C+\sqrt{q}{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^V+{\left(\eta \left({{\rm Y}}_1\right)\right)}^C{\xi }^C\right].\hfill \end{array}$$

If $\eta \left({{\rm Y}}_1\right)=0$, then

$$\begin{array}{ccc}\hfill \tilde{F}{{\rm Y}}_1^V& =& \frac{p}{2}{{\rm Y}}_1^V-A{\left(\varphi {{\rm Y}}_1\right)}^V,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill \tilde{F}{{\rm Y}}_1^C& =& \frac{p}{2}{{\rm Y}}_1^C-A{\left(\varphi {{\rm Y}}_1\right)}^C.\hfill \end{array}$$

(17)

Proof. 

The proof is obtained by applying ${{\rm Y}}_1^C$ and ${{\rm Y}}_1^V$ on $\tilde{F}$ given by (12) and using $\eta \left({{\rm Y}}_1\right)=0$.

Let $g^C$ be the complete lift of the metric g on $TM$. The 2-form on $TM$ defined by

$$\Omega ({\widetilde{{\rm Y}}}_1,\widetilde{{{\rm Y}}_2})=g^C({\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2),$$

(18)

where ${\widetilde{{\rm Y}}}_1$ and $\widetilde{{{\rm Y}}_2}$ are vector fields and $\tilde{F}$ is an $MS$ given by (12) on $TM$. □

Theorem 2.

Let $TM$ be the tangent bundle of M, $g^C$ be the complete lift of g and $\tilde{F}$ be an $MS$ given by (12) on $TM$, then the 2-form Ω is given by

$$\begin{array}{ccc}\hfill \left(i\right)\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^C)& =& \frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^C-A\sqrt{q}\{\eta {\left({{\rm Y}}_1\right)}^V\eta {\left({{\rm Y}}_2\right)}^V+\eta {\left({{\rm Y}}_1\right)}^C\eta {\left({{\rm Y}}_2\right)}^C\}\hfill \\ & -& A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^C,\hfill \\ \hfill \left(ii\right)\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^V)& =& \frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V-A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^C\eta {\left({{\rm Y}}_2\right)}^V-A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V,\hfill \\ \hfill \left(iii\right)\Omega ({{\rm Y}}_1^V,{{{\rm Y}}_2}^C)& =& \frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V-A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^V\eta {\left({{\rm Y}}_2\right)}^C-A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V,\hfill \\ \hfill \left(iv\right)\Omega ({{\rm Y}}_1^V,{{{\rm Y}}_2}^V)& =& -A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^V\eta {\left({{\rm Y}}_2\right)}^V,\hfill \end{array}$$

where ${\widetilde{{\rm Y}}}_1$ and ${\widetilde{{\rm Y}}}_2$ are vector fields on $TM$.

Proof. 

(i) Let ${\widetilde{{\rm Y}}}_1={{\rm Y}}_1^C$ and ${\widetilde{{\rm Y}}}_2={{\rm Y}}_2^C$ in (18) and using (1) and (12), we have

$$\begin{array}{ccc}\hfill \Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^C)& =& g^C({{\rm Y}}_1^C,\tilde{F}{{{\rm Y}}_2}^C)\hfill \\ & =& g^C\left({{\rm Y}}_1^C,\frac{p}{2}{{{\rm Y}}_2}^C-A\left[{\left(\varphi {{\rm Y}}_2\right)}^C+\sqrt{q}{\left(\eta \left({{\rm Y}}_2\right)\right)}^V{\xi }^V+{\left(\eta \left({{\rm Y}}_2\right)\right)}^C{\xi }^C\right]\right)\hfill \\ & =& \frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^C-A\sqrt{q}[{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\left(\eta \left({{\rm Y}}_2\right)\right)}^V+{\left(\eta \left({{\rm Y}}_1\right)\right)}^C{\left(\eta \left({{\rm Y}}_2\right)\right)}^C]\hfill \\ & -& A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^C.\hfill \end{array}$$

(ii) Let ${\widetilde{{\rm Y}}}_1={{\rm Y}}_1^C$ and ${\widetilde{{\rm Y}}}_2={{\rm Y}}_2^V$ in (18) and using (1) and (12), we have

$$\begin{array}{ccc}\hfill \Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^V)& =& g^C({{\rm Y}}_1^C,\tilde{F}{{{\rm Y}}_2}^V)\hfill \\ & =& g^C\left({{\rm Y}}_1^C,\frac{p}{2}{{{\rm Y}}_2}^V-A\left[{\left(\varphi {{\rm Y}}_2\right)}^V+\sqrt{q}{\left(\eta \left({{\rm Y}}_2\right)\right)}^V{\xi }^C\right]\right)\hfill \\ & =& \frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V-A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^C\eta {\left({{\rm Y}}_2\right)}^V-A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V.\hfill \end{array}$$

(iii) Let ${\widetilde{{\rm Y}}}_1={{\rm Y}}_1^V$ and ${\widetilde{{\rm Y}}}_2={{{\rm Y}}_2}^C$ in (18) and using (1) and (12), we have

$$\begin{array}{ccc}\hfill \Omega ({{\rm Y}}_1^V,{{{\rm Y}}_2}^C)& =& g^C({{\rm Y}}_1^V,\tilde{F}{{{\rm Y}}_2}^C)\hfill \\ & =& g^C\left({{\rm Y}}_1^V,\frac{p}{2}{{{\rm Y}}_2}^C-A\left[{\left(\varphi {{\rm Y}}_2\right)}^C+\sqrt{q}{\left(\eta \left({{\rm Y}}_2\right)\right)}^V{\xi }^V+\eta \left({{\rm Y}}_2\right){)}^C{\xi }^C\right]\right)\hfill \\ & =& \frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V-A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^V\eta {\left({{\rm Y}}_2\right)}^C-A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V.\hfill \end{array}$$

(iv) Let ${\widetilde{{\rm Y}}}_1={{\rm Y}}_1^V$ and ${\widetilde{{\rm Y}}}_2={{{\rm Y}}_2}^V$ in (18) and using (1) and (12), we have

$$\begin{array}{ccc}\hfill \Omega ({{\rm Y}}_1^V,{{{\rm Y}}_2}^V)& =& g^C({{\rm Y}}_1^V,\tilde{F}{{{\rm Y}}_2}^V)\hfill \\ & =& g^C\left({{\rm Y}}_1^V,\frac{p}{2}{{{\rm Y}}_2}^V-A\left[{\left(\varphi {{\rm Y}}_2\right)}^V+\sqrt{q}{\left(\eta \left({{\rm Y}}_2\right)\right)}^V{\xi }^C\right]\right)\hfill \\ & =& -A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^V\eta {\left({{\rm Y}}_2\right)}^V.\hfill \end{array}$$

□

Theorem 3. 

Let $TM$ be the tangent bundle of M, $g^C$ be the complete lift of g, and $\tilde{F}$ be an $MS$ given by (12), then the derivative $d\Omega$ is given by

(i) 

$$\begin{gathered} 3d\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^C,{{{\rm Y}}_3}^V)=\frac{p}{2}({\left({{\rm Y}}_{1}g({{\rm Y}}_2,{{\rm Y}}_3)\right)}^V-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,{{\rm Y}}_3)\right)}^V+{\left({{\rm Y}}_{3}g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V) \\ -A({\left({{\rm Y}}_{1}g({{\rm Y}}_2,\varphi {{\rm Y}}_3)\right)}^V-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,\varphi {{\rm Y}}_3)\right)}^V+{\left({{\rm Y}}_{3}g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V) \\ -A\sqrt{q}({{\rm Y}}_1^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_3\right)}^C-{{{\rm Y}}_2}^C{\left(\eta \left({{\rm Y}}_3\right)\right)}^V{\left(\eta \left({{\rm Y}}_1\right)\right)}^C \\ +{{{\rm Y}}_3}^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_1\right)}^C+{{{\rm Y}}_3}^V\eta {\left({{\rm Y}}_2\right)}^C\eta {\left({{\rm Y}}_1\right)}^C)-\frac{p}{2}({\left(g([{{\rm Y}}_1,{{\rm Y}}_2],{{\rm Y}}_3)\right)}^V \\ -{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],{{\rm Y}}_2)\right)}^V+{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],{{\rm Y}}_1)\right)}^V)+A({\left(g([{{\rm Y}}_1,{{\rm Y}}_2],\varphi {{\rm Y}}_3)\right)}^V \\ -{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],\varphi {{\rm Y}}_2)\right)}^V+{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],\varphi {{\rm Y}}_1)\right)}^V)+A\sqrt{q}({\left(\eta \left({{\rm Y}}_3\right)\right)}^V\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^C \\ -\eta {\left({{\rm Y}}_2\right)}^C\eta {\left([{{\rm Y}}_1,{{\rm Y}}_3]\right)}^V+\eta {\left({{\rm Y}}_1\right)}^C\eta {\left([{{\rm Y}}_2,{{\rm Y}}_3]\right)}^V). \end{gathered}$$

(ii) 

$$\begin{gathered} 3d\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^V,{{{\rm Y}}_3}^V)=\frac{p}{2}(({\left({{\rm Y}}_{3}g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,{{\rm Y}}_3)\right)}^V) \\ +A({\left({{\rm Y}}_{3}g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,\varphi {{\rm Y}}_3)\right)}^V)-A\sqrt{q}({{\rm Y}}_1^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_3\right)}^V \\ +{{{\rm Y}}_2}^V\eta {\left({{\rm Y}}_3\right)}^V\eta {\left({{\rm Y}}_1\right)}^V+{{{\rm Y}}_3}^V\eta {\left({{\rm Y}}_1\right)}^C\eta {\left({{\rm Y}}_2\right)}^V). \end{gathered}$$

(iii) 

$$\begin{gathered} 3d\Omega ({{\rm Y}}_1^V,{{{\rm Y}}_2}^V,{{{\rm Y}}_3}^V)=-A\sqrt{q}({{\rm Y}}_1^V\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_3\right)}^V+{{{\rm Y}}_2}^V\eta {\left({{\rm Y}}_3\right)}^V\eta {\left({{\rm Y}}_1\right)}^V \\ -{{{\rm Y}}_3}^V\eta {\left({{\rm Y}}_1\right)}^V\eta {\left({{\rm Y}}_2\right)}^V). \end{gathered}$$

(iv) 

$$\begin{gathered} 3d\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^C,{{{\rm Y}}_3}^C)=\frac{p}{2}({\left({{\rm Y}}_{1}g({{\rm Y}}_2,{{\rm Y}}_3)\right)}^C-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,{{\rm Y}}_3)\right)}^C+{\left({{\rm Y}}_{3}g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^C) \\ -A({\left({{\rm Y}}_{1}g({{\rm Y}}_2,\varphi {{\rm Y}}_3)\right)}^C-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,\varphi {{\rm Y}}_3)\right)}^C+{\left({{\rm Y}}_{3}g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^C) \\ -A\sqrt{q}({{\rm Y}}_1^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_3\right)}^V+{{\rm Y}}_1^C\eta {\left({{\rm Y}}_2\right)}^C\eta {\left({{\rm Y}}_3\right)}^C-{{{\rm Y}}_2}^C\eta {\left({{\rm Y}}_3\right)}^V\eta {\left({{\rm Y}}_1\right)}^C \\ -{{{\rm Y}}_2}^C\eta {\left({{\rm Y}}_3\right)}^C\eta {\left({{\rm Y}}_1\right)}^C+{{{\rm Y}}_3}^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_1\right)}^V+{{{\rm Y}}_3}^C\eta {\left({{\rm Y}}_2\right)}^C\eta {\left({{\rm Y}}_1\right)}^C) \\ -\frac{p}{2}({\left(g([{{\rm Y}}_1,{{\rm Y}}_2],{{\rm Y}}_3)\right)}^C-{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],{{\rm Y}}_2)\right)}^C+{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],{{\rm Y}}_1)\right)}^C) \\ +A({\left(g([{{\rm Y}}_1,{{\rm Y}}_2],\varphi {{\rm Y}}_3)\right)}^C-{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],\varphi {{\rm Y}}_2)\right)}^C+{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],\varphi {{\rm Y}}_1)\right)}^C) \\ +A\sqrt{q}(\eta {\left({{\rm Y}}_3\right)}^V\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V-\eta {\left({{\rm Y}}_2\right)}^C\eta {\left([{{\rm Y}}_1,{{\rm Y}}_3]\right)}^V+\eta {\left({{\rm Y}}_1\right)}^C\eta {\left([{{\rm Y}}_2,{{\rm Y}}_3]\right)}^V) \\ +A\sqrt{q}(\eta {\left({{\rm Y}}_3\right)}^C\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^C-\eta {\left({{\rm Y}}_2\right)}^C\eta {\left([{{\rm Y}}_1,{{\rm Y}}_3]\right)}^C+\eta {\left({{\rm Y}}_1\right)}^C\eta {\left([{{\rm Y}}_2,{{\rm Y}}_3]\right)}^C). \end{gathered}$$

Proof. 

We have

$$\begin{array}{ccc}\hfill 3d\Omega (\widetilde{{{\rm Y}}_1},\widetilde{{{\rm Y}}_2},\widetilde{{{\rm Y}}_3})& =& \{\widetilde{{{\rm Y}}_1}(\Omega (\widetilde{{{\rm Y}}_2},\widetilde{{{\rm Y}}_3}))-\widetilde{{{\rm Y}}_2}(\Omega (\widetilde{{{\rm Y}}_1},\widetilde{{{\rm Y}}_3}))+\widetilde{{{\rm Y}}_3}(\Omega (\widetilde{{{\rm Y}}_1},\widetilde{{{\rm Y}}_2}))\hfill \\ & -& \Omega ([\widetilde{{{\rm Y}}_1},\widetilde{{{\rm Y}}_2}],\widetilde{{{\rm Y}}_3})+\Omega ([\widetilde{{{\rm Y}}_1},\widetilde{{{\rm Y}}_3}],\widetilde{{{\rm Y}}_2})-\Omega ([\widetilde{{{\rm Y}}_2},\widetilde{{{\rm Y}}_3}],\widetilde{{{\rm Y}}_1})\},\hfill \end{array}$$

called coboundary formula [35]. Here $\widetilde{{{\rm Y}}_1},\widetilde{{{\rm Y}}_2},\widetilde{{{\rm Y}}_3}$ are arbitrary vector fields on $TM$.

Applying (1)–(7), (12)–(15), Theorem 2 and using $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$, we have

$$\begin{array}{ccc}\hfill \left(i\right)& & 3d\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^C,{{{\rm Y}}_3}^V)={{\rm Y}}_1^C(\Omega ({{{\rm Y}}_2}^C,{{{\rm Y}}_3}^V))-{{{\rm Y}}_2}^C(\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_3}^V))\hfill \\ & & +{{{\rm Y}}_3}^V(\Omega ({{\rm Y}}_1^C,{{{\rm Y}}_2}^C))-\Omega ([{{\rm Y}}_1^C,{{{\rm Y}}_2}^C],{{{\rm Y}}_3}^V)+\Omega ([{{\rm Y}}_1^C,{{{\rm Y}}_3}^V],{{{\rm Y}}_2}^C)\hfill \\ & & -\Omega ([{{{\rm Y}}_2}^C,{{{\rm Y}}_3}^C],{{\rm Y}}_1^V)={{\rm Y}}_1^C(\frac{p}{2}{\left(g({{\rm Y}}_2,{{\rm Y}}_3)\right)}^V-A{\left(g({{\rm Y}}_2,\varphi {{\rm Y}}_3)\right)}^V\hfill \\ & & -{{{\rm Y}}_2}^C(\frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_3)\right)}^V-A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_3)\right)}^V-A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^C\eta {\left({{\rm Y}}_3\right)}^V)\hfill \\ & & +{{{\rm Y}}_3}^C(\frac{p}{2}{\left(g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V-A{\left(g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V-A\sqrt{q}\eta {\left({{\rm Y}}_1\right)}^C\eta {\left({{\rm Y}}_2\right)}^V)\hfill \\ & & -(\frac{p}{2}{\left(g([{{\rm Y}}_1,{{\rm Y}}_2],{{\rm Y}}_3)\right)}^V-A{\left(g([{{\rm Y}}_1,{{\rm Y}}_2],\varphi {{\rm Y}}_3)\right)}^V-A\sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^C\eta {\left({{\rm Y}}_3\right)}^V)\hfill \\ & & +(\frac{p}{2}{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],{{\rm Y}}_2)\right)}^V-A{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],\varphi {{\rm Y}}_2)\right)}^V-A\sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_3]\right)}^C\eta {\left({{\rm Y}}_2\right)}^V)\hfill \\ & & -(\frac{p}{2}{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],{{\rm Y}}_1)\right)}^V-A{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],\varphi {{\rm Y}}_1)\right)}^V-A\sqrt{q}\eta {\left([{{\rm Y}}_2,{{\rm Y}}_3]\right)}^C\eta {\left({{\rm Y}}_1\right)}^V)\hfill \\ & & =\frac{p}{2}({\left({{\rm Y}}_{1}g({{\rm Y}}_2,{{\rm Y}}_3)\right)}^V-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,{{\rm Y}}_3)\right)}^V+{\left({{\rm Y}}_{3}g({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V)\hfill \\ & & -A({\left({{\rm Y}}_{1}g({{\rm Y}}_2,\varphi {{\rm Y}}_3)\right)}^V-{\left({{\rm Y}}_{2}g({{\rm Y}}_1,\varphi {{\rm Y}}_3)\right)}^V+{\left({{\rm Y}}_{3}g({{\rm Y}}_1,\varphi {{\rm Y}}_2)\right)}^V)\hfill \\ & & -A\sqrt{q}({{\rm Y}}_1^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_3\right)}^C-{{{\rm Y}}_2}^C\eta {\left({{\rm Y}}_3\right)}^V\eta {\left({{\rm Y}}_1\right)}^C+{{{\rm Y}}_3}^C\eta {\left({{\rm Y}}_2\right)}^V\eta {\left({{\rm Y}}_1\right)}^C\hfill \\ & & +{{{\rm Y}}_3}^V\eta {\left({{\rm Y}}_2\right)}^C\eta {\left({{\rm Y}}_1\right)}^C)-\frac{p}{2}({\left(g([{{\rm Y}}_1,{{\rm Y}}_2],{{\rm Y}}_3)\right)}^V-{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],{{\rm Y}}_2)\right)}^V\hfill \\ & & +{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],{{\rm Y}}_1)\right)}^V)+A({\left(g([{{\rm Y}}_1,{{\rm Y}}_2],\varphi {{\rm Y}}_3)\right)}^V-{\left(g([{{\rm Y}}_1,{{\rm Y}}_3],\varphi {{\rm Y}}_2)\right)}^V\hfill \\ & & +{\left(g([{{\rm Y}}_2,{{\rm Y}}_3],\varphi {{\rm Y}}_1)\right)}^V)+A\sqrt{q}(\eta {\left({{\rm Y}}_3\right)}^V\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^C\hfill \\ & & -\eta {\left({{\rm Y}}_2\right)}^C\eta {\left([{{\rm Y}}_1,{{\rm Y}}_3]\right)}^V+\eta {\left({{\rm Y}}_1\right)}^C\eta {\left([{{\rm Y}}_2,{{\rm Y}}_3]\right)}^V).\hfill \end{array}$$

Other results were obtained by using similar devices. □

Theorem 4.

A metallic structure $\tilde{F}$, defined by (12), is integrable on $TM$ over $(\varphi ,\xi ,\eta )$ if and only if $N_{\tilde{F}}=0$, which is equivalent to the conditions

$$\eta \left([\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)=0,\eta \left([\varphi {{\rm Y}}_1,\xi ]\right)=0,\eta \left([{{\rm Y}}_1,{{\rm Y}}_2]\right)=0,\eta \left([{{\rm Y}}_1,\xi ]\right)=0,$$

and $(\varphi ,\xi ,\eta )$ is integrable i.e. $N_{\varphi }=0.$

Proof. 

Let $N_{\tilde{F}}$ stand for the Nijenhuis tensor of $\tilde{F}$. Then

$$N_{\tilde{F}}({\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2)=[\tilde{F}{\widetilde{{\rm Y}}}_1,\tilde{F}{\widetilde{{\rm Y}}}_2]-\tilde{F}[\tilde{F}{\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2]-\tilde{F}[{\widetilde{{\rm Y}}}_1,\tilde{F}{\widetilde{{\rm Y}}}_2]+{\tilde{F}}^2[{\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2],$$

(19)

where ${\widetilde{{\rm Y}}}_1$ and ${\widetilde{{\rm Y}}}_2$ are vector fields on $TM$.

Applying (1)–(7), (12)–(15) on (19), and using $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$, we have

$$\begin{array}{ccc}\hfill N_{\tilde{F}}({{\rm Y}}_1^V,{{\rm Y}}_2^V)& =& 0,\hfill \\ \hfill N_{\tilde{F}}({{\rm Y}}_1^V,{{\rm Y}}_2^C)& =& A^2{\left(N_{\varphi }({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V+A^2\sqrt{q}\eta {\left([\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^C\hfill \\ & +& A\sqrt{q}\eta {\left([{{\rm Y}}_1,\varphi {{\rm Y}}_2]\right)}^V{\xi }^C+q\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^V\hfill \\ & +& q\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^C,\hfill \\ \hfill N_{\tilde{F}}({{\rm Y}}_1^C,{{\rm Y}}_2^C)& =& A^2{\left(N_{\varphi }({{\rm Y}}_1,{{\rm Y}}_2)\right)}^C-A^2\sqrt{q}\eta {\left([\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^V\hfill \\ & -& A^2\sqrt{q}\eta {\left([{{\rm Y}}_1,\varphi {{\rm Y}}_2]\right)}^V{\xi }^V\hfill \\ & -& A^2\sqrt{q}\eta {\left([\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^C{\xi }^C-A^2\sqrt{q}\eta {\left([{{\rm Y}}_1,\varphi {{\rm Y}}_2]\right)}^C{\xi }^C,\hfill \\ \hfill N_{\tilde{F}}({{\rm Y}}_1^V,{\xi }^V)& =& 0,\hfill \\ \hfill N_{\tilde{F}}({{\rm Y}}_1^V,{\xi }^C)& =& A^2{\left({\varphi }^2[{{\rm Y}}_1,\xi ]\right)}^V+A^{2}q\left(\eta [{{\rm Y}}_1,{{\rm Y}}_2]\right){\xi }^V\hfill \\ & -& A^2{\left(\varphi [\varphi {{\rm Y}}_1,\xi ]\right)}^V-A^2\sqrt{q}{\left([\varphi {{\rm Y}}_1,\xi ]\right)}^V{\xi }^C,\hfill \\ \hfill N_{\tilde{F}}({{\rm Y}}_1^C,{\xi }^V)& =& A^2{\left({\varphi }^2[{{\rm Y}}_1,\xi ]\right)}^V+A^2\sqrt{q}{\left([\varphi {{\rm Y}}_1,\xi ]\right)}^C-A^2{\left(\varphi [\varphi {{\rm Y}}_1,\xi ]\right)}^V\hfill \\ & -& A^2\sqrt{q}\eta {\left([\varphi {{\rm Y}}_1,\xi ]\right)}^V{\xi }^C+-A\sqrt{q}{\left(\varphi [{{\rm Y}}_1,\xi ]\right)}^C-\frac{p^2}{2}{[{{\rm Y}}_1,\xi ]}^V\hfill \\ & -& A^{2}q\eta {\left([{{\rm Y}}_1,\xi ]\right)}^C{\xi }^C+Ap{\left(\varphi [{{\rm Y}}_1,\xi ]\right)}^V+pA\sqrt{q}\eta {\left([{{\rm Y}}_1,\xi ]\right)}^V{\xi }^C,\hfill \\ \hfill N_{\tilde{F}}({{\rm Y}}_1^C,{\xi }^C)& =& A^2{\left({\varphi }^2[{{\rm Y}}_1,\xi ]\right)}^C+A^{2}q{\left(\eta [{{\rm Y}}_1,\xi ]\xi \right)}^C+A^2\sqrt{q}{[\varphi {{\rm Y}}_1,\xi ]}^V\hfill \\ & -& A^2{\left(\varphi [\varphi {{\rm Y}}_1,\xi ]\right)}^C+A^2\sqrt{q}{\left(\eta [\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^V+A^2\sqrt{q}{\left(\eta [\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^C{\xi }^C\hfill \\ & -& A^2\sqrt{q}{\left(\varphi [{{\rm Y}}_1,\xi ]\right)}^V-A^{2}q{\left(\eta [{{\rm Y}}_1,\xi ]\right)}^V{\xi }^C,\hfill \\ \hfill N_{\tilde{F}}({\xi }^V,{\xi }^C)& =& 0.\hfill \end{array}$$

Let ${\widetilde{{\rm Y}}}_1$ and $\tilde{F}$ be a vector field and an $MS$, respectively, on $TM$. The Lie derivative of $\tilde{F}$ with respect to ${\widetilde{{\rm Y}}}_1$ is given by ([8], p. 113)

$$\left({\pounds }_{{\widetilde{{\rm Y}}}_1}\tilde{F}\right)\widetilde{{{\rm Y}}_2}=[{\widetilde{{\rm Y}}}_1,\tilde{F}{\widetilde{{\rm Y}}}_2]-\tilde{F}[{\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2],$$

(20)

where $\widetilde{{{\rm Y}}_2}$ is a vector field on $TM$. □

Theorem 5.

Let $\tilde{F}$ be an $MS$ on $TM$ given by (12) and ${{\rm Y}}_1$ and ${{\rm Y}}_2$ be vector fields on M such that $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$, then

$$\begin{array}{ccc}\hfill \left(i\right)\left({\pounds }_{{{{\rm Y}}_2}^V}\tilde{F}\right){{\rm Y}}_1^V& =& 0,\hfill \\ \hfill \left(ii\right)\left({\pounds }_{{{{\rm Y}}_2}^V}\tilde{F}\right){{\rm Y}}_1^C& =& A\left({\left(\varphi [{{\rm Y}}_2,{{\rm Y}}_1]\right)}^V-{[{{\rm Y}}_2,\varphi {{\rm Y}}_1]}^V+\sqrt{q}{\left(\eta [{{\rm Y}}_2,{{\rm Y}}_1]\right)}^V\right),\hfill \\ \hfill \left(iii\right)\left({\pounds }_{{{{\rm Y}}_2}^V}\tilde{F}\right){\xi }^V& =& -Aq{[{{\rm Y}}_2,\xi ]}^V,\hfill \\ \hfill \left(iv\right)\left({\pounds }_{{{{\rm Y}}_2}^V}\tilde{F}\right){\xi }^C& =& A\left(\varphi {[{{\rm Y}}_2,\xi ]}^V-\sqrt{q}{\left(\eta [{{\rm Y}}_2,\xi ]\right)}^V{\xi }^C\right),\hfill \\ \hfill \left(v\right)\left({\pounds }_{{{{\rm Y}}_2}^C}\tilde{F}\right){{\rm Y}}_1^V& =& A\left(\varphi {[{{\rm Y}}_2,{{\rm Y}}_1]}^V-{[{{\rm Y}}_2,\varphi {{\rm Y}}_1]}^V-\sqrt{q}(\eta {\left([{{\rm Y}}_2,{{\rm Y}}_1]\right)}^V{\xi }^C\right),\hfill \\ \hfill \left(vi\right)\left({\pounds }_{{{{\rm Y}}_2}^C}\tilde{F}\right){{\rm Y}}_1^C& =& A\left({\left(\varphi [{{\rm Y}}_2,{{\rm Y}}_1]\right)}^C-{[{{\rm Y}}_2,\varphi {{\rm Y}}_1]}^C\right)\hfill \\ & -& A\sqrt{q}\left({\left(\eta ({{\rm Y}}_2,{{\rm Y}}_1)\right)}^V{\xi }^V-{\left(\eta ({{\rm Y}}_2,{{\rm Y}}_1)\right)}^C{\xi }^C\right),\hfill \\ \hfill \left(vii\right)\left({\pounds }_{{{{\rm Y}}_2}^C}\tilde{F}\right){\xi }^V& =& A\left({\left(\varphi [{{\rm Y}}_2,\xi ]\right)}^V+\sqrt{q}{\left(\eta [{{\rm Y}}_2,\xi ]\right)}^V{\xi }^C-\sqrt{q}{[{{\rm Y}}_2,\xi ]}^C\right),\hfill \\ \hfill \left(viii\right)\left({\pounds }_{{{{\rm Y}}_2}^C}\tilde{F}\right){\xi }^C& =& A\left(\varphi [{{\rm Y}}_2,\xi ]{)}^C-\sqrt{q}{[{{\rm Y}}_2,\xi ]}^V\right)\hfill \\ & +& A\sqrt{q}\left((\eta {[{{\rm Y}}_2,\xi ]}^V{\xi }^V+{\left(\eta [{{\rm Y}}_2,\xi ]\right)}^C{\xi }^C\right),\hfill \end{array}$$

Proof. 

Applying (1)–(7), (12)–(15), and (20), and using $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$.

$$\begin{array}{ccc}\hfill \left(i\right){\pounds }_{{{{\rm Y}}_2}^V}\tilde{F}{{\rm Y}}_1^V& =& {\pounds }_{{{{\rm Y}}_2}^V}(\frac{p}{2}{{{\rm Y}}_1}^V-A\left[{\left(\varphi {{\rm Y}}_1\right)}^V+\sqrt{q}{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^C)\right]\hfill \\ \hfill \left({\pounds }_{{{{\rm Y}}_2}^V}\tilde{F}\right){{\rm Y}}_1^V+\tilde{F}{\pounds }_{{{{\rm Y}}_2}^V}{{\rm Y}}_1^V& =& \frac{p}{2}{\pounds }_{{{{\rm Y}}_2}^V}{{{\rm Y}}_1}^V\hfill \\ & -& A{\pounds }_{{{{\rm Y}}_2}^V}{\left(\varphi {{\rm Y}}_1\right)}^V-A\sqrt{q}{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\pounds }_{{{{\rm Y}}_2}^V}{\xi }^C\hfill \\ & =& 0.\hfill \end{array}$$

Others results are obtained by using similar devices. □

4. Proposed Theorems for the Horizontal Lift of Metallic Structures on the Tangent Bundle Over $(\mathit{\varphi },\mathit{\xi },\mathit{\eta })$

In this section, we study $(\varphi ,\xi ,\eta )$ geometrically using a horizontal lift on $TM$. A tensor field $F^{*}$ on the tangent bundle is defined and shows that it is an $MS$ by using the horizontal lift on $TM$ over $(\varphi ,\xi ,\eta )$. Furthermore, the integrability condition and Lie derivative of an $MS$$F^{*}$ by using the horizontal lift on $TM$ over $(\varphi ,\xi ,\eta )$ are established.

Let M be an n dimensional differentiable manifold and $\varphi$, $\eta$, and $\xi$ be the tensor field of type (1,1), a 1-form, and a vector field on M. Let ${\varphi }^H$, ${\eta }^H$, and ${\xi }^H$ be horizontal lifts of $\varphi$, $\eta$, and $\xi$, respectively, on $TM$. Applying horizontal lifts on (9), (10), and using (1), we obtain

$$\begin{array}{ccc}\hfill {\left({\varphi }^H\right)}^2& =& p{\varphi }^H+qI-q({\eta }^V\otimes {\xi }^H+{\eta }^H\otimes {\xi }^V),\hfill \\ \hfill {\eta }^H\left({\xi }^H\right)& =& {\eta }^V\left({\xi }^V\right)=0,{\eta }^V\left({\xi }^H\right)={\eta }^H\left({\xi }^V\right)=1,\hfill \\ \hfill {\eta }^H\circ {\varphi }^H& =& {\eta }^V\circ {\varphi }^H={\eta }^H\circ {\varphi }^V={\eta }^V\circ {\varphi }^V=0,\hfill \\ \hfill {\varphi }^H\left({\xi }^V\right)& =& {\varphi }^V\left({\xi }^H\right)={\varphi }^H\left({\xi }^V\right)={\varphi }^V\left({\xi }^V\right)=0.\hfill \end{array}$$

From Azami [20] and Khan [21], let us introduced a new tensor field $F^{*}$ of type (1,1) on $TM$ as

$$F^{*}=\frac{p}{2}I-B\left[{\varphi }^H+\sqrt{q}({\eta }^V\otimes {\xi }^V+{\eta }^H\otimes {\xi }^H)\right],$$

(21)

where $B=\frac{2{\sigma }_p^q-p}{2\sqrt{p{\varphi }^H+q}}$. Since $p,q$ are natural numbers and $\varphi$ is non-singular, therefore $p{\varphi }^H+q>0$ and $A\ne 0$.

Theorem 6.

Let the tangent bundle $TM$ of M be immersed with $(\varphi ,\xi ,\eta )$. Then the metallic structure $F^{*}$, given by (21), is an $MS$ on $TM$.

Proof. 

Let ${{\rm Y}}_1$ be a vector field on M and ${{\rm Y}}_1^H$, ${{\rm Y}}_1^C$, and ${{\rm Y}}_1^V$ be horizontal, complete, and vertical lifts of ${{\rm Y}}_1$, respectively, on $TM$. Applying ${\xi }^H$, ${\xi }^V$, ${\xi }^C$, and ${\varphi }^H$ on (21), we obtain

$$\begin{array}{ccc}\hfill \left(i\right)F^{*}\left({\xi }^H\right)& =& \frac{p}{2}{\xi }^H-B\sqrt{q}{\xi }^V,\hfill \\ \hfill \left(ii\right)F^{*}\left({\xi }^V\right)& =& \frac{p}{2}{\xi }^V-B\sqrt{q}{\xi }^H,\hfill \\ \hfill \left(iii\right)F^{*}\left({\xi }^C\right)& =& \frac{p}{2}{\xi }^C-B[{\varphi }^H\left({\nabla }_{\gamma }{{\rm Y}}_1\right)+\sqrt{q}{\varphi }^C\left({\nabla }_{\gamma }{{\rm Y}}_1\right){\xi }^H],\hfill \\ \hfill \left(iv\right)F^{*}\left({\varphi }^H{\widetilde{{\rm Y}}}_1\right)& =& \frac{p}{2}{\varphi }^H\left({\widetilde{{\rm Y}}}_1\right)-B[p{\varphi }^H{\widetilde{{\rm Y}}}_1+q{\widetilde{{\rm Y}}}_1-q({\eta }^V\left({\widetilde{{\rm Y}}}_1\right){\xi }^H+{\eta }^H\left({\widetilde{{\rm Y}}}_1\right){\xi }^V)].\hfill \end{array}$$

(22)

In the view of (21) and (22), we obtain

$$\begin{array}{ccc}\hfill {\left(F^{*}\right)}^2\left({\widetilde{{\rm Y}}}_1\right)& =& \frac{p}{2}{F}^{*}\left({\widetilde{{\rm Y}}}_1\right)-B\left[F^{*}{\phi }^H{\widetilde{{\rm Y}}}_1+\sqrt{q}({\eta }^V\left({\widetilde{{\rm Y}}}_1\right)F^{*}\left({\xi }^V\right)+{\eta }^H\left({\widetilde{{\rm Y}}}_1\right)F^{*}\left({\xi }^H\right))\right],\hfill \\ & =& p{F}^{*}\left({\widetilde{{\rm Y}}}_1\right)+q\left({\widetilde{{\rm Y}}}_1\right).\hfill \end{array}$$

This shows that $F^{*}$ is an $MS$. □

Corollary 2.

Let ${{\rm Y}}_1$ and ${{\rm Y}}_2$ be the vector fields on M and $F^{*}$ be an $MS$ on $TM$ given by (21) such that $\eta \left({{\rm Y}}_1\right)=0$. Then

$$\begin{array}{ccc}\hfill \left(i\right)F^{*}{{\rm Y}}_1^V& =& \frac{p}{2}{{\rm Y}}_1^V-B\left[{\left(\varphi {{\rm Y}}_1\right)}^V+\sqrt{q}{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^H\right],\hfill \\ \hfill \left(ii\right)F^{*}{{\rm Y}}_1^H& =& \frac{p}{2}{{\rm Y}}_1^H-B\left[{\left(\varphi {{\rm Y}}_1\right)}^H+\sqrt{q}{\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^V\right],\hfill \\ \hfill \left(iii\right)F^{*}{{\rm Y}}_1^C& =& \frac{p}{2}{{\rm Y}}_1^C-B[{\left(\varphi {{\rm Y}}_1\right)}^H+{\varphi }^H\left({\nabla }_{\gamma }{{\rm Y}}_1\right)\hfill \\ & +& \sqrt{q}({\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^V+{\eta }^C\left({\nabla }_{\gamma }{{\rm Y}}_1\right){\xi }^H)],\hfill \\ \hfill \left(iv\right)F^{*}{{\rm Y}}_1^C& =& \frac{p}{2}{{\rm Y}}_1^C-B[{\left(\varphi {{\rm Y}}_1\right)}^H+{\varphi }^C\left({\nabla }_{\gamma }{{\rm Y}}_1\right)\hfill \\ & +& \sqrt{q}({\left(\eta \left({{\rm Y}}_1\right)\right)}^V{\xi }^V+{\eta }^C\left({\nabla }_{\gamma }{{\rm Y}}_1\right){\xi }^H)].\hfill \end{array}$$

(23)

If $\eta \left({{\rm Y}}_1\right)=0$, then

$$\begin{array}{ccc}\hfill \left(i\right)F^{*}{{\rm Y}}_1^H& =& \frac{p}{2}{{\rm Y}}_1^H-B{\left(\varphi {{\rm Y}}_1\right)}^H,\hfill \\ \hfill \left(ii\right)F^{*}{{\rm Y}}_1^V& =& \frac{p}{2}{{\rm Y}}_1^V-B{\left(\varphi {{\rm Y}}_1\right)}^V,\hfill \\ \hfill \left(iii\right)F^{*}{{\rm Y}}_1^C& =& \frac{p}{2}{{\rm Y}}_1^C-B[{\left(\varphi {{\rm Y}}_1\right)}^H+{\varphi }^H\left({\nabla }_{\gamma }{{\rm Y}}_1\right)\hfill \\ & +& \sqrt{q}{\eta }^C\left({\nabla }_{\gamma }{{\rm Y}}_1\right){\xi }^H].\hfill \end{array}$$

(24)

Proof. 

The proof is obtained by applying ${{\rm Y}}_1^C$ and ${{\rm Y}}_1^V$ on $F^{*}$ given by (21) and using $\eta \left({{\rm Y}}_1\right)=0$. □

Theorem 7.

The metallic structure $F^{*}$ given by (21) is integrable on $TM$ over $(\varphi ,\xi ,\eta )$ if and only if $N_{F^{*}}=0$, which is equivalent to the conditions

$$\eta \left([\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)=0,\eta \left([\varphi {{\rm Y}}_1,\xi ]\right)=0,\eta \left([{{\rm Y}}_1,{{\rm Y}}_2]\right)=0,$$

$$\eta \left([{{\rm Y}}_1,\xi ]\right)=0,\nabla {{\rm Y}}_1=0,\widehat{R}=0,$$

and $(\varphi ,\xi ,\eta )$ is integrable, i.e., $N_{\varphi }=0.$

Proof. 

Let $N_{F^{*}}$ be the Nijenhuis tensor of the metallic structure $F^{*}$, then

$$N_{F^{*}}({\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2)=[F^{*}{\widetilde{{\rm Y}}}_1,F^{*}{\widetilde{{\rm Y}}}_2]-F^{*}[F^{*}{\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2]-F^{*}[{\widetilde{{\rm Y}}}_1,F^{*}{\widetilde{{\rm Y}}}_2]+{\left(F^{*}\right)}^2[{\widetilde{{\rm Y}}}_1,{\widetilde{{\rm Y}}}_2],$$

(25)

where ${\widetilde{{\rm Y}}}_1$ and ${\widetilde{{\rm Y}}}_2$ are vector fields on $TM$.

Applying (3)–(7), (21), (23), and (16) on (25), and using $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$.

$$\begin{array}{ccc}\hfill N_{F^{*}}({{{\rm Y}}_1}^V,{{{\rm Y}}_2}^V)& =& -B^2\left({\left(\varphi {\nabla }_{{{\rm Y}}_2}\xi \right)}^V+\sqrt{q}{\left(\eta \left({\nabla }_{{{\rm Y}}_2}\xi \right)\right)}^V{\xi }^H\right)+Bq{\left(\eta [{{\rm Y}}_1,\xi ]\right)}^V{\xi }^H.\hfill \\ \hfill N_{F^{*}}({{{\rm Y}}_1}^V,{{{\rm Y}}_2}^H)& =& B^2{\left(N_{\varphi }({{\rm Y}}_1,{{\rm Y}}_2)\right)}^V+Bp\left({\left(\varphi [{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V+\sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^H\right)\hfill \\ & -& Bp\left({\left(\varphi {\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V+\sqrt{q}\eta {\left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V{\xi }^H\right)-B^2\left({\nabla }_{\varphi {{\rm Y}}_1}\varphi {{\rm Y}}_2\right)\hfill \\ & -& B^2\sqrt{q}\eta {\left([\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^H-B^2\sqrt{q}\eta {\left([{{\rm Y}}_1,\varphi {{\rm Y}}_2]\right)}^V{\xi }^H\hfill \\ & +& \left((\varphi {\left({\nabla }_{\varphi {{\rm Y}}_1}{{\rm Y}}_2\right)}^V+\sqrt{q}\eta {\left({\nabla }_{\varphi {{\rm Y}}_1}{{\rm Y}}_2\right)}^V{\xi }^H\right)-\frac{p^2}{4}{\left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V-q{\left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V\hfill \\ & +& B^2\left({\left(\varphi \left({\nabla }_{{{\rm Y}}_1}\varphi {{\rm Y}}_2\right)\right)}^V+\sqrt{q}\eta {\left({\nabla }_{{{\rm Y}}_1}\varphi {{\rm Y}}_2\right)}^V{\xi }^H\right)-Bp({\left(\varphi [{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V\hfill \\ & +& \sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^H)+B\left({\left(\varphi \left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)\right)}^V+\sqrt{q}\eta {\left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V{\xi }^H)\right).\hfill \\ \hfill N_{F^{*}}({{\rm Y}}_1^H,{{{\rm Y}}_2}^H)& =& B^2{\left(N_{\varphi }({{\rm Y}}_1,{{\rm Y}}_2)\right)}^H+B^{2}q\left({\left(\eta [{{\rm Y}}_1,{{\rm Y}}_2]\xi \right)}^H+{\left(\eta [{{\rm Y}}_1,{{\rm Y}}_2]\xi \right)}^V\right)\hfill \\ & -& B^2(p{\varphi }^H+q)\gamma \widehat{R}({{\rm Y}}_1,{{\rm Y}}_2)+\frac{p}{2}B\gamma \widehat{R}({{\rm Y}}_1,\varphi {{\rm Y}}_2)+\frac{p}{2}B\widehat{R}(\varphi {{\rm Y}}_1,{{\rm Y}}_2)\hfill \\ & -& B^2\gamma \widehat{R}(\varphi {{\rm Y}}_1,\varphi {{\rm Y}}_2)-B\sqrt{q}{\left(\eta [\varphi {{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^V-B{F}^{*}\gamma \widehat{R}(\varphi {{\rm Y}}_1,{{\rm Y}}_2)\hfill \\ & -& B^2\sqrt{q}{\left(\eta [{{\rm Y}}_1,\varphi {{\rm Y}}_2]\right)}^V{\xi }^V-B{F}^{*}\gamma \widehat{R}({{\rm Y}}_1,\varphi {{\rm Y}}_2).\hfill \\ \hfill N_{F^{*}}({{\rm Y}}_1^V,{\xi }^V)& =& B^2\sqrt{q}\left({[\varphi {{\rm Y}}_1,\xi ]}^V-\varphi {[{{\rm Y}}_1,\xi ]}^V-{\left({\nabla }_{\varphi {{\rm Y}}_1}\xi \right)}^V-{\left(\varphi {\nabla }_X\xi \right)}^V\right)\hfill \\ & +& B^{2}q\left(\eta {\left({\nabla }_{{{\rm Y}}_1}\xi \right)}^V{\xi }^H-{\left(\eta [{{\rm Y}}_1,\xi ]\right)}^V{\xi }^H\right).\hfill \\ \hfill N_{F^{*}}({{{\rm Y}}_1}^V,{\xi }^H)& =& B^2\left({\left({\varphi }^2[{{\rm Y}}_1,\xi ]\right)}^V-q{\left({\nabla }_{{{\rm Y}}_1}\xi \right)}^V\right).\hfill \\ \hfill N_{F^{*}}({{\rm Y}}_1^H,{\xi }^V)& =& -B^2{\left((p\varphi +q)[\xi ,{{\rm Y}}_1]\right)}^V+B^2{\left((p\varphi +q)\left({\nabla }_{\xi }{{\rm Y}}_1\right)\right)}^V\hfill \\ & +& \frac{p}{2}B\sqrt{q}\gamma \widehat{R}({{\rm Y}}_1,\xi )+B^2\sqrt{q}\left({[\varphi {{\rm Y}}_1,\xi ]}^H-\gamma \widehat{R}(\varphi {{\rm Y}}_1,\xi )\right)\hfill \\ & +& B^2{\left(\varphi [\xi ,\varphi {{\rm Y}}_1]\right)}^V+B^2\sqrt{q}{\left(\eta [\xi ,\varphi {{\rm Y}}_1]\right)}^V{\xi }^H-B^2{\left(\varphi \left({\nabla }_{\xi }\varphi {{\rm Y}}_1\right)\right)}^V\hfill \\ & -& B^2\sqrt{q}{\left(\eta \left({\nabla }_{\xi }\varphi {{\rm Y}}_1\right)\right)}^V{\xi }^H-B^2\sqrt{q}\varphi {[{{\rm Y}}_1,\xi ]}^H\hfill \\ & -& B^{2}q{\left(\eta [{{\rm Y}}_1,\xi ]\right)}^V{\xi }^V-B\sqrt{q}{F}^{*}\gamma \widehat{R}({{\rm Y}}_1,\xi ).\hfill \\ \hfill N_{F^{*}}({{\rm Y}}_1^H,{\xi }^H)& =& B^2{\varphi }^2{[{{\rm Y}}_1,\xi ]}^H+B^2\eta {[{{\rm Y}}_1,\xi ]}^H+B^2{\left(\eta [{{\rm Y}}_1,\xi ]\xi \right)}^V-\frac{p^2+4q}{4}(\gamma \widehat{R}({{\rm Y}}_1,\xi )\hfill \\ & +& pB\sqrt{q}{[\xi ,{{\rm Y}}_1]}^V+\frac{p}{2}B\gamma \widehat{R}(\varphi {{\rm Y}}_1,\xi )-B^2\varphi [\varphi {{\rm Y}}_1,\xi ]-B^2\sqrt{q}{\left(\eta [\varphi {{\rm Y}}_1,\xi ]\right)}^V{\xi }^V\hfill \\ & -& B^2\sqrt{q}{\left(\varphi [\xi ,{{\rm Y}}_1]\right)}^V-B^{2}q{\left(\eta [\xi ,{{\rm Y}}_1]\right)}^V{\xi }^H+B^2\sqrt{q}{\left(\varphi {\nabla }_{\xi }{{\rm Y}}_1\right)}^V\hfill \\ & +& B^{2}q{\left(\eta \left({\nabla }_{\xi }{{\rm Y}}_1\right)\right)}^V{\xi }^H-Bp\sqrt{q}{\left(\eta [{{\rm Y}}_1,\xi ]\right)}^V{\xi }^V.\hfill \\ \hfill N_{F^{*}}({\xi }^V,{\xi }^H)& =& -(\frac{p^2}{2}+q){\left({\nabla }_{\xi }\xi \right)}^V+B^{2}q{\left({\nabla }_{\xi }\xi \right)}^V+\frac{p}{2}B\sqrt{q}{\left({\nabla }_{\xi }\xi \right)}^H.\hfill \end{array}$$

□

Theorem 8.

Let $F^{*}$ be a $MS$ in $TM$ given by (21) and ${{\rm Y}}_1$ and ${{\rm Y}}_2$ be vector fields on M such that $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$, then

$$\begin{array}{ccc}\hfill \left(i\right)\left({\pounds }_{{{{\rm Y}}_2}^H}{F}^{*}\right){{\rm Y}}_1^H& =& B\left({[\varphi {{\rm Y}}_1,{{\rm Y}}_2]}^H-\gamma \widehat{R}(\varphi {{\rm Y}}_1,{{\rm Y}}_2)\right)+\frac{p}{2}\gamma \widehat{R}({{\rm Y}}_1,{{\rm Y}}_2)\hfill \\ & -& B\left({\left(\varphi [{{\rm Y}}_1,{{\rm Y}}_2]\right)}^H-\sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^V\right)-F^{*}\gamma \widehat{R}({{\rm Y}}_1,{{\rm Y}}_2),\hfill \\ \hfill \left(ii\right)\left({\pounds }_{{{{\rm Y}}_2}^V}{F}^{*}\right){{\rm Y}}_1^H& =& B\left({\left(\varphi [{{\rm Y}}_2,{{\rm Y}}_1]\right)}^V+\sqrt{q}{\left(\eta \left([{{\rm Y}}_2,{{\rm Y}}_1]\right)\right)}^V{\xi }^H-{[{{\rm Y}}_2,\varphi {{\rm Y}}_1]}^V\right)\hfill \\ & -& B\left({\left(\varphi \left({\nabla }_{{{\rm Y}}_2}{{\rm Y}}_1\right)\right)}^V+\sqrt{q}{(\eta {\left({\nabla }_{{{\rm Y}}_2}{{\rm Y}}_1\right)}^V{\xi }^H-{\nabla }_{{{\rm Y}}_2}\varphi {{\rm Y}}_1)}^V\right),\hfill \\ \hfill \left(iii\right)\left({\pounds }_{{{{\rm Y}}_2}^V}{F}^{*}\right){{\rm Y}}_1^V& =& 0,\hfill \\ \hfill \left(iv\right)\left({\pounds }_{{{{\rm Y}}_2}^H}{F}^{*}\right){{{\rm Y}}_1}^V& =& B\left({\left(\varphi {\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V+\sqrt{q}\eta {\left({\nabla }_{{{\rm Y}}_1}{{\rm Y}}_2\right)}^V{\xi }^H)-{[\varphi {{\rm Y}}_1,{{\rm Y}}_2]}^V\right)\hfill \\ & -& B\left({\left(\varphi [{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V=\sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^H-{\nabla }_{\varphi {{\rm Y}}_1}{{\rm Y}}_2\right).\hfill \end{array}$$

Proof. 

Applying (21), (23), (16), and (20), and using $\eta \left({{\rm Y}}_1\right)=\eta \left({{\rm Y}}_2\right)=0$.

$$\begin{array}{ccc}\hfill \left(i\right){\pounds }_{{{{\rm Y}}_2}^V}\left(F^{*}{{\rm Y}}_1^H\right)& =& \frac{p}{2}({\pounds }_{{{{\rm Y}}_2}^V}{{\rm Y}}_1^H-B{\pounds }_{{{{\rm Y}}_2}^V}{\left(\varphi {{\rm Y}}_1\right)}^H-B\sqrt{q}{\left(\eta {{\rm Y}}_1\right)}^V{\pounds }_{{{{\rm Y}}_2}^V}{\xi }^V)\hfill \\ \hfill \left({\pounds }_{{{\rm Y}}_2}^V{F}^{*}\right){{\rm Y}}_1^H+F^{*}\left({\pounds }_{{{{\rm Y}}_2}^V}{X}^H\right)& =& \frac{p}{2}[{{\rm Y}}_2^H,{{\rm Y}}_1^H]-B[{{\rm Y}}_2^H,{\left(\varphi {{\rm Y}}_1\right)}^H]\hfill \\ & -& B\sqrt{q}(\eta {{{\rm Y}}_1{)}^V[{{\rm Y}}_2}^H,{\xi }^H]\hfill \\ \hfill \left({\pounds }_{{{{\rm Y}}_2}^H}{F}^{*}\right){{\rm Y}}_1^H& =& B\left({[\varphi {{\rm Y}}_1,{{\rm Y}}_2]}^H-\gamma \widehat{R}(\varphi {{\rm Y}}_1,{{\rm Y}}_2)\right)+\frac{p}{2}\gamma \widehat{R}({{\rm Y}}_1,{{\rm Y}}_2)\hfill \\ & -& B\left({\left(\varphi [{{\rm Y}}_1,{{\rm Y}}_2]\right)}^H-\sqrt{q}\eta {\left([{{\rm Y}}_1,{{\rm Y}}_2]\right)}^V{\xi }^V\right)-F^{*}\gamma \widehat{R}({{\rm Y}}_1,{{\rm Y}}_2),\hfill \end{array}$$

Others results are obtained by using similar devices. □

Example 1.

Setting $p=q=1$ in (8), then $F^2-F-I=0$ is obtained and named as the Golden Structure. Also, from (21), we have

$$F^{*}=\frac{1}{2}-B\left[{\varphi }^H+{\eta }^V\otimes {\xi }^V+{\eta }^H\otimes {\xi }^H\right].$$

(26)

Using (22), we infer

$$\begin{array}{ccc}\hfill \left(i\right)F^{*}\left({\xi }^H\right)& =& \frac{1}{2}{\xi }^H-B{\xi }^V,\hfill \\ \hfill \left(ii\right)F^{*}\left({\xi }^V\right)& =& \frac{1}{2}{\xi }^V-B{\xi }^H,\hfill \\ \hfill \left(iii\right)F^{*}\left({\xi }^C\right)& =& \frac{1}{2}{\xi }^C-B[{\varphi }^H\left({\nabla }_{\gamma }{{\rm Y}}_1\right)+{\varphi }^C\left({\nabla }_{\gamma }{{\rm Y}}_1\right){\xi }^H],\hfill \\ \hfill \left(iv\right)F^{*}\left({\varphi }^H{\widetilde{{\rm Y}}}_1\right)& =& \frac{1}{2}{\varphi }^H\left({\widetilde{{\rm Y}}}_1\right)-B[{\varphi }^H{\widetilde{{\rm Y}}}_1+{\widetilde{{\rm Y}}}_1-({\eta }^V\left({\widetilde{{\rm Y}}}_1\right){\xi }^H+{\eta }^H\left({\widetilde{{\rm Y}}}_1\right){\xi }^V)].\hfill \end{array}$$

(27)

Apply $F^{*}\left({\widetilde{{\rm Y}}}_1\right)$ in (26), we infer

$$\begin{array}{ccc}\hfill {\left(F^{*}\right)}^2\left({\widetilde{{\rm Y}}}_1\right)& =& \frac{1}{2}{F}^{*}\left({\widetilde{{\rm Y}}}_1\right)-B\left[F^{*}{\phi }^H{\widetilde{{\rm Y}}}_1+{\eta }^V\left({\widetilde{{\rm Y}}}_1\right)F^{*}\left({\xi }^V\right)+{\eta }^H\left({\widetilde{{\rm Y}}}_1\right)F^{*}\left({\xi }^H\right)\right],\hfill \\ & =& F^{*}\left({\widetilde{{\rm Y}}}_1\right)+\left({\widetilde{{\rm Y}}}_1\right).\hfill \end{array}$$

This shows that $F^{*}$ is a golden structure.

5. Examples of Almost Quadratic $\mathit{\varphi }$-Manifolds

In this section, we prove the existence of almost quadratic $\varphi$-manifolds on the tangent bundle with non-trivial examples.

Example 2.

Let $M=\left\{\right(x,y,z):x,y,z\in \Re ,z\ne 0\}$ be a differentiable manifold of dimension 3, ℜ is a set of real numbers. We suppose that $e_i^C$ and $e_i^V;i=1,2,3$ be complete and vertical lifts on $TM$ of independent vector fields $e_i;i=1,2,3$ on M, then they form a basis $\{e_i^C,e_i^V;i=1,2,3\}$ for $TM$ of M. Let $g^C$ be the complete lift of a Riemannian metric g such that $g_{ij}={\delta }_{ij}$, where ${\delta }_{ij}$ is Kronecker delta. That is,

$$\begin{array}{ccc}\hfill g^C({{\rm Y}}_1^V,e_3^C)& =& {\left(g({{\rm Y}}_1,e_3)\right)}^V={\left(\eta \left(e_3\right)\right)}^V,\hfill \\ \hfill g^C({{\rm Y}}_1^C,e_3^C)& =& {\left(g({{\rm Y}}_1,e_3)\right)}^C={\left(\eta \left(e_3\right)\right)}^C,\hfill \\ \hfill g^C(e_3^C,e_3^C)& =& 1,g^V({{\rm Y}}_1^V,e_3^C)=0,g^V(e_3^V,e_3^V)=0,\hfill \end{array}$$

where ${{\rm Y}}_1$ is a vector field on M. If ϕ represents the (1,1) symmetric tensor on M such that

$${\varphi }^V\left(e_i^V\right)=(1+\sqrt{2})e^z{e}_i^V,i=1,2,$$

$${\varphi }^C\left(e_i^C\right)=(1+\sqrt{2})e^z{e}_i^C,i=1,2,$$

$${\varphi }^V\left(e_3^V\right)={\varphi }^C\left(e_3^C\right)=0,$$

Then we can easily verify that

$${\left({\varphi }^C\right)}^2=p{\varphi }^C+qI-q({\eta }^V\otimes {\xi }^C+{\eta }^C\otimes {\xi }^V),$$

where $p=2e^z,q=e^{2z}\Rightarrow p^2+4q=8e^{2z}\ne 0$. This shows that M is an almost quadratic ϕ-manifold and the structure $(\varphi ,\xi ,\eta )$ is an almost quadratic ϕ-structure on M.

Again, from the straightforward calculations, we prove that

$$g^C({\left(\varphi e_i\right)}^C,e_j^C)=g^C(e_i^C,{\left(\varphi e_j\right)}^C),$$

$$g^C({\left(\varphi e_i\right)}^V,e_j^C)=g^C(e_i^V,{\left(\varphi e_j\right)}^C),$$

and

$$\begin{array}{ccc}\hfill g^C({\left(\varphi e_i\right)}^C,{\left(\varphi e_j\right)}^C)& =& p{g}^C({\left(\varphi e_i\right)}^C,e_j^C)+q(g^C(e_i^C,e_j^C)\hfill \\ & -& {\left(\eta \left(e_i\right)\right)}^C{\left(\eta \left(e_j\right)\right)}^V-{\left(\eta \left(e_i\right)\right)}^V{\left(\eta \left(e_j\right)\right)}^C),\forall i,j=1,2,3.\hfill \end{array}$$

The manifold M is an almost quadratic metric ϕ-manifold and the structure $(\varphi ,\xi ,\eta ,g)$ is an almost quadratic metric ϕ-structure on M.

Example 3.

A paracontact structure $(\varphi ,\eta ,\xi )$ on M such that [22]

$${\varphi }^2=I-\eta \otimes \xi$$

is an almost quadratic ϕ-structure when $p=0,q=1$ in (9). The new tensor $\tilde{F}$ of type (1,1) given by (12) becomes

$$\tilde{F}=-{\varphi }^C+{\eta }^V\otimes {\xi }^V+{\eta }^C\otimes {\xi }^C.$$

It can be easily proved that $\tilde{F}$ is almost a product structure.

Remark 1.

For the horizontal lift, we can obtain the similar examples of almost quadratic ϕ-manifolds.

6. Conclusions

In this work, we have characterized a metallic structure by using the complete and horizontal lifts over an almost quadratic $\varphi$-structure $(\varphi ,\xi ,\eta )$. Tensor fields $\tilde{F}$ and $F^{*}$ are defined on $TM$ over the structure $(\varphi ,\xi ,\eta )$ and we proved that they are metallic structures, which generalizes the notion of an almost complex structure $\tilde{J}$ introduced by Tanno [10]. The fundamental geometrical properties of fundamental 2-Form and its derivative on $TM$ over the structure $(\varphi ,\xi ,\eta )$ were calculated. The integrability conditions and expressions of the Lie derivative of metallic structures $\tilde{F}$ and $F^{*}$ on $TM$ over the structure $(\varphi ,\xi ,\eta )$ were determined. Finally, we demonstrated that almost quadratic $\varphi$-manifolds exist on $TM$ with non-trivial examples. Future studies could fruitfully explore this issue further by considering the polynomial structure $Q\left(F\right)=F_n+a_n{F}_{n-1}+\cdots +a_{2}F+a_{1}I,$ where F is the tensor field of type (1,1).