# Alternative Equations for Horizontal Lifts of the Metallic Structures from Manifold onto Tangent Bundle

Authors: Mohammad Nazrul Islam Khan; Majid Ali Choudhary; Sudhakar K. Chaubey
Journal: Journal of Mathematics
 (2022)

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## Abstract

We explore “the horizontal lift” of the structureJ satisfying J2−αJ−βI=0 and establish that it as a kind of metallic structure. An analysis of Nijenhuis tensor of metallic structure JH is presented, and a new tensor field J˜ of 1,1-type is introduced and demonstrated to be metallic structure. Some results on the Nijenhuis tensor and the Lie derivative of J˜ in TM are proved and explicit examples are given. Moreover, the metallic structure J˜ endowed with projection operators l˜ and m˜ in TM is studied.


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## Body

## 1. Introduction

Tangent bundle geometry’s framework typically includes geometric structures such as vertical, complete, and horizontal lifts transforming structures on the base manifoldM to TM. These lifts of an almost product structure along with TM over an almost r-contact structure have been studied by Das and Khan [1]. Numerous researchers made valuable contributions on different geometric structures and connections. For example, Das and Nivas [2], Khan [3–5], Mağden et al. [6], Omran et al. [7], Tekkoyun [8], Yano and Ishihara [9], etc.Let us consider(1)QJ=Jn+anJn−1+⋯+a2J+a1I.In this scenario, J stands for 1,1-tensor field, ai,i=1,…,n are the real numbers, and I means identity map [10]. Such a structure satisfying(2)J2=αJ+βI,is called a metallic structure [11]. The set of positive solutions of J2−αJ−βI=0, denoted by σαβ=1/2α+α2+4β, is named Metallic Means Family (MMF) [12–14]. The different values of α and β refer to different members of MMF. For instance, α=1,β=1, Gold mean, applies to architecture. For α=2,β=1, Silver mean, its role can be found in various applications such as civil design, architecture, and physics. For α=3,β=1, Bronze mean, it has uses in quasicrystals.For positive integersα and β, Hretcanu and Crasmareanu [15] introduced and studied the metallic structures satisfying J2−αJ−βI=0, on the Riemannian manifold. In 2018, taking the help of horizontal and vertical lifts, Azami [16] investigated metallic type structures, and integrability results on these structures had been taken into research by Khan [17]. For further developments in the area, one refers to [18–28].The paper covers the following:(i)
“Horizontal lift” applied to a metallic structureJ satisfying J2−αJ−βI=0 and proved to be metallic structure(ii)
The Nijenhuis tensor (N.T.) ofJH is established(iii)
It is proved that the fieldJ˜ is metallic on TM by introducing a new 1,1-tensor field(iv)
N.T. ofJ˜ in TM is calculated(v)
The Lie derivative ofJ˜ in TM is determined(vi)
The existence of metallic structures inTM is shown by nontrivial examples(vii)
The projection operators inTM are discussed as an application


## 2. Preliminaries

We interpretMn as a differentiable manifold and TM=∪p∈MTpM,p∈M as its tangent bundle. Assume that ℑsrM and ℑsrTM stand for set of all r,s-type tensor fields in M and TM, respectively.


### 2.1. Vertical Lifts

One writes “the vertical lift” as(3)fV=f°π,where π:TM⟶M means the standard projection and f stands for differentiable function on M.LetX∈ℑ01M and “the vertical lift” XV of X defined in the components xh,h=1,…,n form as(4)XV:0∂xh.Now, we state the following proposition involving ‘the vertical lifts’ for later use [29]:Proposition 1.
For allG,ℋ∈ℑ00M,X,Y∈ℑ01M,η∈ℑ10M. Then,(5)G.ℋV=GVℋV,G+ℋV=GV+ℋV,X+YV=XV+YV,G⋅XV=GVXV,XVGV=0,XV,YV=0,G⋅ηV=GVηV,ηVXV=0,where η represents 1-form in M.


### 2.2. Complete Lifts

fC in TM represented as(6)fC=idf,Which gives complete lift of f. Locally,(7)fC=yi∂if=∂f.In this case, ∂f means yi∂if. For X∈ℑ01M, one has(8)XCfC=XfC.With respect to componentsxh in M, one can write(9)XC:xh∂xh,associated to induced coordinates in TM.


### 2.3. Horizontal Lifts

We assume that∇ is an affine connection and U represents an arbitrary coordinate neighborhood in M, and there are tensor fields γS and ∇γS with affine connection ∇ given by(10)S=Sk…ji…h∂∂xi⊗⋯⊗∂∂xh⊗dxk⊗⋯⊗dxj,∇γS=yl∇γSk…ji…h∂∂xi⊗⋯⊗∂∂yh⊗dxk⊗⋯⊗dxj,corresponding to xh,yh in π−1U, where γ is an operator on tensor field [29].For differentiable functionf and 1-form η on M, we define:(1)
ηH of η by(11)ηH=XC−∇γη,where∇γη=γ∇η.(2)
fH of f with(12)fH=fC−∇γf,∇γf=γ∇f,and∇f is a gradient of f.(3)
XH of X is(13)XH=XC−∇γX,∇γX=γ∇X.(4)
FH of F with(14)FH=FC−∇γF,∇γF=γ∇F.Here,X means a vector field and F a 1,1-tensor field on M. We recall the following propositions involving “the horizontal lifts” for later use [29]:Proposition 2.
For allf∈ℑ00M,η∈ℑ10M,(15)XHfV=XfV,FVXH=FXV,FHXH=FXH,ηVXH=ηXH,ηHXH=0,∀X,Y∈ℑ01M,F∈ℑ11M.Proposition 3.
∀X,Y in TM, we observe(16)XV,YH=X,YV−∇XYV=−∇^XYV,XV,YV=0,XH,YH=X,YH−γR^X,Y.In this case,XH represents the horizontal lift and XV stands for vertical lift of X in TM. Furthermore, for affine connection ∇^,R^ indicates the curvature tensor of ∇^ that satisfies(17)∇^XY=∇XY+X,Y.Proposition 4.
For the torsion tensors of∇ and ∇H in M and TM denoted by T and T˜, respectively, we have(18)T˜XV,YV=∇XVHYV−∇YVHXV−XV,YV,T˜XV,YH=∇XHHYH−∇YHHXV−XV,YH,T˜XH,YH=∇XHHYH−∇YHHXH−XH,YH,∀X,Y∈ℑ01M.


### 2.4. Projection Tensor

Any tensor fieldm in manifold M satisfying m2=m is known as projection tensor [29, 30].Definition 1.
[31, 32] If the metallic structure J defined in (2) is covariant constant w.r.t. ∇, i.e., ∇J=0, then ∇ is said be J connection.Definition 2.
[23] We write the N.T. N of J as(19)NX,Y=JX,JY−JJX,Y−JX,JY+J2X,Y,∀X,Y∈ℑ01M,where J becomes integrable if NX,Y=0.


## 2.1. Vertical Lifts

One writes “the vertical lift” as(3)fV=f°π,where π:TM⟶M means the standard projection and f stands for differentiable function on M.LetX∈ℑ01M and “the vertical lift” XV of X defined in the components xh,h=1,…,n form as(4)XV:0∂xh.Now, we state the following proposition involving ‘the vertical lifts’ for later use [29]:Proposition 1.
For allG,ℋ∈ℑ00M,X,Y∈ℑ01M,η∈ℑ10M. Then,(5)G.ℋV=GVℋV,G+ℋV=GV+ℋV,X+YV=XV+YV,G⋅XV=GVXV,XVGV=0,XV,YV=0,G⋅ηV=GVηV,ηVXV=0,where η represents 1-form in M.


## 2.2. Complete Lifts

fC in TM represented as(6)fC=idf,Which gives complete lift of f. Locally,(7)fC=yi∂if=∂f.In this case, ∂f means yi∂if. For X∈ℑ01M, one has(8)XCfC=XfC.With respect to componentsxh in M, one can write(9)XC:xh∂xh,associated to induced coordinates in TM.


## 2.3. Horizontal Lifts

We assume that∇ is an affine connection and U represents an arbitrary coordinate neighborhood in M, and there are tensor fields γS and ∇γS with affine connection ∇ given by(10)S=Sk…ji…h∂∂xi⊗⋯⊗∂∂xh⊗dxk⊗⋯⊗dxj,∇γS=yl∇γSk…ji…h∂∂xi⊗⋯⊗∂∂yh⊗dxk⊗⋯⊗dxj,corresponding to xh,yh in π−1U, where γ is an operator on tensor field [29].For differentiable functionf and 1-form η on M, we define:(1)
ηH of η by(11)ηH=XC−∇γη,where∇γη=γ∇η.(2)
fH of f with(12)fH=fC−∇γf,∇γf=γ∇f,and∇f is a gradient of f.(3)
XH of X is(13)XH=XC−∇γX,∇γX=γ∇X.(4)
FH of F with(14)FH=FC−∇γF,∇γF=γ∇F.Here,X means a vector field and F a 1,1-tensor field on M. We recall the following propositions involving “the horizontal lifts” for later use [29]:Proposition 2.
For allf∈ℑ00M,η∈ℑ10M,(15)XHfV=XfV,FVXH=FXV,FHXH=FXH,ηVXH=ηXH,ηHXH=0,∀X,Y∈ℑ01M,F∈ℑ11M.Proposition 3.
∀X,Y in TM, we observe(16)XV,YH=X,YV−∇XYV=−∇^XYV,XV,YV=0,XH,YH=X,YH−γR^X,Y.In this case,XH represents the horizontal lift and XV stands for vertical lift of X in TM. Furthermore, for affine connection ∇^,R^ indicates the curvature tensor of ∇^ that satisfies(17)∇^XY=∇XY+X,Y.Proposition 4.
For the torsion tensors of∇ and ∇H in M and TM denoted by T and T˜, respectively, we have(18)T˜XV,YV=∇XVHYV−∇YVHXV−XV,YV,T˜XV,YH=∇XHHYH−∇YHHXV−XV,YH,T˜XH,YH=∇XHHYH−∇YHHXH−XH,YH,∀X,Y∈ℑ01M.


## 2.4. Projection Tensor

Any tensor fieldm in manifold M satisfying m2=m is known as projection tensor [29, 30].Definition 1.
[31, 32] If the metallic structure J defined in (2) is covariant constant w.r.t. ∇, i.e., ∇J=0, then ∇ is said be J connection.Definition 2.
[23] We write the N.T. N of J as(19)NX,Y=JX,JY−JJX,Y−JX,JY+J2X,Y,∀X,Y∈ℑ01M,where J becomes integrable if NX,Y=0.


## 3. Horizontal Lifts of Metallic Structures inTM

In this section, a study is done on ‘the horizontal lifts’ of a metallic structureJ in TM.Theorem 1.
We assume thatJ∈ℑ01M represents M.S. in M. Then, JH also provides a M.S. in TM, i.e.,(20)JH2−αJH−βI=0.Proof.
The proof is trivial with the help of (2).Theorem 2.
LetM be any C∞ manifold with TM admitting metallic structure J, and ∇ be any J connection. The Nijenhuis tensor N∗ of the horizontal lift JH on TM is represented as(21)N∗XV,YV=0,N∗XV,YH=NX,YV−∇JXJY−J∇XJYJ∇JXY−J2∇XYV,N∗XH,YH=NX,YH−γR^JX,JY−JHR^X,JY−JHR^JX,Y+JH2R^X,Y,∀X,Y∈ℑ01M,J∈ℑ11M,N being N.T. of J.Proof.
Let(22)N∗X˜,Y˜=JHX˜,JHY˜−JHJHX˜,Y˜−JHX˜,JHY˜+JH2X˜,Y˜,∀X˜,Y˜∈ℑ01TM.
SettingX˜=XV,Y˜=YV in (22) and Proposition 2 and Proposition 3, we get(23)N∗XV,YV=JHXV,JHYV−JHJHXV,YV−JHXV,JHYV+JH2XV,YV,=JXV,JYV−JHJXV,YV−JHXV,JYV+JH2XV,YV,=0.
SettingX˜=XV,Y˜=YH in (22) and Proposition 2 and Proposition 3, we get(24)N∗XV,YH=JHXV,JHYH−JHJHXV,YH−JHXV,JHYH+JH2XV,YH,=JXV,JYH−JHJXV,YH−JHXV,JYH+JH2XV,YH,=JX,JYV−∇JXJYV−JHX,JYV+JH∇XJYV−JHJX,YV+JH∇JXY+JH2X,YV−JH2∇XYV,=NX,YV−∇JXJYV+JH∇XJYV+JH∇JXY−JH2∇XYV,=NX,YV−∇JXJYV+JH∇XJYV+JH∇JXY−JH2∇XYV,=NX,YV−∇JXJY−J∇XJY−J∇JXY−J2∇XYV.
SettingX˜=XH,Y˜=YH in (22), using similar devices, the following equation is obtained:(25)N∗XH,YH=NX,YH−γR^JX,JY−JHR^X,JY−JHR^JX,Y+JH2R^X,Y.
This completes the proof.


## 4. Proposed Theorems onJ˜ for the M.S. on TM

Here,J˜ is introduced and shown to be metallic structure in TM. Moreover, the Nijenhuis tensor and the Lie derivative of J˜ in TM are calculated.


### 4.1. Metallic Structures inTM

In [29], Yano and Ishihara defined J˜ in TM with(26)J˜XV=−XH,J˜XH=XV,∀X∈ℑ01M.One can easily establish thatJ˜ produces an a.c.s. on TM.Also, Gezer, and Kamran [22] defined J˜ in TM by means of(27)J˜XH=12αXH+2σαβ−αX⊗E˜V,J˜XV=12αX⊗E˜V+2σαβ−αXH,J˜AV=σαβAV,where X is any vector field; A represents a 1,1-tensor field, E˜=g°E, and g stands for a Riemannian metric on M.Motivated from the above studies, we define a new tensorJ˜ in TM in the following way:(28)J˜XH=12αXH+2σαβ−αXV,J˜XV=12αXV+2σαβ−αXH,∀X∈ℑ01M.Theorem 3.
We considerM.S.J in M associated with ∇, then J˜ written by(29)J˜XH=12αXH+2σαβ−αXV,J˜XV=12αXV+2σαβ−αXH,represents a M.S. in TM.Proof.
By the virtue of (28), we observe(30)J˜2−αJ˜−βIXH=J˜J˜XH−αJ˜XH−βXH,=J˜12αXH+2σαβ−αXV−α12αXH+2σαβ−αXV−βXH.
On solving, the obtained equation isJ˜2−αJ˜−βIXH=0.
Similarly,J˜2−αJ˜−βIXV=0,X∈ℑ01M, which implies(31)J˜2−αJ˜−βI=0.
It is therefore concluded thatJ˜ is a metallic structure in TM.


### 4.2. Nijenhuis TensorN˜ of M.S. in TM

We note that(32)N˜X˜,Y˜=J˜X˜,J˜Y˜+J˜2X˜,Y˜−J˜X˜,J˜Y˜−J˜J˜X˜,Y˜,∀X˜,Y˜∈ℑ01TM.SettingX˜=XV and Y˜=YV in (32) and using (28), the obtained equation is(33)N˜XV,YV=J˜2XV,YV+J˜XV,J˜YV−J˜XV,J˜YV−J˜J˜XV,YV.Using (28) and Proposition 3 in (33) results(34)N˜XV,YV=12αXV+2σαβ−αXH,12αYV+2σαβ−αYH,−J˜12αXV+2σαβ−αXH,YV−J˜XV,12αYV+2σαβ−αYH+J˜2XV,YV,=α42σαβ−αX,YV−∇XYV−Y,XV+∇YXV+2σαβ−α24X,YH−γR^X,Y+α2σαβ−α4Y,XV+2σαβ−α24Y,XH−α2σαβ−α4α∇YXV−2σαβ−α24∇YXH−α2σαβ−α4X,YV−2σαβ−α24Y,XH+α2σαβ−α4∇XYV+2σαβ−α24∇XYH+J˜2XV,YV,=2σαβ−α24Y,XH−2σαβ−α24∇YXH+2σαβ−α24∇XYH−2σαβ−α24γR^X,Y,=2σαβ−α24∇XYH−∇YXH−X,YH−2σαβ−α24γR^X,Y,SinceTX,YH=∇XYH−∇YXH−X,YH.The obtained equation is(35)N˜XV,YV=2σαβ−α24TX,YH−2σαβ−α24γR^X,Y.Using similar techniques, the following equations can be easily obtained:(36)N˜XV,YH=−2σαβ−α24TX,YV+2σαβ−α2J˜γR^X,Y−α2σαβ−α4γR^X,Y,N˜XH,YH=2σαβ−α24TX,YH−2σαβ−α24γR^X,Y,∀X,Y∈ℑ01M.This procedure can be summarized as follows:Theorem 4.
The metallic structureJ˜ defined in (28) is integrable, i.e., N˜=0, iff R^=0 and T=0. In this case, T stands for the torsion tensor of ∇, and R^ is used for curvature tensor of an affine connection ∇^.


### 4.3. The Lie Derivative of the Metallic Structure inTM

We write the Lie derivative ofJ˜ as [29](37)EX˜J˜Y˜=X˜,J˜Y˜−J˜X˜,Y˜,∀X˜,Y˜∈ℑ01TM.OperatingEYH on both sides of (28) and taking account of Proposition 1 and Proposition 3, it reduces to(38)EYHJ˜XH=α2EYHXH+2σαβ−α2£YHXH,since EYHXH=YH,XH,(39)EYHJ˜XH+J˜EYHXH=α2YH,XH+2σαβ−α2YH,XV,EYHJ˜XH=−J˜EYHXH+α2YH,XH+2σαβ−α2YH,XV,asYH,XH=−XH,YH,=J˜X,YH−J˜γR^X,Y−α2X,YH−γR^X,Y−2σαβ−α2X,YV−∇XYV,EYHJ˜XH=2σαβ−α2∇XYV−J˜γR^X,Y−α2γR^X,Y,∀X,Y∈ℑ01M.Using similar devices, the following equations are obtained:(40)EYVJ˜XH=2σαβ−α2∇^XYH,(41)EYVJ˜XV=−2σαβ−α2∇^XYV,(42)EYHJ˜XV=−2σαβ−α2∇XYH+2σαβ−α2γR^X,Y,∀X,Y∈ℑ01M.Equations (39)–(42) produce the following:Theorem 5.
LetY be V.F. in M, YV and YH be its “vertical” and “horizontal lifts” in TM, respectively. Also, we suppose that J˜ represents a metallic structure defined with the help of (28). Then, EYVJ˜=0 iff ∇^Y=0, and EYHJ˜=0 iff ∇^Y=0 and γR^X,Y=0. Here, X∈ℑ01M, and R^ stands for the curvature tensor of ∇^.


### 4.4. Explicit Examples

Example 1.
Consideringα=1,β=1 in (2) implies J2−J−I=0. It is referred to as the golden structure and can be derived from a metallic structure [21, 33].
We defineJ˜ in TM by J˜2=J˜+I, then (28) become(43)J˜XH=12XH+5XV,J˜XV=12XV+5XH,for any X∈ℑ01M.Example 2.
Let us considerα=2,β=3,J˜XH=3XH, and J˜XV=3XV.
On solving, we obtainJ˜2−2J˜−3IXH=0.
Similarly,J˜2−2J˜−3IXV=0 for any X∈ℑ01M, which implies(44)J˜2−2J˜−3I=0,
thereby establishing the fact thatJ˜ is metallic structure in TM.


## 4.1. Metallic Structures inTM

In [29], Yano and Ishihara defined J˜ in TM with(26)J˜XV=−XH,J˜XH=XV,∀X∈ℑ01M.One can easily establish thatJ˜ produces an a.c.s. on TM.Also, Gezer, and Kamran [22] defined J˜ in TM by means of(27)J˜XH=12αXH+2σαβ−αX⊗E˜V,J˜XV=12αX⊗E˜V+2σαβ−αXH,J˜AV=σαβAV,where X is any vector field; A represents a 1,1-tensor field, E˜=g°E, and g stands for a Riemannian metric on M.Motivated from the above studies, we define a new tensorJ˜ in TM in the following way:(28)J˜XH=12αXH+2σαβ−αXV,J˜XV=12αXV+2σαβ−αXH,∀X∈ℑ01M.Theorem 3.
We considerM.S.J in M associated with ∇, then J˜ written by(29)J˜XH=12αXH+2σαβ−αXV,J˜XV=12αXV+2σαβ−αXH,represents a M.S. in TM.Proof.
By the virtue of (28), we observe(30)J˜2−αJ˜−βIXH=J˜J˜XH−αJ˜XH−βXH,=J˜12αXH+2σαβ−αXV−α12αXH+2σαβ−αXV−βXH.
On solving, the obtained equation isJ˜2−αJ˜−βIXH=0.
Similarly,J˜2−αJ˜−βIXV=0,X∈ℑ01M, which implies(31)J˜2−αJ˜−βI=0.
It is therefore concluded thatJ˜ is a metallic structure in TM.


## 4.2. Nijenhuis TensorN˜ of M.S. in TM

We note that(32)N˜X˜,Y˜=J˜X˜,J˜Y˜+J˜2X˜,Y˜−J˜X˜,J˜Y˜−J˜J˜X˜,Y˜,∀X˜,Y˜∈ℑ01TM.SettingX˜=XV and Y˜=YV in (32) and using (28), the obtained equation is(33)N˜XV,YV=J˜2XV,YV+J˜XV,J˜YV−J˜XV,J˜YV−J˜J˜XV,YV.Using (28) and Proposition 3 in (33) results(34)N˜XV,YV=12αXV+2σαβ−αXH,12αYV+2σαβ−αYH,−J˜12αXV+2σαβ−αXH,YV−J˜XV,12αYV+2σαβ−αYH+J˜2XV,YV,=α42σαβ−αX,YV−∇XYV−Y,XV+∇YXV+2σαβ−α24X,YH−γR^X,Y+α2σαβ−α4Y,XV+2σαβ−α24Y,XH−α2σαβ−α4α∇YXV−2σαβ−α24∇YXH−α2σαβ−α4X,YV−2σαβ−α24Y,XH+α2σαβ−α4∇XYV+2σαβ−α24∇XYH+J˜2XV,YV,=2σαβ−α24Y,XH−2σαβ−α24∇YXH+2σαβ−α24∇XYH−2σαβ−α24γR^X,Y,=2σαβ−α24∇XYH−∇YXH−X,YH−2σαβ−α24γR^X,Y,SinceTX,YH=∇XYH−∇YXH−X,YH.The obtained equation is(35)N˜XV,YV=2σαβ−α24TX,YH−2σαβ−α24γR^X,Y.Using similar techniques, the following equations can be easily obtained:(36)N˜XV,YH=−2σαβ−α24TX,YV+2σαβ−α2J˜γR^X,Y−α2σαβ−α4γR^X,Y,N˜XH,YH=2σαβ−α24TX,YH−2σαβ−α24γR^X,Y,∀X,Y∈ℑ01M.This procedure can be summarized as follows:Theorem 4.
The metallic structureJ˜ defined in (28) is integrable, i.e., N˜=0, iff R^=0 and T=0. In this case, T stands for the torsion tensor of ∇, and R^ is used for curvature tensor of an affine connection ∇^.


## 4.3. The Lie Derivative of the Metallic Structure inTM

We write the Lie derivative ofJ˜ as [29](37)EX˜J˜Y˜=X˜,J˜Y˜−J˜X˜,Y˜,∀X˜,Y˜∈ℑ01TM.OperatingEYH on both sides of (28) and taking account of Proposition 1 and Proposition 3, it reduces to(38)EYHJ˜XH=α2EYHXH+2σαβ−α2£YHXH,since EYHXH=YH,XH,(39)EYHJ˜XH+J˜EYHXH=α2YH,XH+2σαβ−α2YH,XV,EYHJ˜XH=−J˜EYHXH+α2YH,XH+2σαβ−α2YH,XV,asYH,XH=−XH,YH,=J˜X,YH−J˜γR^X,Y−α2X,YH−γR^X,Y−2σαβ−α2X,YV−∇XYV,EYHJ˜XH=2σαβ−α2∇XYV−J˜γR^X,Y−α2γR^X,Y,∀X,Y∈ℑ01M.Using similar devices, the following equations are obtained:(40)EYVJ˜XH=2σαβ−α2∇^XYH,(41)EYVJ˜XV=−2σαβ−α2∇^XYV,(42)EYHJ˜XV=−2σαβ−α2∇XYH+2σαβ−α2γR^X,Y,∀X,Y∈ℑ01M.Equations (39)–(42) produce the following:Theorem 5.
LetY be V.F. in M, YV and YH be its “vertical” and “horizontal lifts” in TM, respectively. Also, we suppose that J˜ represents a metallic structure defined with the help of (28). Then, EYVJ˜=0 iff ∇^Y=0, and EYHJ˜=0 iff ∇^Y=0 and γR^X,Y=0. Here, X∈ℑ01M, and R^ stands for the curvature tensor of ∇^.


## 4.4. Explicit Examples

Example 1.
Consideringα=1,β=1 in (2) implies J2−J−I=0. It is referred to as the golden structure and can be derived from a metallic structure [21, 33].
We defineJ˜ in TM by J˜2=J˜+I, then (28) become(43)J˜XH=12XH+5XV,J˜XV=12XV+5XH,for any X∈ℑ01M.Example 2.
Let us considerα=2,β=3,J˜XH=3XH, and J˜XV=3XV.
On solving, we obtainJ˜2−2J˜−3IXH=0.
Similarly,J˜2−2J˜−3IXV=0 for any X∈ℑ01M, which implies(44)J˜2−2J˜−3I=0,
thereby establishing the fact thatJ˜ is metallic structure in TM.


## 5. Application

Here, the metallic structureJ˜ endowed with projection operators l˜ and m˜ in TM is investigated.Letl˜ and m˜ be the projection operators in TM and defined by(45)l˜=J˜2−αJ˜β,(46)m˜=I−J˜2−αJ˜β.Theorem 6.
IfJ˜ represents the metallic structure and l˜ and m˜ stand for the projection operators in TM, then(47)l˜+m˜=0,l˜2=l˜,m˜2=m˜,l˜m˜=m˜l˜=0,J˜l˜=l˜J˜=J˜,J˜m˜=m˜J˜=0.Proof.
One can easily write the proof with the help of (45) and (46).Theorem 7.
LetXH be the horizontal and XV be the vertical lift of v.f. X in TM. Then,(48)l˜XH=XH,l˜XV=XV,m˜XH=0,m˜XV=0,where l˜ and m˜ are the projection operators in TM.Proof.
Equations (28) and (45) produce(49)l˜XH=1βJ˜2XH−αβJ˜XH,βl˜XH=J˜12αXH+2σαβ−αXV−α12αXH+2σαβ−αXV,2βl˜XH=α2αXH+2σαβ−αXV−α2σαβ−α2αXV+2σαβ−αXH−α2XH−α2σαβ−αXV,l˜XH=XH.



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*Source: 5037620-2022-04-26.xml*