# Almost Bronze Structures on Differentiable Manifolds

Authors: Mustafa Özkan; Seher Doğan
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study introduces a novel structure that is not included in the metallic structure family. This new structure, which is called an almost bronze structure, has been defined using a1,1 type tensor field φ which fulfills the requirement φ2=mφ−Id on a differentiable manifold. We investigated the parallelism and integrability conditions of these almost bronze structures by use of an almost product structure corresponding to them. Also, we have defined an almost bronze Riemannian manifold.


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## Body

## 1. Introduction

Several polynomial structures on a differentiable manifold are defined using1,1 type C∞–tensor fields. The following structures can be listed as examples of these polynomial structures: almost tangent structures, almost complex structures, almost product structures, golden structures, silver structures, bronze structures, and metallic structures. These structures have been recently studied by many authors (see [1–12]).The term “metallic ratio” has been defined by Spinadel [13] as a generalized form of the golden proposition in 1999 and coined the concept of the “metallic means family” or “metallic propositions.” The author has revealed the relationship between this metallic means family and the generalization of the Fibonacci numbers, i.e., the generalized secondary Fibonacci sequence.There are several important generalizations of the Fibonacci and Lucas numbers such as Horadam,k-Fibonacci, m-Fibonacci, bivariate Fibonacci, m-Lucas, and p-Lucas numbers. As a result, members of the metallic ratio family have been obtained. The silver Fibonacci, silver Lucas, bronze Fibonacci, and bronze Lucas numbers [14] are examples of these generalizations. In [14], Kalia defined the silver Fibonacci numbers as a generalized form of the Fibonacci numbers and the silver Lucas numbers as a generalized form of the Lucas numbers. The author has revealed that silver Fibonacci numbers and silver Lucas numbers have been related to the golden ratio. Then, the author defined the bronze Fibonacci numbers and the bronze Lucas numbers as well.As a member of this metallic ratio family, the bronze ratio was defined in [7]. In [12], the bronze structure was studied considering the bronze ratio, which is a member of this metallic ratio family. In [14], Kalia introduced a new bronze ratio which is related to the bronze Lucas and bronze Fibonacci numbers, which are not members of the metallic mean family. In [15], Şahin defined an almost poly-Norden structure by use of this new bronze ratio, examined several geometric properties of this structure by using a corresponding almost complex structure, and investigated poly-Norden manifolds in terms of their maps with other manifolds having different structures.In this study, we investigate new almost bronze structures by using the new bronze mean. The new almost bronze structures are polynomial structures with a structure polynomial ofQφ=φ2−mφ+Id for m∈ℝ∖−2,2 on differentiable manifolds. The major novelty introduced by this paper is that an almost product structure has been used to examine a new almost bronze structure’s geometry on a differentiable manifold. To the best of our knowledge, the new almost bronze structure on manifolds has not been studied yet before the current study in the literature. In the study, the method in [2] was used.The present study has been designed as follows: Section2 provides preliminary knowledge about the new bronze means, bronze Fibonacci numbers, and bronze Lucas numbers. In Section 3, the new almost bronze structure on a differentiable manifold is introduced. Several properties of these structures are obtained in relation to the bronze Fibonacci and bronze Lucas numbers. Moreover, the relationships among the bronze ratio, complex bronze ratio, and tangent real bronze ratio were determined. In Section 4, several examples of almost bronze structures are presented. The connections in principal fibre bundles and tangent bundles are explored in Section 5 in terms of almost bronze structure. Then in Section 6, the integrability feature of the almost bronze structure is studied, and the parallelism of the almost bronze structure is investigated considering the Schouten connection and the Vrǎnceanu connection. In the last section, an almost bronze Riemannian manifold is defined, and several features of the defined manifold are studied. This section also includes an illustration of the defined bronze structure on the manifold ℝ2 manifold.


## 2. Preliminaries

This section provides brief information about the new mean, i.e., the bronze mean, using the related bronze Fibonacci and bronze Lucas numbers defined in [14].The bronze Fibonacci numbersfm,n are a family of sequencing numbers defined by the recurrence presented below:(1)fm,n+2=mfm,n+1−fm,n;fm,0=0,fm,1=1.On the other hand, the bronze Lucas numberslm,n refer to a family of sequencing numbers defined by the following recurrence:(2)lm,n+2=mlm,n+1−lm,n;lm,0=2,lm,1=m.Different from the bronze means presented in [7, 10–12], a new bronze mean is defined as follows:(3)ρm=m+m2−42.This is obtained as the positive root of the following equation:(4)x2−mx+1=0.For brevity’s sake, we will refer to this mean as the bronze mean.The bronze Fibonacci numbers and bronze Lucas numbers have the following relationship:(5)lm,n=fm,n+1−fm,n−1=2fm,n+1−mfm,n,m2−4fm,n=lm,n+1−lm,n−1.The bronze means’ continued fractions are defined asm−1;1,m−2¯, while the recurrence relationship is defined as follows:(6)ρmn+2=mρmn+1−ρmn.The following relationship is another one:(7)ρmn=lm,n+fm,nm2−42.


## 3. Almost Bronze Structures on Manifolds

In this paper,M refers to a C∞–class differentiable manifold, and all tensor fields and connections on this manifold are considered to be of class C∞. We denote by XM the Lie algebra of the vector fields on M.Definition 1.
(see [16]). Let M be a differentiable manifold and F be a 1,1 type tensor field on M. If F satisfies the following equation it is defined as a polynomial structure:(8)QX=Xk+akXk−1+⋯+a2X+a1Id=0.In this equation, the identity operator on XM is denoted by Id, while Fk−1q, Fk−2q, …, Fq, Id are linear independent for each point q in M. In this case, the polynomial QX is said to be a structure polynomial.As stated in [17], F, which is an almost product (resp., almost complex, almost tangent) structure, satisfies the condition of F2−Id=0 (resp., J2+Id=0, T2=0). Then M,F is named as an almost product (resp., almost complex, almost tangent) manifold.Being inspired by the bronze mean given in (3), we can introduce the almost bronze structure which is a new structure on a differentiable manifold M.Definition 2.
LetM be a differentiable manifold and φ be a 1,1 tensor field that satisfies the equation below:(9)φ2=mφ−Idwhere m∈ℝ∖−2,2. Here, φ is said to be a new almost bronze structure on manifold M. For brevity’s sake, we will refer to this structure as the almost bronze structure.Several properties of almost bronze structures regarding number sequences are as follows:Proposition 1.
The power of an almost bronze structure on the manifoldM is defined as follows for any integer n:(10)φn=fm,nφ−fm,n−1Id,φn=lm,n+1−lm,n−1m2−4φ−lm,n−lm,n−2m2−4Id,where fm,n stands for the bronze Fibonacci numbers, and lm,n stands for the bronze Lucas numbers.Letρm be a bronze ratio. Binet’s formulas of the bronze Fibonacci sequence and the bronze Lucas sequence are defined as follows:(11)fm,n=ρmn−m−ρmnm2−4,lm,n=ρmn+m−ρmn,respectively [14].From (10) and (11), we have a new form for equality (10)(12)φn=ρmn−m−ρmnm2−4φ+ρmn−1−m−ρmn−1m2−4Id,φn=ρmn+1−ρmn−1+m−ρmn+1−m−ρmn−1m2−4φ−ρmn−ρmn−2+m−ρmn−m−ρmn−2m2−4Id.Unless otherwise stated, we will takem∈ℝ∖−2,2 throughout the study.A simple calculation results in the following.Proposition 2.
The properties of an almost bronze structureφ are as follows:(i)
The bronze ratioρm and ρ¯m=m−ρm are the eigenvalues of φ.(ii)
On the tangent space of the manifoldTpM, φ is an isomorphism for each p∈M.(iii)
φ is an invertible structure, and its inverse φ^ is an almost bronze structure on M, and it can be calculated as follows: φ^=mId−φ.A polynomial structure on a manifoldM induces a generalized almost product structure P, as described in [16]. Thus, the almost product structure and almost bronze structure on M are connected structures.Theorem 1.
(i)
Ifφ is an almost bronze structure on M, then Pφ is an almost product structure on M, and it is defined as follows:(13)Pφ=1m2−42φ−mId.We call thatPφ is an almost product structure induced by φ.(ii)
IfP is an almost product structure on M, then φP is an almost bronze structure on M, and it is defined as follows:(14)φP=12mId+m2−4P.Thus,φP is named as an almost bronze structure induced by P.
SinceφPφ=φ and PφP=P, almost product structures and almost bronze structures on M have a one-to-one correspondence.Proof.
(i)
Assume thatφ is an almost bronze structure on the manifold M. In this case, the structure Pφ=1/m2−42φ−mId obtained from the almost bronze structure φ is an almost product structure since the following condition is satisfied:(15)Pφ2=4φ2−4mφ+m2Idm2−4=4φ2−mφ+m2Idm2−4=m2−4Idm2−4=Id.(ii)
Assume thatP is an almost product structure on the manifold M. The structure φP=1/2mId+m2−4P, which is induced by the almost product structure P, is an almost bronze structure since the following condition is satisfied:(16)φP2=m2Id+2mm2−4P+m2−4Id4=m2−2Id+mm2−4P2=12m2−2Id+m2φP−mId=mφP−Id.We getφPφ=φ and PφP=P by straightforward calculations from (13) and (14).
Using the above-mentioned literature and Theorem1, we can give the following definitions:(i)
Assume thatM,T is an almost tangent manifold. Then, the tensor field φT, which is induced by T, is defined as follows:(17)φT=12mId+m2−4T,m∈ℝ∖−2,2,and it is called an almost tangent bronze structure on manifoldM.Then, the equation verified by an almost tangent bronze structure(18)φT2−mφT+m24Id=0.The tangent real bronze ratioρmt=m/2 is then calculated using the associated equation in the real field ℝ, that is, x2−mx+m2/4=0.(ii)
LetJ be an almost complex structure on manifold M. The tensor field φJ which is induced by J is defined using the equation(19)φJ=12mId+m2−4J,m∈ℝ∖−2,2,is called an almost complex bronze structure on manifoldM.φJ satisfies the following polynomial equation:(20)φJ2−mφJ+m2−22Id=0.
ForM=ℝ2, we obtain the following equation:(21)x2−mx+m2−22=0
with solutions(22)x1=m2+m2−42i,x2=m2−m2−42i.Definition 3.
The complex number(23)ρmc=m+m2−4i2
will be called complex bronze ratio.If we take−2<m<2 in (19), then we have an almost poly-Norden structure defined in [15].(24)φJ2−mφJ+Id=0.


## 4. Examples of the Almost Bronze Structures

Several examples of the almost bronze structure will be presented in this section.Example 1.
(Clifford algebras). LetCℓn be the real Clifford algebra of the Euclidean space ℝn [18]. The standard base ei1≤i≤n of ℝn satisfies the multiplication rules according to the Clifford product(25)eiei=1,eiej=−ejeifori≠j.
Therefore, by using(26)φei=12m+m2−4ei,where m∈ℝ∖−2,2 and (25), we can obtain a new representation of the Clifford algebra as follows:(27)φeialmost bronze structure,φeiφej+φejφei=mφei+φej−m22fori≠j.
In [18], Cℓ2 is constructed as(28)1=1001,e1=100−1,e2=0110,
and then we get(29)iφe1=12m1+m2−4e1=ρm00m−ρm,iiφe2=12m1+m2−4e2=12m2ρm−m2ρm−mm.Example 2.
(Quaternion algebras). There is a quaternion algebraℍ with a base 1,i,j,k satisfying(30)i2=−1,j2=−1,k2=−1,ij=−ji=k,jk=−kj=i,ki=−ik=j.Any quaternion can be written as follows:(31)q=Sq+V⟶q=a0+a1i+a2j+a3k,where Sq=a0 denote the scalar part of q and V⟶q=a1i+a2j+a3k denote the vectorial part of q.
Forq≠0, q0=q/Nq is called unit quaternion where Nq=a02+a12+a22+a32 is the norm of quaternion q. We can express each unit quaternion in the following form q0=cosα+S⟶0sinα, where ε⟶0stands for a unit vector that satisfies S⟶02=−1.
Thus, inspired by [19], we have the following.(a)
An almost bronze hyperbolic quaternion structure can be defined as follows:(32)φhq=m2+m2−42S⟶0,S⟶0,S⟶0=1,S⟶02=1,where.,.L is the inner product and S⟶0 is a unit hyperbolic vector.(b)
An almost bronze biquaternion structure can be defined as follows:(33)φbq=m2+m2−4i2S⟶0,S⟶0,S⟶0=1,S⟶02=−1,where.,.L is the inner product and i2=−1.(c)
An almost bronze split quaternion structure can be defined as follows:(34)φsq=m2+m2−42S⟶0,S⟶0,S⟶0L=1,S⟶02=1,where.,.L is Lorentzian inner product and S⟶0 is a spacelike unit vector in the Minkowski 3-space E13.(d)
An almost bronze dual split quaternion structure can be defined as follows:(35)φdsq=m2+m2−42S⟶0,S⟶0,S⟶0L=1,S⟶02=1,where.,.is Lorentzian inner product and S⟶0is a spacelike unit dual vector in E13.(e)
An almost bronze hyperbolic split quaternion structure can be defined as follows:(36)φhsq=m2+m2−42S⟶0,S⟶0,S⟶0L=1,S⟶02=1,where.,.is Lorentzian inner product and S⟶0is a spacelike unit hyperbolic vector in E13.Example 3.
(Bronze matrices). LetMatn,ℝ be a matrix algebra of real n×n –matrices and φ¯∈Matn,ℝ. If φ¯ satisfies the following equation:(37)φ¯2=mφ¯−In,where In is the identity matrix on Matn,ℝ, then this matrix is called an almost bronze matrix.
By solving (37) for n=2, we can obtain the almost bronze structure in Mat2,ℝ.(i)
Fora∈ℝ, b∈ℝ∖0, and d∈ℝ(38)φ¯a,b=a−1ba2−ma+1bm−a,φ¯d,b=m−d−1bd2−md+1bd.(ii)
Fora=ρm and b∈ℝ,(39)φ¯ρm,b=ρm0bm−ρm,φ¯m−ρm,b=m−ρm0bρmφ¯ρm,b=ρmb0m−ρm,φ¯m−ρm,b=m−ρmb0ρm.(iii)
Fora=ρm and b=0,(40)φ¯ρm,0=ρm00m−ρmφ¯m−ρm,0=m−ρm00ρm.
Then, from (29) and (38), we obtain(41)φ¯1=limb⟶0φ¯ρm,b,φ¯2=φ¯m/2,m2−4/2.
Also, from (38), we get the sequence of trace Trφ¯a,bkk≥0 is the bronze Lucas sequence: 2,m,m2−2,mm2−3,….Example 4.
(Bronze reflections). As also stated in [20], the equation of the reflection in accordance with a hyperplane ℋ with the normal u∈∖0in Euclidean space E,<,> is as follows:(42)ruα=α−2α,uu,uuforα∈E.
In this equation, it is obvious thatru2=IE where IE is the identity on E.
Thus, the bronze reflection with respect tou can be defined as follows:(43)φu=mIE+m2−4ru2,
and thenu is an eigenvector of φu with the corresponding eigenvalue m−ρm. Then, the following equation is obtained from [20, p.314](44)XφuX−1=φXu,where X is an orthogonal transformation on E. Thus, the following equation can be written as an explicit expression of the linear transformation(45)φuα=ρmα+m−2ρmα,uu,uu.Example 5.
(Triple structures with respect to almost bronze structures). Given two1,1 tensor fields ℱ and P on the manifold M and K=P°ℱ, we called that the triple ℱ,P,K=P°ℱ is as follows [21]:(1)
an almost hyperproduct structure: ifℱ, P are almost product structures and P°ℱ=ℱ°P, then K is an almost product structure,(2)
an almost biproduct complex structure: ifℱ, P are almost product structures and P°ℱ=−ℱ°P, then K is an almost complex structure,(3)
an almost product bicomplex structure: ifℱ, P are almost complex structures and P°ℱ=ℱ°P, then K is an almost product structure,(4)
an almost hypercomplex structure: ifℱ, P are almost complex structures and P°ℱ=−ℱ°P, then K is an almost complex structure.
Taking into account (14), we get(46)φℱ=m2Id+2ρm−m2ℱ,φP=m2Id+2ρm−m2P,φK=m2Id+2ρm−m2K.
Then, we find a relation betweenφℱ, φP, and φK as(47)m2−4φK=2φPφℱ−mφP−mφℱ+ρm2Id+Id.Hence, the triple φℱ,φP,φK is as follows:(i)
an almost hyperproduct structure: if and only ifφℱ , φP are almost bronze structures and φℱφP−φPφℱ=0, then φK is an almost bronze structure,(ii)
an almost biproduct complex structure: if and only ifφℱ, φP are almost bronze structures and φPφℱ+φℱφP=mφP+φℱ−1/2m2Id, then φK is an almost complex bronze structure,(iii)
an almost product bicomplex structure: if and only ifφℱ, φP are almost complex bronze structures and φℱφP−φPφℱ=0, then φK is an almost bronze structure,(iv)
an almost hypercomplex structure: if and only ifφℱ, φP are almost complex bronze structures and φPφℱ+φℱφP=mφP+φℱ−1/2m2Id, then φK is an almost complex bronze structure.Example 6.
(Almost bronze structures from symplectic distributions). Given any symplectic vector spaceV,μ, we have V=U+Uμ where Uμ=v∈V:μv,u=0for everyu∈Uif U is a subspace of V,μ. A subspace U of V,μ is symplectic if and only if μ|U×U is nondegenerate (or U∩Uμ=0) [22]. Consequently, if R is a symplectic distribution on a symplectic manifold N,μ (i.e., Rx is a symplectic subspace of the tangent space at x∈N), then another symplectic distribution S=Rμ is obtained complementary to R. In this case, P=r−s is an almost product structure where r and s are the corresponding projection tensors. Then, an associated symplectic almost bronze structure is obtained as follows by using (14)(48)φR=ρmr+m−ρms.


## 5. Connection as Almost Bronze Structure

### 5.1. Connections in the Principal Fibre Bundles

Assume thatPM,G is a principal fibre bundle on a manifold M, where P is the total space, M is the base space, G is the structure group, and π is the projection. Let V denote a vertical distribution (i.e., V=kerπ∗), ℋ denote a horizontal distribution (i.e., TP=V⊕ℋ), and ℋ be a G-invariant. Thus, v and h become the corresponding projectors of V and ℋ, respectively. Therefore, 1,1 type tensor field can be defined as follows:(49)P=v−h,and it is an almost product structure on P. According to [2], P defines a connection if and only if the following conditions are satisfied:(a)
PX=X⇔X is a vertical vector field,(b)
dRe°Pu=Pue°dRe for each u∈P and e∈G.We can get the following proposition by using the relation between the almost bronze structure and the almost product structure:Proposition 3.
An almost bronze structureφ on P specifies a connection if and only if the following conditions are satisfied:(a)
ForX∈XP, φX=ρmX if and only if X∈V.(b)
dRe°φu=φue°dRe for each u∈P and e∈G.Assumingthatϖ∈∧P,g1 is a connection 1-form of horizontal distribution ℋ and Ω∈∧2P,g is the curvature form of ϖ where g stands for the Lie algebra of G, we can obtain the following relation [2]:(50)ΩX,Y=−14ϖNPX,Y,where NP stands for the Nijenhuis tensor of P, i.e.,(51)NPX,Y=X,Y+PX,PY−PPX,Y−PX,PY,for allX, Y vector fields on M.Thus, the following proposition can be stated by straightforward calculations from (13) and (51).Proposition 4.
Letφ be an almost bronze structure on the manifold M and Pφ be an almost product structure induced by φ. Then(52)NPφX,Y=4m2−4NφX,Y(53)ΩX,Y=−1m2−4ϖNφX,Y,where NPφ and Nφ stand for the Nijenhuis tensors of Pφ and φ, respectively.Therefore, it can be stated that the integrability of the structuresφ and Pφ is equivalent.Proposition 5.
The connection is flat (i.e.,Ω=0) if and only if the associated almost bronze structure is integrable, which means Nφ=0.Given two vector fieldsX˜, Y˜ on the manifold M and a connection, the lift ℒϖ:XM⟶XP is determined by this connection if the following condition is met [2]:(54)ℒϖX˜,ℒϖY˜−ℒϖX˜,Y˜=NPℒϖX˜,ℒϖY˜.Thus, considering (52) and (54), we have the following proposition.Proposition 6.
The liftℒϖ, which is defined by ϖ, is a morphism if and only if the associated almost bronze structure is integrable.


### 5.2. Connection in the Tangent Bundles

LetTM=∪p∈MTpM be the tangent bundle of the manifold M, πM be the projection, πM∗ be its differential, and VTM=kerπM∗ be the vertical distribution of M. For any coordinate neighborhood U,xi in M, TM,xi,yi stands for the induced coordinate neighborhood in TM, i.e., xiu=xiπMu and yiu=dxiu for all u∈πM−1U. For an atlas on TM with these local coordinates, the almost tangent structure of TM is T=∂/∂yi⊗dxi, i.e., T2=0.Definition 4.
(see [2]). Given an almost tangent structure T of TM and a 1,1 tensor field ν on M, ν is called a vertical projector when the following conditions are met:(55)ν°T=T,T°ν=0.Definition 5.
(see [2]). N, which is complementary distribution to VTM, i.e.,(56)XM=VTM⊕N,is called a normalization or a nonlinear connection or a horizontal distribution.Knowing that a vertical projectorν is C∞M linear with Imν=VTM, we can state the following proposition.Proposition 7 (see [2]).
We obtain a nonlinear connectionNν from the vertical projector ν by using kerν=Nν. Otherwise, with the respect to the separation (56), νN and hN are vertical and horizontal projectors, respectively, if N is a nonlinear connection.Thus, the next proposition can be given as follow.Proposition 8.
(see [2]).νN is a vertical projector provided that NνN=N.Definition 6.
(see [2]). A 1,1 type tensor field Γ is called a nonlinear connection of an almost product type if the following relations are provided:(57)T°Γ=T,Γ°T=−T.Proposition 9.
(see [2]). The following assertions hold true if Γ is a nonlinear connection of an almost product type:(i)
νΓ=1/2Id−Γ is a vertical projector,(ii)
VTM is the −1-eigenspace of Γ when NνΓ is the +1 -eigenspace of Γ.Corollary 1.
(see [2]). Any vertical projector v induces an almost product structure on manifold M as follows: Γ=Id−2v.Thus,thisresult has been associated with the almost bronzestructure.Proposition 10.
Obtained by the vertical projectorv, a nonlinear connection N on M can also be defined by an almost bronze structure φ=φΓ(58)φ=ρmId−m2−4νwith VTM the m−ρm-eigenspace and N the ρm-eigenspace.


## 5.1. Connections in the Principal Fibre Bundles

Assume thatPM,G is a principal fibre bundle on a manifold M, where P is the total space, M is the base space, G is the structure group, and π is the projection. Let V denote a vertical distribution (i.e., V=kerπ∗), ℋ denote a horizontal distribution (i.e., TP=V⊕ℋ), and ℋ be a G-invariant. Thus, v and h become the corresponding projectors of V and ℋ, respectively. Therefore, 1,1 type tensor field can be defined as follows:(49)P=v−h,and it is an almost product structure on P. According to [2], P defines a connection if and only if the following conditions are satisfied:(a)
PX=X⇔X is a vertical vector field,(b)
dRe°Pu=Pue°dRe for each u∈P and e∈G.We can get the following proposition by using the relation between the almost bronze structure and the almost product structure:Proposition 3.
An almost bronze structureφ on P specifies a connection if and only if the following conditions are satisfied:(a)
ForX∈XP, φX=ρmX if and only if X∈V.(b)
dRe°φu=φue°dRe for each u∈P and e∈G.Assumingthatϖ∈∧P,g1 is a connection 1-form of horizontal distribution ℋ and Ω∈∧2P,g is the curvature form of ϖ where g stands for the Lie algebra of G, we can obtain the following relation [2]:(50)ΩX,Y=−14ϖNPX,Y,where NP stands for the Nijenhuis tensor of P, i.e.,(51)NPX,Y=X,Y+PX,PY−PPX,Y−PX,PY,for allX, Y vector fields on M.Thus, the following proposition can be stated by straightforward calculations from (13) and (51).Proposition 4.
Letφ be an almost bronze structure on the manifold M and Pφ be an almost product structure induced by φ. Then(52)NPφX,Y=4m2−4NφX,Y(53)ΩX,Y=−1m2−4ϖNφX,Y,where NPφ and Nφ stand for the Nijenhuis tensors of Pφ and φ, respectively.Therefore, it can be stated that the integrability of the structuresφ and Pφ is equivalent.Proposition 5.
The connection is flat (i.e.,Ω=0) if and only if the associated almost bronze structure is integrable, which means Nφ=0.Given two vector fieldsX˜, Y˜ on the manifold M and a connection, the lift ℒϖ:XM⟶XP is determined by this connection if the following condition is met [2]:(54)ℒϖX˜,ℒϖY˜−ℒϖX˜,Y˜=NPℒϖX˜,ℒϖY˜.Thus, considering (52) and (54), we have the following proposition.Proposition 6.
The liftℒϖ, which is defined by ϖ, is a morphism if and only if the associated almost bronze structure is integrable.


## 5.2. Connection in the Tangent Bundles

LetTM=∪p∈MTpM be the tangent bundle of the manifold M, πM be the projection, πM∗ be its differential, and VTM=kerπM∗ be the vertical distribution of M. For any coordinate neighborhood U,xi in M, TM,xi,yi stands for the induced coordinate neighborhood in TM, i.e., xiu=xiπMu and yiu=dxiu for all u∈πM−1U. For an atlas on TM with these local coordinates, the almost tangent structure of TM is T=∂/∂yi⊗dxi, i.e., T2=0.Definition 4.
(see [2]). Given an almost tangent structure T of TM and a 1,1 tensor field ν on M, ν is called a vertical projector when the following conditions are met:(55)ν°T=T,T°ν=0.Definition 5.
(see [2]). N, which is complementary distribution to VTM, i.e.,(56)XM=VTM⊕N,is called a normalization or a nonlinear connection or a horizontal distribution.Knowing that a vertical projectorν is C∞M linear with Imν=VTM, we can state the following proposition.Proposition 7 (see [2]).
We obtain a nonlinear connectionNν from the vertical projector ν by using kerν=Nν. Otherwise, with the respect to the separation (56), νN and hN are vertical and horizontal projectors, respectively, if N is a nonlinear connection.Thus, the next proposition can be given as follow.Proposition 8.
(see [2]).νN is a vertical projector provided that NνN=N.Definition 6.
(see [2]). A 1,1 type tensor field Γ is called a nonlinear connection of an almost product type if the following relations are provided:(57)T°Γ=T,Γ°T=−T.Proposition 9.
(see [2]). The following assertions hold true if Γ is a nonlinear connection of an almost product type:(i)
νΓ=1/2Id−Γ is a vertical projector,(ii)
VTM is the −1-eigenspace of Γ when NνΓ is the +1 -eigenspace of Γ.Corollary 1.
(see [2]). Any vertical projector v induces an almost product structure on manifold M as follows: Γ=Id−2v.Thus,thisresult has been associated with the almost bronzestructure.Proposition 10.
Obtained by the vertical projectorv, a nonlinear connection N on M can also be defined by an almost bronze structure φ=φΓ(58)φ=ρmId−m2−4νwith VTM the m−ρm-eigenspace and N the ρm-eigenspace.


## 6. Integrability and Parallelism of Almost Bronze Structures

This section examines the almost bronze structure’s integrability and parallelism.Proposition 11.
LetM,φ be an almost bronze manifold. There are Dk and Dt complementary distributions on M corresponding to the following projection operators:(59)k=12ρm−mφ−m−ρm2ρm−mId,t=−12ρm−mφ+ρm2ρm−mId.Remark 1.
The operatorsk and t obtained in Proposition 11 verify the following equations:(60)k+t=Id,kt=kt=0,k2=k,t2=t,(61)φk=kφ=ρmk,φt=tφ=m−ρmt.As a result, k and t operators define Dk and Dt complementary distributions corresponding to these projections.From (61), we get(62)tkX,kY=12ρm−m2tNφkX,kY,ktX,tY=12ρm−m2kNφtX,tY.As also stated by [23, 24], we have the following:(i)
A polynomial structureφ is integrable if and only if Nφ=0 or its equivalent ∇φ=0 where ∇ is a torsion-free linear connection.(ii)
For any vector fieldsX,Y in XM, the distribution Dk (resp. Dt) is integrable if and only if tkX,kY=0 (resp., ktX,tY=0).The following proposition can be stated with the help of Proposition4.Proposition 12.
The almost bronze structureφ is integrable if and only if the almost product structure Pφ induced by φ is integrable.Using the above-mentioned literature and (62), we can give the following proposition.Proposition 13.
The following claims are true:(i)
Dk is an integrable distribution if and only if tNφkX,kY=0.(ii)
Dt is an integrable distribution if and only if kNφtX,tY=0.(iii)
The almost bronze structureφ is integrable if and only if both of the distributions Dk and Dt are integrable.Let us consider a fixed linear connection∇ on manifold M. We can define the following two linear connections associated with the pair φ,∇(63)∇XScY=k∇XkY+t∇XtY,(64)∇XVrY=k∇kXkY+t∇tXtY+ktX,kY+tkX,tY,for any X, Y vector fields of the manifold M. ∇Sc and ∇Vr are known as the Schouten connection and the Vrǎnceanu connection, respectively [25, 26].Recall that a1,1 tensor field F is parallel in accordance with the linear connection ∇ if its covariant derivative ∇F vanishes.Theorem 2.
The following claims are true:(i)
Both of the projectorsk and t are parallel in accordance with the connections ∇Sc and ∇Vr.(ii)
The almost bronze structureφ is parallel regarding the connections ∇Sc and ∇Vr.Proof.
(i)
With the help of (60), we can express the following equations for each vector field X, Y∈XM:(65)∇XSckY=∇XSckY−k∇XScY=k∇XkY−k∇XkY=0,∇XVrkY=∇XVrkY−k∇XVrY=k∇kXkY+ktX,kY−k∇kXkY−ktX,kY=0.Therefore, the projectork is parallel in accordance with the connections ∇Sc and ∇Vr.Likewise, it can be shown that the projectort is parallel according to the connections ∇Sc and ∇Vr.(ii)
By direct computation, we get from (59) that φ is parallel in accordance with the connections ∇Sc and ∇Vr.As is known, a distributionD on manifold M is said parallel in accordance with the linear connection ∇ provided that ∇XY belongs to D for each vector field X∈XM and Y∈D.Definition 7.
(see [27]). For any vector fields X∈Dk (resp., Dt) and Y∈XM, if the vector field ΔφX,Y belongs to Dk (resp., Dt) where(66)ΔφX,Y=φ∇XY−φ∇YX−∇φXY+∇YφX
then the distributionDk (resp., Dt) is named half-parallel.Definition 8.
(see [27]). For any vector fields X∈Dk (resp., Dt) and Y∈XM, if the vector field ΔφX,Y belongs to Dt (resp., Dk) then Dk (resp., Dt) is named anti-half-parallel.Theorem 3.
According to the connections∇Sc and ∇Vr, both distributions Dk and Dt are parallel.Proof.
GivenX∈XM and Y∈Dk one has tY=0 and kY=Y; then, taking into account (60), (63), and (64) we obtain(67)t∇XScY=0,t∇XVrY=0.
Thus, the distributionDk is parallel in accordance with Schouten connection and Vrănceanu connection.
Likewise, it is seen that similar relations are satisfied byDt.Proposition 14.
The connection∇Sc is equal to the connection ∇ if and only if the distributions of almost bronze structure φ (i.e., Dk and Dt) are parallel in terms of the connection ∇.Proof.
If the connections∇Sc and ∇ are equal, then it follows from (63) that(68)k∇XtY+t∇XkY=0and from (60)(69)k∇XtY=0,t∇XkY=0.
Therefore,Dk and Dt are parallel regarding the connection ∇.
The other direction of the proof can be shown easily.Proposition 15.
For anyX vector field in Dk and any Y vector field on M, if the vector field kX,tY belongs to the distribution Dk then Dk is half-parallel according to the connection ∇Vr.Proof.
GivenX∈Dk and Y∈XM, we get the following equation by using (66) for ∇Vr(70)tΔφX,Y=tφ∇XVrY−tφ∇YVrX−t∇φXVrY+t∇YVrφX.
Finally, we get the following equation by using (61) and (64),(71)tΔφX,Y=m−2ρmtkX,tY,which proves the proposition.Likewise, the following proposition can be presented for the distributionDt.Proposition 16.
For anyX vector field in Dt and any Y vector field on M, if the vector field tX,kY belongs to the distribution Dt then Dt is half-parallel in accordance with the Vrănceanu connection ∇Vr.Proposition 17.
According to the connection∇Vr, both distributions Dk and Dt are anti-half-parallel.Proof.
GivenX∈Dk and Y∈XM, taking into consideration equation (66) for ∇Vr, we can obtain(72)kΔφX,Y=kφ∇XVrY−kφ∇YVrX−k∇φXVrY+k∇YVrφX.
By using the equations of (61) and (64), we have(73)kΔφX,Y=2ρm−mktX,kY.Because of tX=0, one can obtain kΔφX,Y=0. Therefore, ΔφX,Y∈Dt.Similarly, it can be obtained that Dt is anti-half-parallel regarding ∇Vr.


## 7. Almost Bronze Riemannian Metrics

Consider the fact that an almost product Riemannian structure is aP,g pair where P is an almost product structure on manifold M and g is a Riemannian metric on M, which is related to(74)gPX,Y=gX,PYor its equivalent(75)gPX,PY=gX,Y,for all X,Y vector fields on M. Thus, the Riemannian metric g is called pure in accordance with the almost product structure P .Definition 9.
An almost bronze Riemannian structure is a pairφ,g, which is satisfies(76)gφX,Y=gX,φYor its equivalent(77)gφX,φY=mgX,φY−gX,Y,for all X,Y vector fields of the manifold M. Also, the triple M,φ,g is called an almost bronze Riemannian manifold.From Theorem 1 and Definition 9, we get the followingproposition.Proposition 18.
LetP be an almost product Riemannian structure and let φ be an almost bronze Riemannian structure on M. The Riemannian metric g is pure in accordance with the operator φ if and only if g is pure in accordance with the operator Pφ. Also, the Riemannian metric g is pure in accordance with the operator P if and only if g is pure in accordance with the operator φP.Corollary 2.
On an almost bronze Riemannian manifold:(i)
With respect to the projectorsk and t, the Riemannian metric g is pure, which means(78)gX,kY=gkX,Y,gX,tY=gtX,Y.(ii)
Dk, Dt are g-orthogonal distributions, which means(79)gkX,tY=0.(iii)
The almost bronze structure isNφ-symmetric, which means(80)NφφX,Y=NφX,φY.If an almost product structureP is parallel in accordance with the Levi-Civita connection ∇g of g, an almost product Riemannian structure is a locally product structure, which means ∇gP=0. Also if the torsion tensor of linear connection ∇ vanishes then the Nijenhuis tensor of P satisfies the following equation:(81)NPX,Y=∇PXPY−P∇XPY−∇PYPX+P∇YPX.Thus, we have the following proposition.Proposition 19.
The almost bronze structureφ is integrable if M,φ,g is a locally product bronze Riemannian manifold.Considering this finding, we can give the linear connections by making them parallel with the almost bronze structure given as follows.Theorem 4.
For∇φ=0, the set of linear connections ∇ is defined as follows:(82)∇XY=1m2−4m2−2∇˘XY+2φ∇˘XφY−mφ∇˘XY−m∇˘XφY+OPφQX,Y.
In this equation,∇˘ stands for a linear connection while Q stands for 1,2 type tensor field where OPφQ is an associated Obata operator(83)OPφQX,Y=12QX,Y+PφQX,PφY,
for each fieldX, Y on the manifold M.We complete the study of the almost bronze structure with the following example.Example 7.
For anyC∞ differentiable functions f and g depending on x,y,(84)k=f2f2+g2∂∂x⊗dx+fgf2+g2∂∂x⊗dy+fgf2+g2∂∂y⊗dx+g2f2+g2∂∂y⊗dy,t=g2f2+g2∂∂x⊗dx−fgf2+g2∂∂x⊗dy−fgf2+g2∂∂y⊗dx+f2f2+g2∂∂y⊗dywhere f2+g2≠0, are projection operators in ℝ2 and they satisfy the conditions in (60).(85)Dk=Spf∂∂x+g∂∂yandDt=Spg∂∂x−f∂∂y
are complementary distributions that correspond to thek and t projection operators, respectively. In terms of the Euclidean metric of ℝ2, the distributions Dk and Dt are orthogonal. Furthermore, these distributions are connected to the almost bronze structure(86)φ∂∂x=ρmf2+ρ¯mg2f2+g2∂∂x+m−2ρ¯mfgf2+g2∂∂y,φ∂∂y=m−2ρ¯mfgf2+g2∂∂x+ρ¯mf2+ρmg2f2+g2∂∂y,which is integrable since Nφ∂/∂x,∂/∂y=0.



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