Almost Bronze Structures on Differentiable Manifolds

Özkan, Mustafa; Doğan, Seher

doi:https://doi.org/10.1155/2022/6940387

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 6940387 | https://doi.org/10.1155/2022/6940387

Almost Bronze Structures on Differentiable Manifolds

Mustafa Özkan¹and Seher Doğan²

Academic Editor: Francisco J. Garcia Pacheco

Received13 May 2022

Accepted26 Jul 2022

Published24 Sept 2022

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 a type tensor field which fulfills the requirement 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.

1. Introduction

Several polynomial structures on a differentiable manifold are defined using type –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, -Fibonacci, -Fibonacci, bivariate Fibonacci, -Lucas, and -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 of for 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: Section 2 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 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 numbers are a family of sequencing numbers defined by the recurrence presented below:

On the other hand, the bronze Lucas numbers refer to a family of sequencing numbers defined by the following recurrence:

Different from the bronze means presented in [7, 10–12], a new bronze mean is defined as follows:This is obtained as the positive root of the following equation: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:

The bronze means’ continued fractions are defined as , while the recurrence relationship is defined as follows:The following relationship is another one:

3. Almost Bronze Structures on Manifolds

In this paper, refers to a –class differentiable manifold, and all tensor fields and connections on this manifold are considered to be of class . We denote by the Lie algebra of the vector fields on .

Definition 1. (see [16]). Let be a differentiable manifold and be a type tensor field on . If satisfies the following equation it is defined as a polynomial structure:In this equation, the identity operator on is denoted by , while , , , , are linear independent for each point in . In this case, the polynomial is said to be a structure polynomial.

As stated in [17], , which is an almost product (resp., almost complex, almost tangent) structure, satisfies the condition of (resp., , ). Then 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 .

Definition 2. Let be a differentiable manifold and be a tensor field that satisfies the equation below:where . Here, is said to be a new almost bronze structure on manifold . 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 manifold is defined as follows for any integer :where stands for the bronze Fibonacci numbers, and stands for the bronze Lucas numbers.

Let be a bronze ratio. Binet’s formulas of the bronze Fibonacci sequence and the bronze Lucas sequence are defined as follows:

respectively [14].

From (10) and (11), we have a new form for equality (10)

Unless otherwise stated, we will take throughout the study.

A simple calculation results in the following.

Proposition 2. The properties of an almost bronze structure are as follows:(i)The bronze ratio and are the eigenvalues of .(ii)On the tangent space of the manifold , is an isomorphism for each .(iii) is an invertible structure, and its inverse is an almost bronze structure on , and it can be calculated as follows: .

A polynomial structure on a manifold induces a generalized almost product structure , as described in [16]. Thus, the almost product structure and almost bronze structure on are connected structures.

Theorem 1. (i)If is an almost bronze structure on , then is an almost product structure on , and it is defined as follows: We call that is an almost product structure induced by .(ii)If is an almost product structure on , then is an almost bronze structure on , and it is defined as follows: Thus, is named as an almost bronze structure induced by .Since and , almost product structures and almost bronze structures on have a one-to-one correspondence.

Proof. (i)Assume that is an almost bronze structure on the manifold . In this case, the structure obtained from the almost bronze structure is an almost product structure since the following condition is satisfied:(ii)Assume that is an almost product structure on the manifold . The structure , which is induced by the almost product structure , is an almost bronze structure since the following condition is satisfied: We get and by straightforward calculations from (13) and (14).Using the above-mentioned literature and Theorem 1, we can give the following definitions:(i)Assume that is an almost tangent manifold. Then, the tensor field , which is induced by , is defined as follows: and it is called an almost tangent bronze structure on manifold . Then, the equation verified by an almost tangent bronze structure The tangent real bronze ratio is then calculated using the associated equation in the real field , that is, .(ii)Let be an almost complex structure on manifold . The tensor field which is induced by is defined using the equation is called an almost complex bronze structure on manifold . satisfies the following polynomial equation:For , we obtain the following equation:with solutions

Definition 3. The complex numberwill be called complex bronze ratio.

If we take in (19), then we have an almost poly-Norden structure defined in [15].

4. Examples of the Almost Bronze Structures

Several examples of the almost bronze structure will be presented in this section.

Example 1. (Clifford algebras). Let be the real Clifford algebra of the Euclidean space [18]. The standard base of satisfies the multiplication rules according to the Clifford productTherefore, by usingwhere and (25), we can obtain a new representation of the Clifford algebra as follows:In [18], is constructed asand then we get

Example 2. (Quaternion algebras). There is a quaternion algebra with a base satisfyingAny quaternion can be written as follows:where denote the scalar part of and denote the vectorial part of .
For , is called unit quaternion where is the norm of quaternion . We can express each unit quaternion in the following form , where stands for a unit vector that satisfies .
Thus, inspired by [19], we have the following.(a)An almost bronze hyperbolic quaternion structure can be defined as follows: where is the inner product and is a unit hyperbolic vector.(b)An almost bronze biquaternion structure can be defined as follows: where is the inner product and .(c)An almost bronze split quaternion structure can be defined as follows: where is Lorentzian inner product and is a spacelike unit vector in the Minkowski 3-space .(d)An almost bronze dual split quaternion structure can be defined as follows: whereis Lorentzian inner product and is a spacelike unit dual vector in .(e)An almost bronze hyperbolic split quaternion structure can be defined as follows: where is Lorentzian inner product and is a spacelike unit hyperbolic vector in .

Example 3. (Bronze matrices). Let be a matrix algebra of real –matrices and . If satisfies the following equation:where is the identity matrix on , then this matrix is called an almost bronze matrix.
By solving (37) for , we can obtain the almost bronze structure in .(i)For , , and (ii)For and ,(iii)For and ,Then, from (29) and (38), we obtainAlso, from (38), we get the sequence of trace is the bronze Lucas sequence: .

Example 4. (Bronze reflections). As also stated in [20], the equation of the reflection in accordance with a hyperplane with the normal in Euclidean space is as follows:In this equation, it is obvious that where is the identity on .
Thus, the bronze reflection with respect to can be defined as follows:and then is an eigenvector of with the corresponding eigenvalue . Then, the following equation is obtained from [20, p.314]where is an orthogonal transformation on . Thus, the following equation can be written as an explicit expression of the linear transformation

Example 5. (Triple structures with respect to almost bronze structures). Given two tensor fields and on the manifold and , we called that the triple is as follows [21]:(1)an almost hyperproduct structure: if , are almost product structures and , then is an almost product structure,(2)an almost biproduct complex structure: if , are almost product structures and , then is an almost complex structure,(3)an almost product bicomplex structure: if , are almost complex structures and , then is an almost product structure,(4)an almost hypercomplex structure: if , are almost complex structures and , then is an almost complex structure.Taking into account (14), we getThen, we find a relation between , , and asHence, the triple is as follows:(i)an almost hyperproduct structure: if and only if , are almost bronze structures and , then is an almost bronze structure,(ii)an almost biproduct complex structure: if and only if , are almost bronze structures and , then is an almost complex bronze structure,(iii)an almost product bicomplex structure: if and only if , are almost complex bronze structures and , then is an almost bronze structure,(iv)an almost hypercomplex structure: if and only if , are almost complex bronze structures and , then is an almost complex bronze structure.

Example 6. (Almost bronze structures from symplectic distributions). Given any symplectic vector space , we have where if is a subspace of . A subspace of is symplectic if and only if is nondegenerate (or ) [22]. Consequently, if is a symplectic distribution on a symplectic manifold (i.e., is a symplectic subspace of the tangent space at ), then another symplectic distribution is obtained complementary to . In this case, is an almost product structure where and are the corresponding projection tensors. Then, an associated symplectic almost bronze structure is obtained as follows by using (14)

5. Connection as Almost Bronze Structure

5.1. Connections in the Principal Fibre Bundles

Assume that is a principal fibre bundle on a manifold , where is the total space, is the base space, is the structure group, and is the projection. Let denote a vertical distribution (i.e., ), denote a horizontal distribution (i.e., ), and be a -invariant. Thus, and become the corresponding projectors of and , respectively. Therefore, type tensor field can be defined as follows:and it is an almost product structure on . According to [2], defines a connection if and only if the following conditions are satisfied:(a) is a vertical vector field,(b) for each and .

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 structure on specifies a connection if and only if the following conditions are satisfied:(a)For , if and only if .(b) for each and .

Assumingthat is a connection 1-form of horizontal distribution and is the curvature form of where stands for the Lie algebra of , we can obtain the following relation [2]:where stands for the Nijenhuis tensor of , i.e.,

for all , vector fields on .

Thus, the following proposition can be stated by straightforward calculations from (13) and (51).

Proposition 4. Let be an almost bronze structure on the manifold and be an almost product structure induced by . Thenwhere and stand for the Nijenhuis tensors of and , respectively.

Therefore, it can be stated that the integrability of the structures and is equivalent.

Proposition 5. The connection is flat (i.e., ) if and only if the associated almost bronze structure is integrable, which means .

Given two vector fields , on the manifold and a connection, the lift is determined by this connection if the following condition is met [2]:

Thus, considering (52) and (54), we have the following proposition.

Proposition 6. The lift , which is defined by , is a morphism if and only if the associated almost bronze structure is integrable.

5.2. Connection in the Tangent Bundles

Let be the tangent bundle of the manifold , be the projection, be its differential, and be the vertical distribution of . For any coordinate neighborhood in , stands for the induced coordinate neighborhood in , i.e., and for all . For an atlas on with these local coordinates, the almost tangent structure of is , i.e., .

Definition 4. (see [2]). Given an almost tangent structure of and a tensor field on , is called a vertical projector when the following conditions are met:

Definition 5. (see [2]). , which is complementary distribution to , i.e.,is called a normalization or a nonlinear connection or a horizontal distribution.

Knowing that a vertical projector is linear with , we can state the following proposition.

Proposition 7 (see [2]). We obtain a nonlinear connection from the vertical projector by using . Otherwise, with the respect to the separation (56), and are vertical and horizontal projectors, respectively, if is a nonlinear connection.

Thus, the next proposition can be given as follow.

Proposition 8. (see [2]). is a vertical projector provided that .

Definition 6. (see [2]). A type tensor field is called a nonlinear connection of an almost product type if the following relations are provided:

Proposition 9. (see [2]). The following assertions hold true if is a nonlinear connection of an almost product type:(i) is a vertical projector,(ii) is the -eigenspace of when is the -eigenspace of .

Corollary 1. (see [2]). Any vertical projector induces an almost product structure on manifold as follows: .

Thus,thisresult has been associated with the almost bronzestructure.

Proposition 10. Obtained by the vertical projector , a nonlinear connection on can also be defined by an almost bronze structure with the -eigenspace and the -eigenspace.

6. Integrability and Parallelism of Almost Bronze Structures

This section examines the almost bronze structure’s integrability and parallelism.

Proposition 11. Let be an almost bronze manifold. There are and complementary distributions on corresponding to the following projection operators:

Remark 1. The operators and obtained in Proposition 11 verify the following equations:As a result, and operators define and complementary distributions corresponding to these projections.

From (61), we get

As also stated by [23, 24], we have the following:(i)A polynomial structure is integrable if and only if or its equivalent where is a torsion-free linear connection.(ii)For any vector fields in , the distribution (resp. ) is integrable if and only if (resp., ).

The following proposition can be stated with the help of Proposition 4.

Proposition 12. The almost bronze structure is integrable if and only if the almost product structure induced by is integrable.

Using the above-mentioned literature and (62), we can give the following proposition.

Proposition 13. The following claims are true:(i) is an integrable distribution if and only if .(ii) is an integrable distribution if and only if .(iii)The almost bronze structure is integrable if and only if both of the distributions and are integrable.

Let us consider a fixed linear connection on manifold . We can define the following two linear connections associated with the pair for any , vector fields of the manifold . and are known as the Schouten connection and the Vrǎnceanu connection, respectively [25, 26].

Recall that a tensor field is parallel in accordance with the linear connection if its covariant derivative vanishes.

Theorem 2. The following claims are true:(i)Both of the projectors and are parallel in accordance with the connections and .(ii)The almost bronze structure is parallel regarding the connections and .

Proof. (i)With the help of (60), we can express the following equations for each vector field , : Therefore, the projector is parallel in accordance with the connections and . Likewise, it can be shown that the projector is parallel according to the connections and .(ii)By direct computation, we get from (59) that is parallel in accordance with the connections and .

As is known, a distribution on manifold is said parallel in accordance with the linear connection provided that belongs to for each vector field and .

Definition 7. (see [27]). For any vector fields (resp., ) and , if the vector field belongs to (resp., ) wherethen the distribution (resp., ) is named half-parallel.

Definition 8. (see [27]). For any vector fields (resp., ) and , if the vector field belongs to (resp., ) then (resp., ) is named anti-half-parallel.

Theorem 3. According to the connections and , both distributions and are parallel.

Proof. Given and one has and ; then, taking into account (60), (63), and (64) we obtainThus, the distribution is parallel in accordance with Schouten connection and Vrănceanu connection.
Likewise, it is seen that similar relations are satisfied by .

Proposition 14. The connection is equal to the connection if and only if the distributions of almost bronze structure (i.e., and ) are parallel in terms of the connection .

Proof. If the connections and are equal, then it follows from (63) thatand from (60)Therefore, and are parallel regarding the connection .
The other direction of the proof can be shown easily.

Proposition 15. For any vector field in and any vector field on , if the vector field belongs to the distribution then is half-parallel according to the connection .

Proof. Given and , we get the following equation by using (66) for Finally, we get the following equation by using (61) and (64),which proves the proposition.

Likewise, the following proposition can be presented for the distribution .

Proposition 16. For any vector field in and any vector field on , if the vector field belongs to the distribution then is half-parallel in accordance with the Vrănceanu connection .

Proposition 17. According to the connection , both distributions and are anti-half-parallel.

Proof. Given and , taking into consideration equation (66) for , we can obtainBy using the equations of (61) and (64), we haveBecause of , one can obtain . Therefore, .Similarly, it can be obtained that is anti-half-parallel regarding .

7. Almost Bronze Riemannian Metrics

Consider the fact that an almost product Riemannian structure is a pair where is an almost product structure on manifold and is a Riemannian metric on , which is related toor its equivalentfor all vector fields on . Thus, the Riemannian metric is called pure in accordance with the almost product structure .

Definition 9. An almost bronze Riemannian structure is a pair , which is satisfiesor its equivalentfor all vector fields of the manifold . Also, the triple is called an almost bronze Riemannian manifold.

From Theorem 1 and Definition 9, we get the followingproposition.

Proposition 18. Let be an almost product Riemannian structure and let be an almost bronze Riemannian structure on . The Riemannian metric is pure in accordance with the operator if and only if is pure in accordance with the operator . Also, the Riemannian metric is pure in accordance with the operator if and only if is pure in accordance with the operator .

Corollary 2. On an almost bronze Riemannian manifold:(i)With respect to the projectors and , the Riemannian metric is pure, which means(ii), are -orthogonal distributions, which means(iii)The almost bronze structure is -symmetric, which means

If an almost product structure is parallel in accordance with the Levi-Civita connection of , an almost product Riemannian structure is a locally product structure, which means . Also if the torsion tensor of linear connection vanishes then the Nijenhuis tensor of satisfies the following equation:

Thus, we have the following proposition.

Proposition 19. The almost bronze structure is integrable if 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 , the set of linear connections is defined as follows:In this equation, stands for a linear connection while stands for type tensor field where is an associated Obata operatorfor each field , on the manifold .

We complete the study of the almost bronze structure with the following example.

Example 7. For any differentiable functions and depending on ,where , are projection operators in and they satisfy the conditions in (60).are complementary distributions that correspond to the and projection operators, respectively. In terms of the Euclidean metric of , the distributions and are orthogonal. Furthermore, these distributions are connected to the almost bronze structurewhich is integrable since .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

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Copyright © 2022 Mustafa Özkan and Seher Doğan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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