Abstract
We explore “the horizontal lift” of the structure satisfying and establish that it as a kind of metallic structure. An analysis of Nijenhuis tensor of metallic structure is presented, and a new tensor field of -type is introduced and demonstrated to be metallic structure. Some results on the Nijenhuis tensor and the Lie derivative of in are proved and explicit examples are given. Moreover, the metallic structure endowed with projection operators and in is studied.
1. Introduction
Tangent bundle geometry’s framework typically includes geometric structures such as vertical, complete, and horizontal lifts transforming structures on the base manifold to . These lifts of an almost product structure along with over an almost -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 considerIn this scenario, stands for -tensor field, are the real numbers, and means identity map [10]. Such a structure satisfyingis called a metallic structure [11]. The set of positive solutions of , denoted by , is named Metallic Means Family (MMF) [12–14]. The different values of and refer to different members of MMF. For instance, , Gold mean, applies to architecture. For , Silver mean, its role can be found in various applications such as civil design, architecture, and physics. For , Bronze mean, it has uses in quasicrystals.
For positive integers and , Hretcanu and Crasmareanu [15] introduced and studied the metallic structures satisfying , 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 structure satisfying and proved to be metallic structure(ii)The Nijenhuis tensor (N.T.) of is established(iii)It is proved that the field is metallic on by introducing a new -tensor field(iv)N.T. of in is calculated(v)The Lie derivative of in is determined(vi)The existence of metallic structures in is shown by nontrivial examples(vii)The projection operators in are discussed as an application
2. Preliminaries
We interpret as a differentiable manifold and as its tangent bundle. Assume that and stand for set of all -type tensor fields in and , respectively.
2.1. Vertical Lifts
One writes “the vertical lift” aswhere means the standard projection and stands for differentiable function on .
Let and “the vertical lift” of defined in the components form as
Now, we state the following proposition involving ‘the vertical lifts’ for later use [29]:
Proposition 1. For all . Then,where represents 1-form in .
2.2. Complete Lifts
in represented asWhich gives complete lift of . Locally,In this case, means . For , one has
With respect to components in , one can writeassociated to induced coordinates in .
2.3. Horizontal Lifts
We assume that is an affine connection and represents an arbitrary coordinate neighborhood in , and there are tensor fields and with affine connection given bycorresponding to in , where is an operator on tensor field [29].
For differentiable function and 1-form on , we define:(1) of by where .(2) of with and is a gradient of .(3) of is(4) of with
Here, means a vector field and a -tensor field on . We recall the following propositions involving “the horizontal lifts” for later use [29]:
Proposition 2. For all ,
Proposition 3. in , we observe
In this case, represents the horizontal lift and stands for vertical lift of in . Furthermore, for affine connection , indicates the curvature tensor of that satisfies
Proposition 4. For the torsion tensors of and in and denoted by and , respectively, we have
2.4. Projection Tensor
Any tensor field in manifold satisfying is known as projection tensor [29, 30].
Definition 1. [31, 32] If the metallic structure defined in (2) is covariant constant w.r.t. , i.e., , then is said be connection.
Definition 2. [23] We write the N.T. of aswhere becomes integrable if .
3. Horizontal Lifts of Metallic Structures in
In this section, a study is done on ‘the horizontal lifts’ of a metallic structure in .
Theorem 1. We assume that represents M.S. in . Then, also provides a M.S. in , i.e.,
Proof. The proof is trivial with the help of (2).
Theorem 2. Let be any manifold with admitting metallic structure , and be any connection. The Nijenhuis tensor of the horizontal lift on is represented as being N.T. of .
Proof. LetSetting in (22) and Proposition 2 and Proposition 3, we getSetting in (22) and Proposition 2 and Proposition 3, we getSetting in (22), using similar devices, the following equation is obtained:This completes the proof.
4. Proposed Theorems on for the M.S. on
Here, is introduced and shown to be metallic structure in . Moreover, the Nijenhuis tensor and the Lie derivative of in are calculated.
4.1. Metallic Structures in
In [29], Yano and Ishihara defined in with
One can easily establish that produces an a.c.s. on .
Also, Gezer, and Kamran [22] defined in by means ofwhere is any vector field; represents a -tensor field, , and stands for a Riemannian metric on .
Motivated from the above studies, we define a new tensor in in the following way:
Theorem 3. We consider M.S. in associated with , then written byrepresents a M.S. in .
Proof. By the virtue of (28), we observeOn solving, the obtained equation is .
Similarly, , which impliesIt is therefore concluded that is a metallic structure in .
4.2. Nijenhuis Tensor of M.S. in
We note that
Setting and in (32) and using (28), the obtained equation is
Using (28) and Proposition 3 in (33) results
Since .
The obtained equation is
Using similar techniques, the following equations can be easily obtained:
This procedure can be summarized as follows:
Theorem 4. The metallic structure defined in (28) is integrable, i.e., , iff and . In this case, stands for the torsion tensor of , and is used for curvature tensor of an affine connection .
4.3. The Lie Derivative of the Metallic Structure in
We write the Lie derivative of as [29]
Operating on both sides of (28) and taking account of Proposition 1 and Proposition 3, it reduces tosince ,
Using similar devices, the following equations are obtained:
Equations (39)–(42) produce the following:
Theorem 5. Let be V.F. in , and be its “vertical” and “horizontal lifts” in , respectively. Also, we suppose that represents a metallic structure defined with the help of (28). Then, iff , and iff and . Here, , and stands for the curvature tensor of .
4.4. Explicit Examples
Example 1. Considering in (2) implies . It is referred to as the golden structure and can be derived from a metallic structure [21, 33].
We define in by , then (28) becomefor any .
Example 2. Let us consider , and .
On solving, we obtain .
Similarly, for any , which impliesthereby establishing the fact that is metallic structure in .
5. Application
Here, the metallic structure endowed with projection operators and in is investigated.
Let and be the projection operators in and defined by
Theorem 6. If represents the metallic structure and and stand for the projection operators in , then
Proof. One can easily write the proof with the help of (45) and (46).
Theorem 7. Let be the horizontal and be the vertical lift of v.f. in . Then,where and are the projection operators in .
Proof. Equations (28) and (45) produce
Data Availability
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Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.