The aim of the present paper is to explore the lifts of an f(\alpha,\beta)(3; 2; 1)-structure and obtain its partial integrability and integrability conditions on the tangent bundle. Also, the prolongation of an f(\alpha,\beta)(3; 2; 1)-structure on the third tangent bundle T3M is studied

Khan, Mohammad Nazrul Islam

English

University of Niš: Facta Universitatis (E-Journals) / Универзитет у Нишу

FACTA UNIVERSITATIS (NIŠ)Ser. Math. Inform. Vol. 38, No 1 (2023), 209 - 218https://doi.org/10.22190/FUMI230121014KOriginal Scientific PaperLIFTS OF F (α, β)(3, 2, 1)-STRUCTURES FROM MANIFOLDS TOTANGENT BUNDLESMohammad Nazrul Islam KhanDepartment of Computer Engineering, College of Computer,Qassim University, Buraydah, Saudia ArabiaAbstract. The aim of the present paper is to explore the lifts of an f(α, β)(3, 2, 1)-structure and obtain its partial integrability and integrability conditions on the tangentbundle. Also, the prolongation of an f(α, β)(3, 2, 1)-structure on the third tangentbundle T3M is studied.Keywords: Lifts, Nijenhuis tensor, Partial differential equations, Projection tensors,Integrability.1. IntroductionThe notion of the polynomial structure of degree nQ(F ) = Fn + anFn−1 + .....+ a2F + a1I,where F is the tensor field of type (1,1) and I is the identity tensor field on adifferentiable manifold was introduced by Goldenberg et. al. [7, 9]. Recently, Göket. al. [8] have defined an f(α, β)(3, 2, 1)-structure on a differentiable manifold,where α, β ∈ R and β 6= 0 and established its some fundamental properties. Theyinvestigated its partial integrability and integrability conditions.On the other hand, let us consider the tangent bundle TM of a manifold M .Tangent bundle is a primary field of differential geometry used to investigate geomet-rical structures and their properties such as integrability, curvature, Lie derivative,Received January 21, 2023. accepted May 01, 2023.Communicated by Uday Chand DeCorresponding Author: Mohammad Nazrul Islam Khan, Department of Computer Engineering,College of Computer, Qassim University, Buraydah, Saudia Arabia | E-mail: m.nazrul@edu.qu.sa,mnazrul@rediffmail.com2010 Mathematics Subject Classification. Primary 53B25, 53C03; Secondary 58A30© 2023 by University of Nǐs, Serbia | Creative Commons License: CC BY-NC-ND209210 M. N. I. Khanetc. Yano and Ishihara [24] introduced and studied structures like almost complexstructures with some basic properties induced in tangent bundles. Das and Khan [3]have researched these lifts of an almost product structure over an almost r-contactstructure along with TM . The work of several scholars on various geometric struc-tures and connections have been extremely beneficial, for instance, Dida et. al.[4, 5, 6], Khan and De [11, 19], Khan [14, 15, 16], Omran at el [20], Peyghan et. al.[21], Tekkoyun [22] and Yano and Ishihara [24].The main purpose of this paper is to study the lifts of an f(α, β)(3, 2, 1)-structureon manifolds on the tangent bundle and establish its partial integrability and inte-grability conditions. Finally, the prolongation of an f(α, β)(3, 2, 1)-structure on thethird tangent bundle T3M is studied.Let M be an n-dimensional differentiable manifold. A non-null tensor field F oftype (1, 1) on M is called an f(α, β)(3, 2, 1)-structure if it satisfies the equationF 3 = αF 2 + βF,(1.1)where α, β ∈ R and β 6= 0. Hence, the f(α, β)(3, 2, 1)-structure F is a polynomialstructure of degree 3 with the structure polynomial x3 − αx2 − βx = 0. Also,the pair (M,F ) is said to be an f(α, β)(3, 2, 1)-manifold. F is of constant rank neverywhere in MLet l and m be operators defined as(a) l =F 2 − αFβ,(b) m =−F 2 + αF + βIβ.(1.2)The operators l and m defined in the equation (1.2) satisfy the following identities:l +m = 0,l2 = l, m2 = m, lm = ml = 0,(1.3)Fl = lF = F, Fm = mF = 0.Thus there exist two complementary distributions Dl and Dm corresponding to theprojection tensors l and m respectively in M .2. The complete lift of an f(α, β)(3, 2, 1)-structure in the tangent bundleLet M be an m-dimensional differentiable manifold of class C∞ and Tp(M) thetangent space at a point p of M then T (M) = ∪p∈MTpM is a tangent bundleover the manifold M . The tangent bundle TM of M is a differentiable manifold ofdimension 2n. Let ℘rs denote the set of tensor field of class C∞ and type (r, s) in Mand ℘rs(T (M)) denote the corresponding set of tensor fields in T (M) [12, 13, 10].Let F,G be elements of ℘11(M). Then we have [23](FG)C = FCGC .(2.1)Lifts of f(α, β)(3, 2, 1)-structures from manifolds to tangent bundles 211Putting F = G in the equation (2.3), we obtain(F 2)C = (FC)2.(2.2)Also,(F +G)C = FC +GC .(2.3)Operating the complete lifts of both sides of the equation (1.1), we get(F 3)C = (αF 2 + βF )C ,(F 3)C = (αF 2)C + (βF )C .In the view of (2.2) and IC = I, we get(FC)3 = α(FC)2 + βFC .(2.4)In the view of equations (1.1), (2.4) and [23], we can easily say that the rank of FCis 2n if and only if the rank of F is n. Therefore, we have the following theorems:Theorem 2.1. Let F ∈ ℘11(M) be a f(α, β)(3, 2, 1)-structure in M , then its com-plete lift FC is also an f(α, β)(3, 2, 1)-structure in TM .Theorem 2.2. The f(α, β)(3, 2, 1)-structure F of rank n in M if and only if itscomplete lift FC is of rank 2n in TM .Let F be a f(α, β)(3, 2, 1)-structure of rank n in M . Then the complete lift lC ofl and mC of m are complementary projection tensors in TM . Thus there exist twocomplementary distributions DlC and DmC determined by lC and mC respectivelyin TM . The distributions DlC and DmC are respectively the complete lifts of DCland DCm of Dl and Dm [3].3. Integrability conditions of an f(α, β)(3, 2, 1)-structure in the tangentbundleLet F be the f(α, β)(3, 2, 1)-structure that is F 3 = αF 2 + βF . Then the Nijenhuistensor N of F is a tensor of type (1,2) given by [17, 18]N(X,Y ) = [FX,FY ]− F [FX, Y ]− F [X,FY ] + F 2[X,Y ].(3.1)Let NC be the Nijenhuis tensor of FC in TM , then we haveNC(XC , Y C) = [FCXC , FCY C ]− FC [FCXC , Y C ]− FC [XC , FCY C ] + (F 2)C [XC , Y C ].(3.2)Let X,Y ∈ Im10(M) and F ∈ ℘11(M), we have[XC , Y C ] = [X,Y ]C ,(X + Y )C = XC + Y C ,(3.3)FCXC = (FX)C .212 M. N. I. KhanIn the view of equations (1.3) and (3.3), we getFC lC = (Fl)C = FC ,FCmC = (Fm)C = 0.(3.4)Theorem 3.1. The following identities hold:NC(mCXC ,mCY C) = αFC [mCXC ,mCY C ]+ β[mCXC ,mCY C ],(3.5)mCNC(XC , Y C) = mC [FCXC , FCY C ],(3.6)mC(lCXC , lCY C) = mC [FCXC , FCY C ],(3.7)mCNC((F 2 − αF )CXC , (F 2 − αF )CY C) = β2mCNC(lCXC , lCY C).(3.8)Proof: The proof of equations (3.5) to (3.8) is followed by virtue of equations (1.3),(3.4) and (3.1).Theorem 3.2. Let X,Y ∈ ℘10(M), the following conditions are equivalent(a) mCNC(XC , Y C) = 0,(b) mCNC(lCXC , lCY C) = 0,(c) mCNC((F 2 − αF )CXC , (F 2 − αF )CY C) = 0.Proof: In consequence of the equation (3.8), we haveNC(lCXC , lCY C) = 0↔ NC((F 2 − αF )CXC , (F 2 − αF )CY C) = 0.Now the right sides of the equations (3.6), (3.7) are equal which in view of the lastequation shows that conditions (a), (b), and (c) are equivalent.Theorem 3.3. The complete lift DCm in TM of a distribution Dm in M is integralif Dm is integrable in M .Proof: The distribution Dm is integral if and only if [23]l[mX,mY ] = 0,(3.9)for all X,Y ∈ ℘(M), where l = I −m. Operating complete lift of both sides andusing (3.5), we getlC [mCXC ,mCY C ] = 0,(3.10)for all X,Y ∈ ℘(M), where lC = (I − m)C = I − mC is the projection tensorcomplementary to mC . Thus the condition (3.9) implies (3.10).Theorem 3.4. The complete lift DCm in TM of a distribution Dm in M is integralif lCNC(mCXC ,mCY C) = 0, or equivalently NC(mCXC ,mCY C) = 0, for allX,Y ∈ ℘10(M).Lifts of f(α, β)(3, 2, 1)-structures from manifolds to tangent bundles 213Proof: The distribution Dm is integral in M if and only if [23]N(mX,mY ) = 0,(3.11)for all X,Y ∈ ℘(M). By virtue of condition (3.5), we haveNC(mCXC ,mCY C) = (F 2)C(mCXC ,mCY C)Multiplying throughout by lC , we getlCNC(mCXC ,mCY C) = (F 2)C lC(mCXC ,mCY C).In view of (3.10), the above relation becomeslCNC(mCXC ,mCY C) = 0.(3.12)Also, we havemCNC(mCXC ,mCY C) = 0.(3.13)Adding equations (3.12) and (3.13), we get(lC +mC)NC(mCXC ,mCY C) = 0.Since lC +mC = IC = I, we haveNC(mCXC ,mCY C) = 0.Theorem 3.5. Let the distribution Dl be integrable in M , that is mN(X,Y ) = 0for all X,Y ∈ ℘10(M).Then the distribution DCl is integrable in TM if and only ifthe one of the conditions of Theorem (3.2) is satisfied.Proof: The distribution Dl is integral in M if and only ifmN(lX, lY ) = 0.(3.14)Thus distribution DCl is integrable in TM if and only ifmCNC(lCXC , lCY C) = 0.Thus the theorem follows by making use of the equation (3.8).Theorem 3.6. Let complete lift FC of a f(α, β)(3, 2, 1)-structure F in M is par-tially integrable in TM if and only if F is partially integrable in M .Proof: The f(α, β)(3, 2, 1)-structure F in M is partially integrable if and only ifN(lX, lY ) = 0,∀X,Y ∈ ℘10(M).(3.15)In view of the equations (1.3) and (3.1), we obtainNC(lCXC , lCY C) = (N(lX, lY ))Cwhich impliesNC(lCXC , lCY C) = 0⇔ N(lX, lY ) = 0.Also from Theorem (3.2), NC(lCXC , lCY C) = 0 is equivalent toNC((F 2 − αF )CXC , (F 2 − αF )CY C) = 0.214 M. N. I. KhanTheorem 3.7. Let complete lift FC of a f(α, β)(3, 2, 1)-structure F in M is par-tially integrable in TM if and only if F is partially integrable in M .Proof: A necessary and sufficient condition for a f(α, β)(3, 2, 1)-structure in M tobe integrable is that(N(X,Y )) = 0(3.16)for all X,Y ∈ ℘10(M).In view of the equation (3.1), we getNC(XC , Y C) = (N(X,Y ))C .Therefore, with the help of the equation (3.16) we obtain the result.4. The horizontal lift of an f(α, β)(3, 2, 1)-structure in the tangentbundleNow, we shall prove some theorems on horizontal lift of the f(α, β)(3, 2, 1)-structure.Suppose that there are tensor fields S and ∇γS in M and TM respectively withaffine connection∇ in the term of partial differential equations are given by [1, 2, 23]S = Si...hk...j∂∂xi⊗ . . .⊗ ∂∂xh⊗ dxk ⊗ . . .⊗ dxj ,(4.1)∇γS = yl∇γSi...hk...j∂∂xi⊗ . . .⊗ ∂∂yh⊗ dxk ⊗ . . .⊗ dxj(4.2)corresponding to the induced coordinates (xh, yh) in π−1(U)[23].Now, we define the horizontal lift SH of a tensor field S in M to TM bySH = SC −∇γS.(4.3)Theorem 4.1. Let F ∈ ℘11 be an f(α, β)(3, 2, 1)-structure in M , then its horizon-tal lift FH is also f(α, β)(3, 2, 1)-structure in TM .Proof: If P (t) is a polynomial in one variable t, then we have [23](P (F ))H = P (FH),(4.4)for all F ∈ ℘11(M).Operating the horizontal lifts of both sides of the equation (1.1), we get(F 3)H = (αF 2 + βF )H ,(F 3)H = (αF 2)H + (βF )H .Lifts of f(α, β)(3, 2, 1)-structures from manifolds to tangent bundles 215In the view of (4.4) and IH = I, we get(FH)3 = (αFH)2 + βFH(4.5)which shows that FH is an f(α, β)(3, 2, 1)-structure in TM [13]. In the view ofequations (1.1) and (4.5), we can easily say that the rank of FH is 2n if and only ifthe rank of F is n. Therefore , we have the following theorem:Theorem 4.2. The f(α, β)(3, 2, 1)-structure F of rank n in M if and only if itscomplete lift FH is of rank 2n in TM .Let m be a projection tensor field of type (1,1) in Mdefined by (1.3), in M thereexists a distribution D determined by m. Alsom2 = m.In view of (4.4), we get(mH)2 = mH .Thus, mH is also a projection in TM . Hence there exists in TM a distribution DHcorresponding to mH , which is called the horizontal lift of the distribution D.5. Prolongation of an f(α, β)(3, 2, 1)-structure on third tangent bundleT3MLet T3M be the third order tangent bundle over M and let FIII be the third lifton F in T3M . Then for any F,G ∈ ℘11(M), we have(GIIIF III)XIII = GIII(F IIIXIII)= (GIII(FX)III)= (G(FX))III= (GF )IIIXIII ,(5.1)for all X ∈ ℘10(M). Thus we haveGIIIF III = (GF )IIIIf P (t) is a polynomial in one variable t, then we have [23](P (F ))III = P (F III).(5.2)for all F ∈ ℘11(M).Theorem 5.1. Let F ∈ ℘11(M) be a f(α, β)(3, 2, 1)-structure in M , then the thirdlift F III is also f(α, β)(3, 2, 1)-structure in T3M .216 M. N. I. KhanProof: If P (t) is a polynomial in one variable t, then we have [23](P (F ))III = P (F III),(5.3)for all F ∈ ℘11(M). Operating the third lifts of both sides of the equation (1.1), weget(F 3 = αF 2 + βF )III ,(F 3)III = (αF 2)III + (βF )III .In the view of (5.3) and IIII = I, we get(F III)3 = α(F III)2 − βF III(5.4)which shows that F III is a f(α, β)(3, 2, 1)-structure in T3M .Theorem 5.2. The third lift F III is integrable in T3M if and only if F is inte-grable in M .Proof: Let N III and N be Nijenhuis tensors of F III and F respectively. Then wehaveN III(X,Y ) = (N(X,Y ))III .(5.5)since f(α, β)(3, 2, 1)-structure is integrable in M if and only if N(X,Y ) = 0. thenfrom (5.5), we getN III(X,Y ) = 0.(5.6)Thus F III is integrable if and only if F is integrable in M .AcknowledgementI would like to thank the Reviewers and the Editor for reviewing the paper carefullyand providing valuable comments that improved the overall quality of the paper.REFERENCES1. M. Altunbas, L. Bilen and A. Gezer: Remarks about the Kaluza-Klein metricon tangent bundle, Int. J. Geo. Met. Mod. Phys., 16 (3) (2019), 1950040.2. S. 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LIFTS OF F(α, β)(3, 2, 1)-STRUCTURES FROM MANIFOLDS TO TANGENT BUNDLES

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LIFTS OF G(α, β)(3, 2, 1)-STRUCTURES FROM MANIFOLDS TO TANGENT BUNDLES

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