Note on the cross-section in the semi-tensor bundle

NTMSCI 5, No. 2, 212-221 217 212 New Trends in Mathematical Sciences http://dx.doi.org/1.2852/ntmsci.217.171 Note on the cross-section in the semi-tensor bundle Furkan Yildirim Narman Vocational Training School, Ataturk University, Erzurum, Turkey Received: 18 June 217, Accepted: 18 July 217 Published online: 15 August 217. Abstract: Using the fiber bundle M over a manifold B, we define a semi-tensor pull-back bundle tb of type p,q. The complete and horizontal lift of projectable geometric objects on M to the semi-tensor pull-back bundle tb of type p,q are presented. The main purpose of this paper is to study the behaviour of complete lift of vector and affinor tensor of type 1,1 fields on cross-sections for pull-back semi-tensor bundle tb of type p,q. Keywords: Vector field, complete lift, horizontal lift, pull-back bundle, cross-section, semi-tensor bundle. 1 Introduction Let M n be an n-dimensional differentiable manifold of class C and π 1 : M n B m the differentiable bundle determined by a submersion π 1. Suppose thatx i =x a,x α, a,b,...= 1,...,n m;α,,...=n m+1,...,n;i, j,...=1,2,...,n is a system of local coordinates adapted to the bundle π 1 : M n B m, where x α are coordinates in B m, and x a are fiber coordinates of the bundle π 1 : M n B m. Ifx i =x a,x α is another system of local adapted coordinates in the bundle, then we have The Jacobian of 1 has components where { x a = x a x b,x, x α = x α x. x i A i j = x j = A α A a b = xα x. Aa A α Let Tq p x B mx=π 1 x, x=x a,x α M n be the tensor space at a point x B m with local coordinatesx 1,...,x m, we have the holonomous frame field x i 1 x i 2... x ip dx j 1 dx j 2... dx j q, for i {1,...,m p, j {1,...,m q, over U B m of this tensor bundle, and for anyp,q-tensor field t we have [[6], p.163]: t U = t i 1...i p j 1... j q x i 1 x i 2... x ip dx j 1 dx j 2... dx j q, then by definition the set of all pointsx I =x a,x α,x α, x α = t i 1...i p j 1... j q,α=α+ m p+q,i,j,...=1,...,n+m p+q is a semi-tensor bundle t p qb m over the manifold M n., 1 c 217 BISKA Bilisim Technology Corresponding author e-mail: furkan.yildirim@atauni.edu.tr

213 F. Yildirim: Note on the cross-section in the semi-tensor bundle The semi-tensor bundle t p qb m has the natural bundle structure over B m, its bundle projection π : t p qb m B m being defined by π : x a,x α,x α x α. If we introduce a mapping π 2 : t p qb m M n by π 2 : x a,x α,x α x a,x α, then t p qb m has a bundle structure over M n. It is easily verified that π = π 1 π 2. On the other hand, let ε = π : E B denote a fiber bundle with fiber F. Given a manifold B and a map f : B B, one can construct in a natural way a bundle over B with the same fiber: Consider the subset f E = { b,e B E f b = πe together with the subspace topology from B E, and denote by π 1 : f E B, π 2 : f E E the projections. f ε = π 1 : f E B is a fiber bundle with fiber F, called the pull-back bundle of ε via f [[5],[7],[11],[13]]. From the above definition it follows that the semi-tensor bundle t p qb m,π 2 is a pull-back bundle of the tensor bundle over B m by π 1. In other words, the semi-tensor bundle induced or pull-back bundle of the tensor bundle T p q B m, π,b m is the bundle t p q B m,π 2,M n over Mn with a total space { tq p B m= x a,x α,x α M n Tq p x B m : π 1 x a,x α = π x α,x α =x α M n Tq p x B m. To a transformation 1 of local coordinates of M n, there corresponds on t p qb m the coordinate transformation x a = x a x b,x, x α = x α x, x α = t 1... p α 1 =...α q A 1... p α 1...α p A 1... q α 1...α tα 1...α p q = A α A α x. 2 The Jacobian of 2 is given by Ā= A I J = A a b A α t α σ A α Aσ α A α A α, 3 where I =a,α,α, J =b,,, I,J...=1,...,n+m p+q, t α σ = tα 1...α p σ 1...σ q, A α = xα. x It is easily verified that the condition DetĀ is equivalent to the condition: DetA a b,detaα,deta α A α. Also, dimt p qb m =n+m p+q. In the special case n=m, t p qb m is a tensor bundle T p q B m [[9], p.118]. In the special case, the semi-tensor bundles t 1 B m p=1, q= and t 1 B m p=, q=1 are semi-tangent and semi-cotangent bundles, respectively. We note that semi-tangent and semi-cotangent bundle were examined in [[1],[1],[12]] and [[14],[15]], respectively. Also, Fattaev studied the special class of semi-tensor bundle [3]. We denote by I p qt p qb m and I p qb m the modules c 217 BISKA Bilisim Technology

NTMSCI 5, No. 2, 212-221 217 / www.ntmsci.com 214 over F t p qb m and FB m of all tensor fields of type p,q on t p qb m and B m respectively, where F t p qb m and FB m denote the rings of real-valued C functions on t p qb m and B m, respectively. 2 Vertical lifts of tensor fields and γ operator If ψ t p qb m, it is regarded, in a natural way, by contraction on t p qb m, which we denote by ıψ. If ψ has the local expression ψ = ψ j 1... j q i 1...i p j1... jq dx i 1... dx i p in a coordinate neighborhood Ux α B m, then ıψ = ψt has the local expression with respect to the coordinatesx a,x α,x α in π 1 U [4]. ıψ = ψ j 1... j q i 1...i p t i 1...i p j 1... j q Suppose that A I p qb m. Then there is a unique vector fields vv A I 1 t p qb m such that for ψ t p qb m [8]: vv Aıψ= v A π 2 = ψa π 1 π 2 = ψa π = vv ψa, 4 where vv ψa is the vertical lift of the function ψa FB m. We note that the vertical lift vv f = f π of the arbitrary function f FB m is constant along each fiber of π : tqb p m B m. If vv A= vv A a a + vv A α α + vv A α α, then we have from 4 vv A= vv A a t α 1...α p a ψ 1... q α 1...α p + vv A α t α 1...α p α ψ 1... q α 1...α p + vv A α ψ θ 1...θ q σ 1...σ p = ψ θ 1...θ q σ 1...σ p A σ 1...σ p θ 1...θ q. But ψ θ 1...θ q σ 1...σ p, a ψ 1... q α 1...α p and α ψ 1... q α 1...α p can take any preassigned valued at each point. Thus we have vv A a t α 1...α p =, vv A α t α 1...α p =, vv A α = A σ 1...σ p θ 1...θ q. Hence vv A α = at all points of tqb p m except possibly those at which all the components x α = t α 1...α p are zero: that is, at points of the base space. Thus we see that the components vv A α are zero a point such that x α, that is, on tqb p m B m. However, tqb p m B m is dense in tqb p m and the components of vv A are continuous at every point of tqb p m. Hence, we have vv A α = at all points of tqb p m. Consequently, the vertical lift vv A of A to tqb p m has components vv A= with respect to the coordinatesx a,x α,x α on t p qb m. vv A a vv A α vv A α = A α 1...α p, 5 Let ϕ I 1 1 B m. We define a vector field γϕ in π 1 U by γϕ = γϕ = q µ=1 tα 1...α p ϕ ε µ ϕ α λ ε x, p 1,q x, p,q 1 6 with respect to the coordinates x b,x,x on t p qb m. From 3 we easily see that the vector fields γϕ and γϕ defined in each π 1 U t p qb m determine respectively global vertical vector fields on t p qb m. We call γϕ or γϕ the verticalvector lift of the tensor field ϕ I 1 1 B m to t p qb m. For any ϕ I 1 1 B m, if we take account of 3 and 6, we can prove c 217 BISKA Bilisim Technology

215 F. Yildirim: Note on the cross-section in the semi-tensor bundle thatγϕ = Āγϕ where γϕ is a vector field defined by γϕ =γϕ I = ϕ α λ ε. 7 Let ϕ I 1 1 B m. On putting γϕ = γϕ I = q µ=1 tα 1...α p ϕ ε µ, 8 we easily see that γϕ = Ā γϕ. 3 Complete lifts of vector fields We now denote by I p qm n the module over FM n of all tensor fields of typep,q on M n, where FM n denotes the ring of real-valued C functions on M n. Let X I 1 M n be a projectable vector field [9] with projection X = X α x α α i.e. X = X a x a,x α a + X α x α α. On putting cc cc X = cc X = cc X X ε X α λ q µ=1 tα 1...α p µ X ε, 9 we easily see that cc X = Ā cc X. The vector field cc X is called the complete lift of X to the semi-tensor bundle tqb p m. 4 Horizontal lifts of vector fields Let X I 1 M n be a projectable vector field [9] with projection X = X α x α α i.e. X = X a x a,x α a + X α x α α. If we take account of 3, we can prove that HH X = Ā HH X, where HH X is a vector field defined by HH X = X X l q µ=1 Γ l ε µ t α 1...α p p λ=1 Γ α λ lε t α 1...ε...α p, 1 with respect to the coordinatesx b,x,x on t p qb m. We call HH X the horizontal lift of the vector field X to t p qb m. Theorem 1. If X I 1 M n then cc X HH X = γ ˆ X γ ˆ X, where the symmetric affine connection ˆ is the given by Γ α θ = Γ α θ. c 217 BISKA Bilisim Technology

NTMSCI 5, No. 2, 212-221 217 / www.ntmsci.com 216 Proof. From 7, 8, 9 and 1, we have cc X HH X = X X ε X α λ q µ=1 tα 1...α p µ X ε X l q µ=1 Γ l ε µ t α 1...α p p λ=1 Γ α λ lε = ε X α λ + Γ α λ lε X l q µ=1 tα 1...α p µ X ε + Γl ε µ X l = ε X α λ + Γ α λ lε X l = = X l ε X α λ + Γ α λ εl {{ ˆ ε X α λ ˆ ε X α λ = γ ˆ ε X α λ γˆ µ X ε = γ ˆ X γ ˆ X. Thus, we have Theorem 1. q µ=1 tα 1...α p µ X ε + Γl ε µ X l q µ=1 tα 1...α p q µ=1 tα 1...α p ˆ µ X ε µ X ε + Γ ε µ l X l {{ ˆ µ X ε t α 1...ε...α p 5 Cross-sections in the semi-tensor bundle Let ξ IqB p m be a tensor field on B m. Then the correspondence x ξ x, ξ x being the value of ξ at x M n, determines a cross-section ξ of semi-tensor bundle. Thus if σ ξ : B m Tq p B m is a cross-section of Tq p B m, π,b m, such that π σ ξ = I Bm, an associated cross-section ξ : M n tqb p m of semi-tensor bundle tqb p m,π 2,M n defined by [[2], p. 217-218], [[9], p. 126-127]: ξ x a,x α = x a,x α,σ ξ π 1 x a,x α = x a,x α,σ ξ x α = x a,x α,ξ α 1...α p x. If the tensor field ξ has the local components ξ α 1...α p x, the cross-section ξ M n of tqb p m is locally expressed by x b = x b, x = x, x = ξ α 1...α p x, with respect to the coordinates x B = x b,x,x in t p qb m. Differentiating 11 by x c, we see that n m tangent vector fields B c c=1,...,n m to ξ M n have components c x b B c = xb x c = cx B = c x, c ξ α 1...α p 11 c 217 BISKA Bilisim Technology

217 F. Yildirim: Note on the cross-section in the semi-tensor bundle which are tangent to the cross-section ξ M n. Thus B c have components B c : B B c δ b c =, with respect to the coordinatesx b,x,x in t p qb m. Where δ b c = A b c = xb x c. Let X I 1 M n be a projectable vector field [9] with projection X = X α x α α i.e. X = X a x a,x α a + X α x α α. We denote by BX the vector field with local components BX : B B c X c = δc b X c A b c X c = = with respect to the coordinatesx b,x,x in t p qb m, which is defined globally along ξ M n., 12 Differentiating 11 by x θ, we have vector fields C θ θ = n m+1,...,n with components which are tangent to the cross-section ξ M n. θ x b C θ = xb x θ = θ x B = θ x, θ ξ α 1...α p Thus C θ have components C θ : Cθ B = A b θ δ θ θ ξ α 1...α p, with respect to the coordinatesx b,x,x in t p qb m. Where A b θ = xb x θ,δ θ = A θ = x x θ. Let X I 1 M n be a projectable vector field [9] with projection X = X α x α α i.e. X = X a x a,x α a + X α x α α. We denote by CX the vector field with local components CX : Cθ B X θ = A b θ X θ X X θ θ ξ α 1...α p with respect to the coordinatesx b,x,x in t p qb m, which is defined globally along ξ M n., 13 c 217 BISKA Bilisim Technology

NTMSCI 5, No. 2, 212-221 217 / www.ntmsci.com 218 On the other hand, the fibre is locally expressed by x b = const., x = const., x = t α 1...α p = t α 1...α p, t α 1...α p being considered as parameters. Thus, on differentiating with respect to x = t α 1...α p, we easily see that the vector fields E θ θ = n+1,...,n+m p+q with components E θ : E θ B θ x b = θ x B = θ x is tangent to the fibre, where δ is the Kronecker symbol. θ t α 1...α p = δ θ 1...δ θ q 1 q δ α 1 γ 1...δ α p γ p Let ξ be a tensor field of typep,q with local components on B m. ξ = ξ γ 1...γ p θ 1...θ q dx θ 1... dx θ q γ1... γp We denote by Eξ the vector field with local components which is tangent to the fibre. Eξ : E θ B ξ γ 1...γ p θ 1...θ q = ξ α 1...α p, 14 According to 12 and 13, we define new projectable vector field H y with respect to the coordinatesx b,x,x in t p qb m, where BX+CX = H X A b c X c A b θ X θ H X = + X = X. 15 X θ θ ξ α 1...α p X θ θ ξ α 1...α p We consider in π 1 U tqb p m, n+m p+q local vector fields B c, C θ and E θ along ξ M n. They form a local family of frames {B c,c θ,e θ along ξ M n, which is called the adaptedb,c,e-frame of ξ M n in π 1 U. We can state following theorem: Theorem 2. Let e a vector field on M n with projection X on B m. We have along ξ M n the formulas i cc X = H X+ E L X ξ, ii vv ξ = Eξ 16 c 217 BISKA Bilisim Technology

219 F. Yildirim: Note on the cross-section in the semi-tensor bundle for any t I p qb m, where L X ξ denotes the Lie derivative of ξ with respect to X. Proof. i Using 9, 14 and 15, we have H X+ E L X ξ = X + X θ θ ξ α 1...α p X θ θ ξ α 1...α p q µ=1 µ X ξ α 1...α p 1... q + p λ=1 X α λ ξ α 1...ε...α p = X = cc X. p λ=1 X α λ ξ α 1...ε...α p q µ=1 µ X ξ α 1...α p 1... q Thus, we have 16. ii This immediately follows from 5. On the other hand, on putting C = E, we write the adapted frame of ξ M n as frame {B b,c,c of ξ M n is given by the matrix Ã= ÃA B = δb a x a δ α ξ σ 1...σ p α 1...α q δ 1 α 1...δ q α q δ σ 1 γ 1...δ σ p γ p {B b,c,c. The adapted. 16 Since the matrix à in 16 is non-singular, it has the inverse. Denoting this inverse byã 1, we have 1 δc 1 à b θ x b = ÃB C = δ θ θ ξ σ 1...σ p δ θ 1...δ θ q 1 q δ σ 1 γ 1...δ σ p γ q 1 1 where Ãà =à A B ÃB C = δ A C = Ĩ, where A=a,α,α, B= b,,, C= c,θ,θ., 17 Proof. In fact, from 16 and 17, we easily see that 1 δ 1 b Ãà a x a δc b θ x b = à A B ÃB C = δ α δ ξ σ 1...σ p α 1...α q δ 1 α 1...δ q α q δ σ 1 γ 1...δ σ θ p γ p θ ξ σ 1...σ p δ θ 1...δ θ q 1 q δ σ 1 γ 1...δ σ p γ q δc a θ x a + θ x a δ = δθ α c a = δ α θ ξ σ 1...σ p α 1...α q θ ξ σ 1...σ p α 1...α q δ θ 1 α 1...δ θ θ =δc A = Ĩ. q α q δα θ Let A IqB p m. If we take account of 5 and 16, we can easily prove that vv A = à vv A, where vv A I 1 t qb p m is a vector field defined by vv A a vv A= vv A α =, vv A α with respect to the adapted frame {B b,c,c of ξ M n. A α 1...α p Taking account of 9, 1 and 16, we see that the complete lift cc X and horizontal lift HH X have respectively c 217 BISKA Bilisim Technology

NTMSCI 5, No. 2, 212-221 217 / www.ntmsci.com 22 components with respect to the adapted frame cc X : X L X ξ α 1...α p, HH X : X X ξ α 1...α p {B b,c,c of ξ M n, where cc X = Ã cc X We now, from equations 7, 8 and 16 see that γϕ and γϕ have respectively components γϕ =γϕ I = p λ=1 ξ α 1...ε...α p ϕ α λ ε with respect to the adapted frame {B b,c,c of ξ M n., γϕ = γϕ I = and HH X = Ã HH X. q µ=1 ξ α 1...α p ϕ ε µ Let S I 1 2 M n now. If we take account of 16, we see thatγs = ÃγS. γs is given by γs=γs I J =, p λ=1 S λ ε ξ 1...ε... p α 1...α q with respect to the adapted frame {B b,c,c, where S λ ε are local componenets of S on B m. BX, CX and Eξ also have components: X α BX =,CX = X α,eξ = ξ α 1...α p respectively, with respect to the adapted frame {B b,c,c of the cross-section ξ M n determined by a tensor field ξ of typep,q in M n. Acknowledgment The authors would like to thanks the referees for useful and improving comments. Competing interests The authors declare that they have no competing interests. Authors contributions All authors have contributed to all parts of the article. All authors read and approved the final manuscript. c 217 BISKA Bilisim Technology

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