Some properties of semi-tensor bundle

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1 NTMSCI 6, No. 3, New Trends in Mathematical Sciences htt://dx.doi.org/1.2852/ntmsci Some roerties of semi-tensor undle Murat Polat and Nejmi Cengiz Deartment of Mathematics, Faculty of Science, Atatürk University, Erzurum, Turkey Received: 19 January 218, Acceted: 1 June 218 Pulished online: 1 August 218. Astract: Using the fier undle M over a manifold B, we define a semi-tensor ull-ack undle tb of tye,. The resent aer is devoted to some results concerning with the vertical and comlete lifts of some tensor fields from manifold B to its semi-tensor undle tb of tye,. Keywords: Vector field, comlete lift, ull-ack undle, semi-tensor undle. 1 Introduction Let M n e an n-dimensional differentiale manifold of class C and π 1 : M n B m the differentiale undle determined y a sumersion π 1. Suose thatx i =x a,x α, a,,...= 1,...,n m;α,,...=n m+1,...,n;i, j,...=1,2,...,n is a system of local coordinates adated to the undle π 1 : M n B m, where x α are coordinates in B m, and x a are fier coordinates of the undle π 1 : M n B m. If x i =x a,x α is another system of local adated coordinates in the undle, then we have [14] { x a = x a x,x, x α = x α x 1. The Jacoian of 1 has comonents where x i A i j = x j = A α A a = xα x. Aa A α Let T x B mx=π 1 x, x=x a,x α M n e the tensor sace at a oint x B m with local coordinatesx 1,...,x m, we have the holonomous frame field x i 1 x i 2... x i dx j 1 dx j 2... dx j, for i {1,...,m}, j {1,...,m}, over U B m of this tensor undle, and for any,-tensor field t we have [[4],.163]: t U = t i 1...i j 1... j x i 1 x i 2... x i dx j 1 dx j 2... dx j, then y definition the set of all ointsx I =x a,x α,x α, x α = t i 1...i j 1... j,α=α+ m +,I,J,...=1,...,n+m + is a semi-tensor undle t B m over the manifold M n [14]. The semi-tensor undle t B m has the natural undle structure over B m, its, Corresonding author murat sel 22@hotmail.com c 218 BISKA Bilisim Technology

2 148 M. Polat and N. Cengiz: Some roerties of semi-tensor undle undle rojection π : t B m B m eing defined y π :x a,x α,x α x α. If we introduce a maing π 2 : t B m M n y π 2 :x a,x α,x α x a,x α, then t B m has a undle structure over M n. It is easily verified that π = π 1 π 2 [14]. On the other hand, let ε = π : E B denote a fier undle with fier F. Given a manifold B and a ma f : B B, one can construct in a natural way a undle over B with the same fier: Consider the suset f E = {,e B E f = πe } together with the susace toology from B E, and denote y π 1 : f E B, π 2 : f E E the rojections. f ε = π 1 : f E B is a fier undle with fier F, called the ull-ack undle of ε via f [[3], [5], [8], [1], [14]]. From the aove definition it follows that the semi-tensor undle tb m,π 2 is a ull-ack undle of the tensor undle over B m y π 1 see, for examle [12], [14]. In other words, the semi-tensor undle induced or ull-ack undle of the tensor undle T B m, π,b m is the undle t B m,π 2,M n over M n with a total sace tb m = { x a,x α,x α M n T x B m : π 1 x a,x α = π x α,x α =x α } M n T x B m.to a transformation 1 of local coordinates of M n, there corresonds on tb m the coordinate transformation The Jacoian of 2 is given y [14]: x a = x a x,x, x α = x α x, x α = t 1... α 1 =...α A 1... α 1...α A 1... α 1...α tα 1...α = A α A α x. Ā= A I J = A a A α t α σ A α Aσ α A α A α where I =a,α,α, J =,,, I,J...=1,...,n+m +, t α σ = tα 1...α σ 1...σ, A α = xα x. It is easily verified that the condition DetĀ is euivalent to the condition: DetA a,detaα,deta α A α. 2, 3 Also, dimt B m =n+m +. In the secial case n=m, t B m is a tensor undle T B m [[6],.118]. In the secial case, the semi-tensor undles t 1 B m =1, = and t 1 B m =, =1 are semi-tangent and semi-cotangent undles, resectively. We note that semi-tangent and semi-cotangent undle were examined in [[1], [7], [9]] and [[11], [13], [15], [16]], resectively. Also, Fattaev studied the secial class of semi-tensor undle [2]. We denote y I t B m and I B m the modules over F t B m and FB m of all tensor fields of tye, on t B m and B m resectively, where F t B m and FB m denote the rings of real-valued C functions on t B m and B m, resectively. 2 Some lifts of tensor fields and γ oerator Let X I 1 M n e a rojectale vector field [9] with rojection X = X α x α α i.e. X = X a x a,x α a + X α x α α. On utting cc X X cc X = cc X = X, 4 cc X tα 1...ε...α ε X λ tα 1...α 1...ε... µ X ε c 218 BISKA Bilisim Technology

3 NTMSCI 6, No. 3, / we easily see that cc X = Ā cc X. The vector field cc X is called the comlete lift of X to the semi-tensor undle tb m [14]. Now, consider A I B m and ϕ I 1 1 B m, then vv A I 1 t B m vertical lift, γϕ I 1 t B m and γϕ I 1 t B m have resectively, comonents on the semi-tensor undle t B m [14] vv A= A α 1...α,γϕ = tα 1...ε...α ϕ α λ ε, γϕ = tα 1...α 1...ε... ϕ ε µ 5 with resect to the coordinatesx a,x α,x α on t B m, where A α 1...α, ϕ α λ ε and ϕ ε µ are local comonents of A and ϕ. On the other hand, vv f the vertical lift of function f I B m on t B m is defined y [14]: vv f = v f π 2 = f π 1 π 2 = f π. 6 Theorem 1. For any vector fields X, Ỹ on M n and f I B m, we have i cc X+Ỹ= cc X+ cc Ỹ, ii cc X vv f = vv X f. Proof. i This immediately follows from 4. ii Let X I 1 M n. Then we get y 4 and 6: cc X vv f = cc X I I vv f= cc X a a vv f + cc X α α vv f+ cc X α α vv f = X α α vv f= vv X f, which gives ii of Theorem 1. Theorem 2. If ϕ I 1 1 B m, f I B m and A I B m, then i vv A vv f =, ii γϕ vv f=, iii γϕ vv f=. Proof. i If A I B m, then, y 5 and 6, we find vv A vv f = vv A I I vv f= vv A a a vv f + vv A α α vv f+ vv A α α vv f =. Thus, we have i of Theorem 2. ii If ϕ I 1 1 B m, then we have y 5 and 6: γϕ vv f=γϕ I I vv f=γϕ a a vv f +γϕ α α vv f+γϕ α α vv f =. Thus, we have ii of Theorem 2. iii If ϕ I 1 1 B m, then we have y 5 and 6: γϕ vv f= γϕ I I vv f= γϕ a a vv f + γϕ α α vv f+ γϕ α α vv f =. c 218 BISKA Bilisim Technology

4 15 M. Polat and N. Cengiz: Some roerties of semi-tensor undle Thus, we have iii of Theorem 2. Theorem 3. Let A,B I B m. For the Lie roduct, we have [ vv A, vv B]=. [ vv A, vv B] Proof. If A,B IB m and [ vv A, vv B] are comonents of [ vv A, vv B] J with resect to the coordinatesx,x,x on [ vv A, vv B] tb m, then we have [ vv A, vv B] J = vv A I I vv B J vv B I I vv A J = vv A a a vv B J + vv A α α vv B J + vv A α α vv B J vv B a a vv A J vv B α α vv A J vv B α α vv A J = A α 1...α α vv B J B α 1...α α vv A J. Firstly, if J =, we have y virtue of 5. Secondly, if J =, we have y virtue of 5. Thirdly, if J =, then we have [ vv A, vv B] = A α 1...α α vv B [ vv A, vv B] = A α 1...α α vv B B α 1...α α vv A B α 1...α =, α vv A =, [ vv A, vv B] = A α 1...α α vv B B α 1...α α vv A = A α 1...α α B 1... θ 1...θ y virtue of 5. Thus, we have Theorem 3. B α 1...α α A 1... θ 1...θ = Theorem 4. Let X and Ỹ e rojectale vector fields on M n with rojections X and Y on B m, resectively. For the Lie roduct, we have [ cc X, cc Ỹ]= cc [ X,Ỹ]i.e.L cc cc XỸ =cc L XỸ. [ cc X, cc Ỹ] Proof. If X and Ỹ are rojectale vector fields on M n with rojection X,Y I 1 B m and [ cc X, cc Ỹ] are comonents [ cc X, cc Ỹ] of[ cc X, cc Ỹ] J with resect to the coordinatesx,x,x on tb m, then we have [ cc X, cc Ỹ] J = cc X I I cc Ỹ J cc Ỹ I I cc X J. c 218 BISKA Bilisim Technology

5 NTMSCI 6, No. 3, / Firstly, if J =, we have [ cc X, cc Ỹ] = cc X I I cc Ỹ cc Ỹ I I cc X = cc X a a cc Ỹ + cc X α α cc Ỹ + cc X α α cc Ỹ cc Ỹ a a cc X cc Ỹ α α cc X cc Ỹ α α cc X = cc X α α cc Y cc Y α α cc X = X α α Ỹ Y α α X = [X,Y] y virtue of 4. Secondly, if J =, we have [ cc X, cc Ỹ] = cc X I I cc Ỹ cc Ỹ I I cc X = cc X a a cc Ỹ + cc X α α cc Ỹ + cc X α α cc Ỹ cc Ỹ a a cc X cc Ỹ α α cc X cc Ỹ α α cc X = cc X α α cc Ỹ cc Ỹ α α cc X = X α α Y Y α α X =[X,Y] y virtue of 4. Thirdly, if J =, then we have [ cc X, cc Ỹ] = cc X I I cc Ỹ cc Ỹ I I cc X = cc X a a cc Ỹ + cc X α α cc Ỹ + cc X α α cc Ỹ cc Ỹ a a cc X cc Ỹ α α cc X cc Ỹ α α cc X = X α α + Y α α = X α α + t α 1...ε...α t α 1...ε...α t α 1...ε...α ε Y λ ε X α λ t α 1...ε...α t α 1...ε...α t α 1...ε...α t α 1...ε...α ε Y α λ t α 1...α ε X λ t α 1...α 1...ε... µ Y ε 1...γ... αµ X γ α t α 1...α ε Y λ X α α ε X α λ α ε Y α λ α t α 1...α 1...ε... µ X ε 1...ε... µ Y ε α t α 1...σ...α δα σ λ t α 1...σ...α δα σ λ t α 1...σ...α t α 1...σ...α t α 1...α 1...ε... µ Y ε σ Y λ Y α α σ X λ + σ Y λ σ X λ t α 1...ε...α t α 1...α 1...γ... µ Y γ t α 1...α 1...γ... µ X γ ε X λ+y α α t α 1...α 1...ε... αµ X ε α t α 1...α 1...γ... δ α γ µ Y γ tα 1...α 1...ε... αµ Xε µ Y α t α 1...α 1...ε... µ X ε c 218 BISKA Bilisim Technology

6 152 M. Polat and N. Cengiz: Some roerties of semi-tensor undle = = t α 1...α 1...ε... αµ Y ε α t α 1...α 1...γ... δ α γ µ X γ tα 1...α 1...ε... αµ Y ε µ X α t α 1...ε...α ε X σ t α 1...ε...α t α 1...ε...α σ Y λ + t α 1...ε...α X α α ε Y λ t α 1...ε...α ε Y σ σ X λ Y α α ε X λ + t α 1...α 1...ε... X α αµ µ Y ε + µ Y α αµ X ε +Y α αµ µ X ε µ X α αµ Y ε tα 1...α 1...ε... µ X α αµ Y ε Y α αµ X ε [X,Y] ε ε [X,Y] λ t α 1...α 1...ε... µ [X,Y] ε y virtue of 4. On the other hand, we know that cc [X,Y] have comonents [X,Y] [X,Y]= [X,Y] cc tα 1...ε...α ε [X,Y] λ tα 1...α 1...ε... µ [X,Y] ε with resect to the coordinatesx,x,x on t B m. Thus Theorem 4 is roved. 3 Acknowledgment The authors would like to thanks the referees for useful and imroving comments. Cometing interests The authors declare that they have no cometing interests. Authors contriutions All authors have contriuted to all arts of the article. All authors read and aroved the final manuscrit. References [1] T.V. Duc, Structure resue-transverse. J. Diff. Geom., , no. 2, [2] H. Fattaev, The Lifts of Vector Fields to the Semitensor Bundle of the Tye 2,, Journal of Qafaz University, 25 29, no. 1, [3] D. Husemoller, Fire Bundles. Sringer, New York, [4] V. Ivancevic and T. Ivancevic, Alied Differential Geometry, A Modern Introduction, World Scientific, Singaore, 27. [5] H.B. Lawson and M.L. Michelsohn, Sin Geometry. Princeton University Press., Princeton, [6] A. Salimov, Tensor Oerators and their Alications. Nova Science Pul., New York, 213. c 218 BISKA Bilisim Technology

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