ON THE LIFTS OF Fa(5; 1) STRUCTURE ON TANGENT AND COTANGENT BUNDLE

This paper consist of three main sections. In the rst part, we obtain the complete lifts of the Fa(5; 1) structure on tangent bundle. We have also obtained the integrability conditions by calculating the Nijenhuis tensors of the complete lifts of Fa(5; 1) structure. Later we get the conditions of to be the almost holomorc vector eld with respect to the complete lifts of Fa(5; 1) structure. Finally, we obtained the results of the Tachibana operator applied to the vector elds with respect to the complete lifts of Fa(5; 1) structure on tangent bundle. In the second part, all results obtained in the rst section investigated according to the horizontal lifts of Fa(5; 1) structure in tangent bundle T (Mn). In nally section, all results obtained in the rst and second section were investigated according to the horizontal lifts of the Fa(5; 1) structure in cotangent bundle T (Mn).


Introduction
The investigation for the integrability of tensorial structures on manifolds and extension to the tangent or cotangent bundle, whereas the de…ning tensor …eld sat-is…es a polynomial identity has been an actively discussed research topic in the last 50 years, initiated by the fundamental works of Kentaro Yano and his collaborators, see for example [17]. Also, the idea of F structure manifold on a di¤erentiable manifold developed by Yano [14], Ish¬hara and Yano [7], Goldberg [6] and among others. Moreover, Yano and Patterson [15,16] studied on the horizontal and complete lifts from a di¤erentiable manifold M n of class C 1 to its cotangent bundles. Andreu has studied the structure de…ned by a tensor …eld F (6 = 0) of type (1; 1) satisfying F 5 + F = 0 [1]. Later Ram Nivas and C.S. Prasad [11] studied on more form F a (5; 1) structure. This paper consist of three main sections. In the …rst part, we obtain the complete lifts of the F a (5; 1) structure on tangent bundle. We have also obtained the integrability conditions by calculating the Nijenhuis tensors of the complete lifts of F a (5; 1) structure. Later we get the conditions of to be the almost holomor…c vector …eld with respect to the complete lifts of F a (5; 1) structure. Finally, we obtained the results of the Tachibana operator applied to the vector …elds with respect to the complete lifts of F a (5; 1) structure on tangent bundle. In the second part, all results obtained in the …rst section investigated according to the horizontal lifts of F a (5; 1) structure in tangent bundle T (M n ). In …nally section, all results obtained in the …rst and second section were investigated according to the horizontal lifts of the F a (5; 1) structure in cotangent bundle T (M n ).
Let M n be an n dimensional di¤erentiable manifold of class C 1 . Suppose there exist on M n , a (1; 1) tensor …eld F (6 = 0) satisfying [11] F 5 a 2 F = 0; (1) where a is a complex number not equal to zero. If a = i where i = p 1, our structure takes the form F 5 + F = 0 studied by Andreou [1].
Let us de…ne on M n , the operators l and m as follows : I being unit tensor …eld. In view of equations (1) and (2), we have l 2 = l, m 2 = m and l + m = I: For a tensor …eld F (6 = 0) of type (1; 1) satisfying (1) the operators l and m de…ned by (2), when applied to the tangent space of M n at a point, are complementary projection operators. Thus there exist complementary distributions L and M corresponding to the projection operators l and m respectively. If the rank of F is constant every where or equal to r, the dimensions of L and M are r and n r respectively [10]. Us call such a structure as F a (5; 1) structure of rank r [11].
1.1. Complete Lift of F a (5; 1) Structure on Tangent Bundle. Let M n be an n dimensional di¤erentiable manifold of class C 1 and T P (M n ) the tangent space at a point p of M n and T (M n ) = U p2M n T P (M n ) 368 F. JABRA · ILZADE is the tangent bundle over the manifold M n : Let us denote by T r s (M n ), the set of all tensor …elds of class C 1 and of type (r; s) in M n and T (M n ) be the tangent bundle over M n . The complete lift of F C of an element of T 1 1 (M n ) with local components F h i has components of the form [16] Now we obtain the following results on the complete lift of F satisfying F 5 a 2 F = 0.
Let F; G 2 T 1 1 (M n ). Then we have [16] (F G) C = F C G C : Replacing G by F in (7) we obtain Now putting G = F 4 in (7) since G is (1; 1) tensor …eld therefore F 4 is also (1; 1) so we obtain (F F 4 ) C = F C (F 4 ) C which in view of (8) becomes Taking complete lift on both sides of equation F 5 a 2 F = 0 we get which in consequence of equation (9) gives Let F satisfying (1; 1) be an F structure of rank r in M n . Then the complete lifts l C = (F 4 ) C of l and m C = I (F 4 ) C of m are complementary projection tensors in T (M n ). Thus there exist in T (M n ) two complementary distributions L C and M C determined by l C and m C , respectively.

1.2.
Horizontal Lift of F a (5; 1) Structure on Tangent Bundle. Let F h i be the component of F at A in the coordinate neighbourhood U of M n . Then the horizontal lift F H of F is also a tensor …eld of type (1; 1) in T (M n ) whose components F A B in 1 (U ) are given by Let F , G be two tensor …elds of type (1; 1) on the manifold M . If F H denotes the horizontal lift of F , we have Taking F and G identical, we get Taking horizontal lift on both sides of equation F 5 a 2 F = 0 we get view of (13), we can write 2. Main Results

The Nijenhuis Tensor
De…nition 1. Let F be a tensor …eld of type (1; 1) admitting F a (5; 1) structure in M n : The Nijenhuis tensor of a (1; 1) tensor …eld F of M n is given by for any X; The Nijenhuis tensor N F is de…ned local coordinates by . De…nition 2. Let X and Y be any vector …elds on a Riemannian manifold (M n ; g), we have [17] where R is the Riemannian curvature tensor of g de…ned by In particular, we have the vertical spray u V and the horizontal spray u H on T (M n ) de…ned by where i = @ i u j s ji @ s : u V is also called the canonical or Liouville vector …eld on T (M n ).

F. JABRA · ILZADE
Theorem 3. The Nijenhuis tensor N (F 5 ) C (F 5 ) C X C ; Y C of the complete lift of F 5 vanishes if the Nijenhuis tensor of the F is zero.
Proof. In consequence of De…nition 1 the Nijenhuis tensor of F 5 C is given by Proof.
De…nition 6. The Sasaki metric S g is a (positive de…nite) Riemannian metric on the tangent bundle T (M n ) which is derived from the given Riemannian metric on M as follows: Theorem 7. The Sasaki metric S g is pure with respect to F 5 C if rF = 0 and F = a 2 I , where I=¬dentity tensor …eld of type (1; 1).
where L Y is the Lie derivation with respect to Y (see [3,5,8]), e) the module of all pure tensor …elds of type (r; s) on M n with respect to the a¢ nor …eld, C is a tensor product with a contraction C [2, 4, 12](see [13] for applied to pure tensor …eld).  Consider the function f = 1: This may be written in many di¤ erent ways as { Y . Indeed taking = dx, we may choose Y = @ @x or Y = @ @x + x @ @y . Nov the right- is ( X)1 0 = 0 in the …rst case, and ( X)1 Xx = Xx in the second case. For X = @ @x ; the latter expression is 1 6 = 0. Therefore, we put r + s > 0 [12].
Theorem 11. Let ' be the Tachibana operator and the structure F 5 C a 2 F C = 0 de…ned by De…nition 8 and (10), respectively. If L Y F = 0, then all results with respect to F 5 C is zero, where X; Y 2 = 1 0 (M ), the complete lifts X C ; Y C 2 = 1 0 (T (M )) and the vertical lift Theorem 12. If L Y F = 0 for Y 2 M , then its complete lift Y C to the tangent bundle is an almost holomor…c vector …eld with respect to the structure F 5 C a 2 F C = 0. Proof.
Proof. Because of X V ; Y V = 0 for X; Y 2 M , easily we get Theorem 16. The Sasakian metric S g is pure with respect to F 5 H if F = a 2 I, where I =identity tensor …eld of type (1; 1).
Let ' be the Tachibana operator and the structure F 5 H a 2 F H = 0 de…ned by De…nition 8 and (14), respectively. if L Y F = 0 and F = a 2 I, then all results with respect to F 5 H is zero, where X; Y 2 = 1 0 (M ), the horizontal lifts X H ; Y H 2 = 1 0 (T (M n )) and the vertical lift Taking horizontal lift, we obtain In view of (22), we can write Proof. The Nijenhuis tensor N (X H ; Y H ) for the horizontal lift of F 5 is given by Let us suppose that F = a 2 I on M . Thus, the equation becomes Proof.
The theorem is proved.
Theorem 20. The Nijenhuis tensor N ( Theorem 21. Let (F 5 ) H be a tensor …eld of type (1; 1) on T (M n ). If the Tachibana operator ' applied to vector and covector …elds according to horizontal lifts of F 5 de…ned by (25) on T (M n ), then we get the following results.
where horizontal lifts X H ; Y H 2 = 1 0 (T (M n )) of X; Y 2 = 1 0 (M n ) and the vertical lift ! V ; V 2 = 1 0 (T (M n )) of !; 2 = 0 1 (M n ) are given, respectively. iii) De…nition 22. A Sasakian metric S g is de…ned on T (M n ) by the three equations For each x 2 M n the scalar product g 1 = (g ij ) is de…ned on the cotangent space 1 (x) = T x (M n ) by g 1 (!; ) = g ij ! i j ; where X; Y 2 = 1 0 (M n ) and !; 2 = 0 1 (M n ). Since any tensor …eld of type (0; 2) on T (M n ) is completely determined by its action on vector …elds of type X H and ! V (see [17], p.280), it follows that S g is completely determined by equations (26), (27) and (28).
Theorem 23. Let (T (M n ); S g) be the cotangent bundle equipped with Sasakian metric S g and a tensor …eld (F 5 ) H of type (1; 1) de…ned by (25). Sasakian metric S g is pure with respect to ( Thus, F = a 2 I, then S g is pure with respect to (F 5 ) H .

Declaration of Competing Interest
There is no con ‡ict of interests regarding the publication of this paper.