Lifts of derivations to the semitangent bundle

The main purpose of this paper is to investigate the complete lifts of derivations for semitangent bundle and to discuss relations between these and lifts already known. | Turk J Math 24 (2000) , 259 – 266. ¨ ITAK ˙ c TUB Lifts of Derivations to the Semitangent Bundle . Salimov and Ekrem Kadıo˘glu Abstract The main purpose of this paper is to investigate the complete lifts of derivations for semitangent bundle and to discuss relations between these and lifts already known. Key Words: Vector, Field, Derivation, Complete lift, Semitangent bundle. 1. Semitangent bundle Let Mn be an n-dimensional differentiable manifold of class C ∞ and π : Mn → Bm the differentiable bundle determined by a submersion π. Suppose that (xa , xα ), a, b, . = 1, ., n−m; α, β, . = n−m+1, ., n; i = 1, 2, ., n is a system of local coordinates adapted to the bundle π : Mn → Bm where xα are coordinates in Bm , xa are fibre coordinates 0 0 of the bundle (see [1, p. 190]). If (xa , xα ) is another system of local coordinates in the bundle, then we have ( 0 0 xa = xa (xa , xα ), 0 (1) 0 xα = xα (xα ). The Jacobian of (1) is given by the matrix 0 (Aii ) 0 = ∂xi ∂xi ! 0 = 0 Aaa Aaα 0 Aα α 0 ! . Subject classification number: Primary 53A45, Secondary 53C55. 259 ˘ SALIMOV, KADIOGLU ∼ ∼ Let Tp (Bm ) (p = π(p ), p = (xa , xα ) ∈ Mn ) be the tangent space at a point p of Bm . If X α = dxα (X) are components of X in tangent space Tp (Bm ) with respect to the natural base {∂α } (∂α = ∂ ), ∂xα then we have the set of all points (xa, xα , xα ), xα = X α , α = α + m is by definition, the semitangent bundle t(Mn ) over the manifold Mn (see [1], [2]), dimt(Mn ) = n + m. In the special case n = m, t(Mn ) is a tangent bundle T (Mn ). To a transformation of local coordinates of Mn (see (1)), there corresponds in t(Mn ) the coordinate transformation 0 a a0 a α x = x (x , x ), 0 0 xα = xα (xα ), xα0 = ∂xα0 xα . (2) ∂xα The Jacobian of (2) is given by A= 0 0 Aaa Aaα 0 Aα α 0 0 σ Aα ασ x 0 0 , 0 0 Aα α (3) where 0 0 Aα ασ = ∂ 2 xα . ∂xα ∂xσ We denote by Tqp (Mn ) the module over F (Mn ) of .