Lifts of Derivations in the Coframe Bundle

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.

Horizontal lifts of projectable linear connection to semi-tangent bundle

“…Let now (T (B m ), π, B m ) be a tangent bundle [14] over base space B m , and let M n be differentiable bundle determined by a submersion (natural projection) π 1 : M n → B m . The semi-tangent bundle (pullback [3,5,9,10,15,16]) of the tangent bundle (T (B m ), π, B m ) is the bundle (t(B m ), π 2 , M n ) over differentiable bundle M n with a total space…”

“…Thus (t(B m ),π 1 • π 2 ) is the step-like bundle [6] or composite bundle [8, p. 9]. Consequently, we notice the semitangent bundle (t(B m ),π 2 ) is a pullback bundle of the tangent bundle over B m by π 1 [9].…”

“…where [9]. The main aim of this article is to study the horizontal lifts of projectable linear connection to semi-tangent (pullback) bundle (t(B m ) , π 2 ) and their properties.…”

On Semi-Tensor Bundle

“…Let now T p q (M n ), π, M n be a tensor bundle [3], [6], [[7], p.118] with base space M n , and let T * (M n ) be cotangent bundle determined by a natural projection (submersion) π 1 : T * (M n ) → M n . The semi-tensor bundle (induced, pull-back [4], [5], [8], [9], [11], [12], [13], [14]) of the tensor bundle T p q (M n ), π, M n is the bundle t p q (M n ), π 2 , T * (M n ) over cotangent bundle T * (M n ) with a total space…”

“…Let now T p q (M n ), π, M n be a tensor bundle [3], [6], [ [7], p.118] with base space M n , and let T * (M n ) be cotangent bundle determined by a natural projection (submersion) π 1 : T * (M n ) → M n . The semi-tensor bundle (induced, pull-back [4], [5], [8], [9], [11], [12], [13], [14]) of the tensor bundle T p q (M n ), π, M n is the bundle t p q (M n ), π 2 , T * (M n ) over cotangent bundle T * (M n ) with a total space and with the projection map π 2 : t p q (M n ) → T * (M n ) defined by π 2 (x α , x α , x α ) = x α , x α , where T p q x (M n ) x = π 1 ( x) , x = x α , x α ∈ T * (M n ) is the tensor space at a point x of M n , where x α = t β1...βp α1...αq α, β, ... = 2n + 1, ..., 2n + n p+q are fiber coordinates of the tensor bundle…”