Abstract
Abstract We analyze some properties of diagonal lift of tensor fields of type (1,1) in semi-cotangent bundles with the help of adapted frames.
Highlights
We analyze some properties of diagonal lift of tensor fields of type (1,1) in semi-cotangent bundles with the help of adapted frames
Defining some structure on the tangent bundles and cotangant bundles to obtain subtle information about the topology and geometry of the manifold is the main way for mathematicians
Many authors have been systematically worked on them [1, 2, 9, 13, 14]. One of these studies is analyzing some properties of diagonal lift of tensor fields of type (1,1) in semi-cotangent bundles with the help of adapted frames Let Mn be an n-dimensional differentiable manifold of class C∞ and T (Mn) the tangent bundle determined by a natural projection π1 : T (Mn) → Mn
Summary
Be the cotangent space at a point x of Mn. If pα are components of p ∈ Tx∗(Mn) with respect to the natural coframe {dxα }, i.e. p = pi dxi, by definition the set t∗(Mn) of all points xI = (xα , xα , xα ), xα = pα ; I, J, ... = 1, ..., 3n with projection π2 : t∗(Mn) → T (Mn) (i.e. π2 : (xα , xα , xα ) → (xα , xα )) is a semi-cotangent (pull-back [12]) bundle of the cotangent bundle by submersion π1 : T (Mn) → Mn (For definition of the pull-back bundle, see for example [3], [5], [6], [7]). Be the cotangent space at a point x of Mn. If pα are components of p ∈ Tx∗(Mn) with respect to the natural coframe {dxα }, i.e. p = pi dxi, by definition the set t∗(Mn) of all points xI = (xα , xα , xα ), xα = pα ; I, J, ... It is clear that the pull-back bundle t∗(Mn) of the cotangent bundle T ∗(Mn) has the natural bundle structure over Mn, its bundle projection π : t∗(Mn) → Mn being defined by π : (xα , xα , xα ) → (xα ), and π = π1 ◦ π2. The main purpose of the present paper is to study complete lift of vector fields and tensor fields of type (1,1) from tangent bundle T (Mn) to semi-cotangent (pull-back) bundle (t∗(Mn), π2)
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