1. bookVolume 6 (2021): Issue 1 (January 2021)
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Some Properties of Diagonal Lifts in Semi-Cotangent Bundles

Published Online: 31 Dec 2020
Page range: 479 - 488
Received: 20 Jul 2019
Accepted: 26 Sep 2019
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
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.

Keywords

MSC 2010

Lifts of Vector Fields on a Cross-Section in the Semi-Cotangent Bundle

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. Due to this feature, 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 (submersion) π1 : T (Mn) → Mn. We use the notation (xi) = (xα̅,xα), where the indices i, j,... run from 1 to 2n, the indices α,β,... from 1 to n and the indices α̅,β̅ ,... from n + 1 to 2n, xα are coordinates in Mn, xα̅ = yα are fibre coordinates of the tangent bundle T (Mn). If (xi)=(xα¯,xα) ({x^{{i^\prime}}}) = ({x^{{{\overline \alpha }^\prime}}},{x^{{\alpha ^\prime}}}) is another system of local adapted coordinates in the tangent bundle T (Mn), then we have {xα¯=xαxβyβ,xα=xα(xβ). \left\{ {\matrix{ {{x^{{{\overline \alpha }^\prime}}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}}{y^\beta },} \hfill & \cr {{x^{{\alpha ^\prime}}} = {x^{{\alpha ^\prime}}}\left( {{x^\beta }} \right).} \hfill & \cr } } \right.

The Jacobian of (1) has components (Aji)=(xixj)=(AβαAβεαyε0Aβα), (A_j^{{i^\prime}}) = \left( {{{\partial {x^{{i^\prime}}}} \over {\partial {x^j}}}} \right) = \left( {\matrix{ {A_\beta ^{{\alpha ^\prime}}} & {A_{\beta \varepsilon }^{{\alpha ^\prime}}{y^\varepsilon }} \cr 0 & {A_\beta ^{{\alpha ^\prime}}} \cr } } \right), where Aβα=xαxβ A_\beta ^{{\alpha ^\prime}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}} , Aβεα=2xαxβxε A_{\beta \varepsilon }^{{\alpha ^\prime}} = {{{\partial ^2}{x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }\partial {x^\varepsilon }}} . Let Tx(Mn)(x=π1(x˜),x˜=(xα¯,xα)T(Mn)) T_x^ * ({M_n})(x = {\pi _1}(\widetilde x),\widetilde x = ({x^{\overline \alpha }},{x^\alpha }) \in T\left( {{M_n}} \right)) be the cotangent space at a point x of Mn. If pα are components of pTx(Mn) p \in T_x^ * ({M_n}) with respect to the natural coframe {dxα}, i.e. p = pi dxi, then by definition the set t*(Mn) of all points (xI)=(xα¯,xα,xα¯¯) \left( {{x^I}} \right) = ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) , xα¯¯=pα {x^{\overline {\overline \alpha } }} = {p_\alpha } ; I,J,... = 1,...,3n with projection π2 : t*(Mn) → T (Mn) (i.e. π2:(xα¯,xα,xα¯¯)(xα¯,xα)) {\pi _2}:({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^{\overline \alpha }},{x^\alpha })) ) 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]). It is remarkable fact that the semi-cotangent (pull-back) bundle has a degenerate symplectic structure [12] ω=(ωAB)=dp=(00000δβα0δαβ0). \omega = ({\omega _{AB}}) = dp = \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & { - \delta _\beta ^\alpha } \cr 0 & {\delta _\alpha ^\beta } & 0 \cr } } \right).

It is clear that the pull-back bundle t*(Mn) of the cotangent bundle T* (Mn) also has the natural bundle structure over Mn, its bundle projection π: t*(Mn) → Mn being defined by π:(xα¯,xα,xα¯¯)(xα) \pi :({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^\alpha }) , and hence π = π1π2. Thus (t*(Mn),π1π2) is the composite bundle [ [14], p.9] or step-like bundle [15].

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).

We denote by qp(T(Mn)) \Im _q^p(T({M_n})) and qp(Mn) \Im _q^p({M_n}) the modules over F (T (Mn)) and F (Mn) of all tensor fields of type (p,q) on T (Mn) and Mn respectively, where F (T (Mn)) and F (Mn) denote the rings of real-valued C−functions on T (Mn) and Mn, respectively.

To a transformation (1) of local coordinates of T (Mn), there corresponds on t*(Mn) the coordinate transformation [10]: {xα¯=xαxβyβ,xα=xα(xβ),xα¯¯=xβxαpβ. \left\{ {\matrix{ {{x^{{{\overline \alpha }^\prime}}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}}{y^\beta },} \cr {\;{x^{{\alpha ^\prime}}} = {x^{{\alpha ^\prime}}}\left( {{x^\beta }} \right),} \cr {{x^{{{\overline {\overline \alpha } }^\prime}}} = {{\partial {x^\beta }} \over {\partial {x^{{\alpha ^\prime}}}}}{p_\beta }.} \cr } } \right.

The Jacobian of (2) has components [10]: A¯=(AJI)=(AβαAβεαyε00Aβα00pσAββAβασAαβ), \overline A = (A_J^{{I^\prime}}) = \left( {\matrix{ {A_\beta ^{{\alpha ^\prime}}} & {A_{\beta \varepsilon }^{{\alpha ^\prime}}{y^\varepsilon }} & 0 \cr 0 & {A_\beta ^{{\alpha ^\prime}}} & 0 \cr 0 & {{p_\sigma }A_\beta ^{{\beta ^\prime}}A_{{\beta ^\prime}{\alpha ^\prime}}^\sigma } & {A_{{\alpha ^\prime}}^\beta } \cr } } \right), where Aβεα=2xαxβxε,Aβαα=2xαxβxα. A_{\beta \varepsilon }^{{\alpha ^\prime}} = {{{\partial ^2}{x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }\partial {x^\varepsilon }}},\quad A_{{\beta ^\prime}{\alpha ^\prime}}^\alpha = {{{\partial ^2}{x^\alpha }} \over {\partial {x^{{\beta ^\prime}}}\partial {x^{{\alpha ^\prime}}}}}.

We denote by qp(T(Mn)) \Im _q^p(T({M_n})) and qp(Mn) \Im _q^p({M_n}) the modules over F (T (Mn)) and F (Mn) of all tensor fields of type (p,q) on T (Mn) and Mn, respectively, where F (T (Mn)) and F (Mn) denote the rings of real-valued C −functions on T (Mn) and Mn, respectively.

Let θ be a covector field on T (Mn). Then the transformation pθp, θp being the value of θ at pT (Mn), determines a cross-section βθ of semi-cotangent bundle. Thus if σ : MnT* (Mn) is a cross-section of (T* (Mn), π̃,Mn), such that π̃σ = I(Mn), an associated cross-section βθ : T (Mn) → t*(Mn) of semi-cotangent (pull-back) bundle (t*(Mn),π2,T (Mn)) of cotangent bundle by using projection (submersion) of the tangent bundle T (Mn) defined by [ [4], p. 217–218], [ [9], p. 301]: βθ(xα¯,xα)=(xα¯,xα,σπ1(xα¯,xα))=(xα¯,xα,σ(xα))=(xα¯,xα,θα(xβ)). {\beta _\theta }\left( {{x^{\overline \alpha }},{x^\alpha }} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },\sigma \circ {\pi _1}\left( {{x^{\overline \alpha }},{x^\alpha }} \right)} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },\sigma \left( {{x^\alpha }} \right)} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },{\theta _\alpha }\left( {{x^\beta }} \right)} \right).

If the covector field θ has the local components θα (xβ), the cross-section βθ (T (Mn)) of t*(Mn) is locally expressed by xα¯=yα=Vα(xβ),xα=xα,xα¯¯=pα=θα(xβ) {x^{\overline \alpha }} = {y^\alpha } = {V^\alpha }\left( {{x^\beta }} \right),\quad {x^\alpha } = {x^\alpha },\quad {x^{\overline {\overline \alpha } }} = {p_\alpha } = {\theta _\alpha }\left( {{x^\beta }} \right) with respect to the coordinates xA=(xα¯,xα,xα¯¯) {x^A} = ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t* (Mn). xα̅ = yα being considered as parameters. Differentiating (4) by xα̅ = yα, we have vector fields B(β̅) (β̅ = 1,...,n) with components B(β¯)=xAxβ¯=β¯xA=(β¯Vαβ¯xαβ¯θα), {B_{\left( {\overline \beta } \right)}} = {{\partial {x^A}} \over {\partial {x^{\overline \beta }}}} = {\partial _{\overline \beta }}{x^A} = \left( {\matrix{ {{\partial _{\overline \beta }}{V^\alpha }} \hfill \cr {{\partial _{\overline \beta }}{x^\alpha }} \hfill \cr {{\partial _{\overline \beta }}{\theta _\alpha }} \hfill \cr } } \right), which are tangent to the cross-section βθ (T (Mn)) [10].

Thus B(β̅) have components B(β¯):(B(β¯)A)=(δβ¯α00) {B_{\left( {\overline \beta } \right)}}:\left( {B_{\left( {\overline \beta } \right)}^A} \right) = \left( {\matrix{ {\delta _{\overline \beta }^\alpha } \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t*(Mn), where δβ¯α=Aβ¯α=xαxβ¯. \delta _{\overline \beta }^\alpha = A_{\overline \beta }^\alpha = {{\partial {x^\alpha }} \over {\partial {x^{\overline \beta }}}}.

Let X01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) , i.e. X = Xαα. We denote by BX the vector field with local components BX:(B(β¯)AXβ¯)=(δβ¯αXβ¯00)=(Aβ¯αXβ¯00)=(Xα00) BX:\left( {B_{\left( {\overline \beta } \right)}^A{X^{\overline \beta }}} \right) = \left( {\matrix{ {\delta _{\overline \beta }^\alpha {X^{\overline \beta }}} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) = \left( {\matrix{ {A_{\overline \beta }^\alpha {X^{\overline \beta }}} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) = \left( {\matrix{ {{X^\alpha }} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t*(Mn), which is defined globally along βθ (T (Mn)). Then a mapping B:01(T(Mn))01(βθ(T(Mn))) B:\Im _0^1(T({M_n})) \to \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) is defined by (5). The mapping B is the differential of βθ : T (Mn) → t* (Mn) and so an isomorphism of 01(T(Mn)) \Im _0^1(T({M_n})) onto 01(βθ(T(Mn))) \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) [10].

Since a cross-section is locally expressed by xα̅ = yα = const., xα¯¯=pα=const. {x^{\overline {\overline \alpha } }} = {p_\alpha } = const. , xα = xα, xα being considered as parameters. Differentiating (4) by xα, we have vector fields C(β) (β = n + 1,...,2n) with components C(β)=xAxβ=βxA=(βVαβxαβθα), {C_{\left( \beta \right)}} = {{\partial {x^A}} \over {\partial {x^\beta }}} = {\partial _\beta }{x^A} = \left( {\matrix{ {{\partial _\beta }{V^\alpha }} \hfill \cr {{\partial _\beta }{x^\alpha }} \hfill \cr {{\partial _\beta }{\theta _\alpha }} \hfill \cr } } \right), which are tangent to the cross-section βθ (T (Mn)).

Thus C(β) have components C(β):(C(β)A)=(βVαδβαβθα) {C_{\left( \beta \right)}}:\left( {C_{\left( \beta \right)}^A} \right) = \left( {\matrix{ {{\partial _\beta }{V^\alpha }} \hfill \cr {\delta _\beta ^\alpha } \hfill \cr {{\partial _\beta }{\theta _\alpha }} \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t* (Mn), where δβα=Aβα=xαxβ. \delta _\beta ^\alpha = A_\beta ^\alpha = {{\partial {x^\alpha }} \over {\partial {x^\beta }}}.

Let X01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) . Then we denote by CX the vector field with local components CX:(C(β)AXβ)=(XββVαXαXββθα) CX:\left( {C_{\left( \beta \right)}^A{X^\beta }} \right) = \left( {\matrix{ {{X^\beta }{\partial _\beta }{V^\alpha }} \hfill \cr {{X^\alpha }} \hfill \cr {{X^\beta }{\partial _\beta }{\theta _\alpha }} \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t* (Mn), which is defined globally along βθ (T (Mn)). Then a mapping C:01(T(Mn))01(βθ(T(Mn))) C:\Im _0^1(T({M_n})) \to \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) is defined by (6). The mapping C is the differential of βθ : T (Mn) → t* (Mn) and so an isomorphism of 01(T(Mn)) \Im _0^1(T({M_n})) onto 01(βθ(T(Mn))) \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) [10].

Now, consider ω10(Mn) \omega \in \Im _1^0({M_n}) and vector field X01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) , then vvω (vertical lift), ccX (complete lift) and HHX (horizontal lift) have respectively, components on the semi-cotangent bundle t*(Mn) [11]: vvω=(00ωα),ccX=(yεεXαXαpσ(αXσ)),HHX=(ΓβαXβXαXβΓβα) ^{^{vv}}\omega = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {{\omega _\alpha }} \hfill \cr } } \right),{\quad ^{cc}}X = \left( {\matrix{ {{y^\varepsilon }{\partial _\varepsilon }{X^\alpha }} \hfill \cr {{X^\alpha }} \hfill \cr { - {p_\sigma }({\partial _\alpha }{X^\sigma })} \hfill \cr } } \right),{\quad ^{HH}}X = \left( {\matrix{ { - \Gamma _\beta ^\alpha {X^\beta }} \hfill \cr {{X^\alpha }} \hfill \cr {{X^\beta }{\Gamma _\beta }_\alpha } \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) , where Γβα=VεΓεαβ,Γβα=θεΓβεα. \Gamma _\beta ^\alpha = {V^\varepsilon }{{\Gamma_\varepsilon^\alpha}_\beta},\quad {\Gamma _{\beta\, \alpha}} = {\theta _\varepsilon }{{\Gamma_{\beta \alpha}^\varepsilon}}.

On the other hand, the fibre is locally represented by xα¯=yα=const.,xα=const.,xα¯=pα=pα, {x^{\overline \alpha }} = {y^\alpha } = const.,\quad {x^\alpha } = const.,\quad x\overline {^\alpha } = {p_\alpha } = {p_\alpha }, pα being considered as parameters. Thus, on differentiating with respect to pα, we easily see that the vector fields E(β¯¯)=vv(dxβ)(β¯¯=2n+1,...,3n) {E_{\left( {\overline {\overline \beta } } \right)}}{ = ^{vv}}\left( {d{x^\beta }} \right)\;(\overline {\overline \beta } = 2n + 1,...,3n) with components E(β¯¯):(E(β¯¯)A)=(β¯¯)xA=(β¯¯yαβ¯¯xαβ¯¯pα)=(00δαβ) {E_{\left( {\overline {\overline \beta } } \right)}}:\left( {E_{\left( {\overline {\overline \beta } } \right)}^A} \right) = {\partial _{\left( {\overline {\overline \beta } } \right)}}{x^A} = \left( {\matrix{ {{\partial _{\overline {\overline \beta } }}{y^\alpha }} \hfill \cr {{\partial _{\overline {\overline \beta } }}{x^\alpha }} \hfill \cr {{\partial _{\overline {\overline \beta } }}{p_\alpha }} \hfill \cr } } \right) = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {\delta _\alpha ^\beta } \hfill \cr } } \right) is tangent to the fibre, where δαβ=Aαβ=xβxα. \delta _\alpha ^\beta = A_\alpha ^\beta = {{\partial {x^\beta }} \over {\partial {x^\alpha }}}.

Let ω be an 1-form with local components ωα on Mn, so that ω is a 1-form with local expression ω = ωαdxα. We denote by the vector field with local components Eω:(E(β¯¯)Aωβ)=(00ωα), E\omega :\left( {E_{\left( {\overline {\overline \beta } } \right)}^A{\omega _\beta }} \right) = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {{\omega _\alpha }} \hfill \cr } } \right), which is tangent to the fibre. Then a mapping E:10(Mn)01(t(Mn)) E:\Im _1^0({M_n}) \to \Im _0^1({t^ * }({M_n})) is defined by (8) and so an isomorphism of 10(Mn) \Im _1^0({M_n}) in to 01(t(Mn)) \Im _0^1({t^ * }({M_n})) [10].

We consider in π1 (U) 3n local vector fields B(β̅), C(β) and E(β¯¯) {E_{\left( {\overline {\overline \beta } } \right)}} along βθ (T (Mn)), which are respectively represented by B(β¯)=Bxβ¯,C(β)=Cxβ,E(β¯¯)=Edxβ. {B_{\left( {\overline \beta } \right)}} = B{\partial \over {\partial {x^{\overline \beta }}}},\quad {C_{\left( \beta \right)}} = C{\partial \over {\partial {x^\beta }}},\quad {E_{\left( {\overline {\overline \beta } } \right)}} = Ed{x^\beta }.

Theorem 1

Let X be a vector field on T (Mn). We have along βθ (T (Mn)) the formula ccX=CX+B(LVX)+E(LXθ), ^{cc}X = CX + B\left( {{L_V}X} \right) + E\left( {\; - {L_X}\theta } \right), where LV X denotes the Lie derivative of X with respect to V, and LX θ denotes the Lie derivative of θ with respect to X [10].

On the other hand, on putting C(β¯¯)=E(β¯¯) {C_{\left( {\overline {\overline \beta } } \right)}} = {E_{\left( {\overline {\overline \beta } } \right)}} , we write the adapted frame of βθ (T (Mn)) as {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} . The adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} of βθ (T (Mn)) is given by the matrix A˜=(A˜BA)=(δβαβVα00δβα00βθαδαβ). \widetilde A = \left( {\widetilde A_B^A} \right) = \left( {\matrix{ {\delta _\beta ^\alpha } & {{\partial _\beta }{V^\alpha }} & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {{\partial _\beta }{\theta _\alpha }} & {\delta _\alpha ^\beta } \cr } } \right).

Since the matrix à in (9) is non-singular, it has the inverse. Denoting this inverse by (Ã)−1, we have (A˜)1=(A˜CB)1=(δθβθVβ00δθβ00θθβδβθ), {\left( {\widetilde A} \right)^{ - 1}} = {\left( {\widetilde A_C^B} \right)^{ - 1}} = \left( {\matrix{ {\delta _\theta ^\beta } & { - {\partial _\theta }{V^\beta }} & 0 \cr 0 & {\delta _\theta ^\beta } & 0 \cr 0 & { - {\partial _\theta }{\theta _\beta }} & {\delta _\beta ^\theta } \cr } } \right), where A˜(A˜)1=(A˜BA)(A˜CB)1=δCA=I˜ \widetilde A{\left( {\widetilde A} \right)^{ - 1}} = (\widetilde A_B^A){\left( {\widetilde A_C^B} \right)^{ - 1}} = \delta _C^A = \widetilde I , where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) , C=(θ¯,θ,θ¯¯) C = \left( {\overline \theta ,\theta ,\overline {\overline \theta } } \right) .

Then we see from Theorem 1 that the complete lift ccX of a vector field X01(T(Mn)) X \in \Im _0^1(T({M_n})) has along βθ (T (Mn)) components of the form ccX:(LVXαXαLXθα) ^{cc}X:\left( {\matrix{ {{L_V}{X^\alpha }} \hfill \cr {{X^\alpha }} \hfill \cr { - {L_X}{\theta _\alpha }} \hfill \cr } } \right) with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} [10].

Theorem 2

The complete lift ccX of a vector field X in Mn to t* (Mn) is tangent to the cross-section βθ (T (Mn)) determined by a 1form θ and vector field V in Mn if and only if LXθ=0,LVX=0, {L_X}\theta = 0,{L_V}X = 0, where LV X denotes the Lie derivative of X with respect to V, and LX θ denotes the Lie derivative of θ with respect to X.

BX, CX and also have components: BX=(Xα00),CX=(0Xα0),Eω=(00ωα) BX = \left( {\matrix{ {{X^\alpha }} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right),\quad CX = \left( {\matrix{ 0 \hfill \cr {{X^\alpha }} \hfill \cr 0 \hfill \cr } } \right),\quad E\omega = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {{\omega _\alpha }} \hfill \cr } } \right) respectively, with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} of the cross-section βθ (T (Mn)) determined by a 1-form θ on T (Mn) [10].

Complete Lift of Tensor Fields of Type (1,1) on a Cross-Section in Semi-Cotangent Bundle

Suppose now that F11(T(Mn)) F \in \Im _1^1(T({M_n})) and F has local components Fβα F_\beta ^\alpha in a neighborhood U of Mn, F=Fβααdxβ F = F_\beta ^\alpha {\partial _\alpha } \otimes d{x^\beta } . Then the semi-cotangent (pull-back) bundle t* (Mn) of cotangent bundle T* (Mn) by using projection of the tangent bundle T (Mn) admits the complete lift ccF of F with components [11]: ccF=(ccFJI)=(FβαyεεFβα00Fβα00pσ(βFασαFβσ)Fαβ), ^{cc}F = ({\;^{^{cc}}}F_J^I) = \left( {\matrix{ {F_\beta ^\alpha } & {{y^\varepsilon }{\partial _\varepsilon }F_\beta ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & {{p_\sigma }({\partial _\beta }F_\alpha ^\sigma - {\partial _\alpha }F_\beta ^\sigma )} & {F_\alpha ^\beta } \cr } } \right), with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) on t* (Mn). Then ccF has components FBA F_B^A given by ccF=(ccFBA)=(FβαLVFβα00Fβα00φFθFαβ) ^{cc}F{ = (^{^{cc}}}F_B^A) = \left( {\matrix{ {F_\beta ^\alpha } & {{L_V}F_\beta ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & {{\varphi _F}\theta } & {F_\alpha ^\beta } \cr } } \right) with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} of the cross-section βθ (T (Mn)) determined by a 1-form θ in T (Mn), where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) [10]. Also, the component ccFβα¯¯ ^{^{cc}}F_\beta ^{\overline {\overline \alpha } } of ccFBA ^{^{cc}}F_B^A is defined as Tachibana operator φFθ of F, i.e., ccFβα¯¯=ϕFθ=(βFασαFβσ)θσFβγγθα+Fαγβθγ, {\;^{^{cc}}}F_\beta ^{\overline {\overline \alpha } } = {\phi _F}\theta = ({\partial _\beta }F_\alpha ^\sigma - {\partial _\alpha }F_\beta ^\sigma ){\theta _\sigma } - F_\beta ^\gamma {\partial _\gamma }{\theta _\alpha } + F_\alpha ^\gamma {\partial _\beta }{\theta _\gamma }, and LVFβα {L_V}F_\beta ^\alpha denotes the Lie derivative of Fβα F_\beta ^\alpha with respect to V, i.e., LVFβα=VγγFβα+FγαβVγFβγγVα. {L_V}F_\beta ^\alpha = {V^\gamma }{\partial _\gamma }F_\beta ^\alpha + F_\gamma ^\alpha {\partial _\beta }{V^\gamma } - F_\beta ^\gamma {\partial _\gamma }{V^\alpha }.

Adapted Frames and Diagonal Lifts of Affinor Fields

Let ∇ be a symmetric affine connection in Mn. In each coordinate neighborhood {U,xα} of Mn, we put X(α)=xα,θ(α)=dxα. {X_{\left( \alpha \right)}} = {\partial \over {\partial {x^\alpha }}},\quad {\theta ^{\left( \alpha \right)}} = d{x^\alpha }.

Then 3n local vector fields Y(α), HHX(α) and vvθ (α) have respectively components of the form Y(α):(δαβ00),HHX(α):(ΓβαδαβΓβα),vvθ(α):(00δβα) {Y_{\left( \alpha \right)}}:\left( {\matrix{ {\delta _\alpha ^\beta } \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right),{\quad ^{HH}}{X_{\left( \alpha \right)}}:\left( {\matrix{ { - \Gamma _\beta ^\alpha } \hfill \cr {\delta _\alpha ^\beta } \hfill \cr {{\Gamma _\beta }_\alpha } \hfill \cr } } \right),{\quad ^{vv}}{\theta ^{\left( \alpha \right)}}:\left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {\delta _\beta ^\alpha } \hfill \cr } } \right) with respect to the induced coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in π−1 (U), where we have used (7). We call the set {Y(α),HH X(α),vv θ (α)} the frame adapted to the symmetric affine connection ∇ in π−1 (U). On putting e^(α¯)=Y(α),e^(α)=HHX(α),e^(α¯¯)=vvθ(α) {\widehat e_{\left( {\overline \alpha } \right)}} = {Y_{\left( \alpha \right)}},\quad {\widehat e_{\left( \alpha \right)}}{ = ^{HH}}{X_{\left( \alpha \right)}},\quad {\widehat e_{\left( {\overline {\overline \alpha } } \right)}}{ = ^{vv}}{\theta ^{\left( \alpha \right)}} we write the adapted frame as {e^(B)}={e^(α¯),e^(α),e^(α¯¯)}. \left\{ {{{\widehat e}_{\left( B \right)}}} \right\} = \left\{ {{{\widehat e}_{\left( {\overline \alpha } \right)}},{{\widehat e}_{\left( \alpha \right)}},{{\widehat e}_{\left( {\overline {\overline \alpha } } \right)}}} \right\}.

The adapted frame {e^(B)}={e^(α¯),e^(α),e^(α¯¯)} \left\{ {{{\widehat e}_{\left( B \right)}}} \right\} = \left\{ {{{\widehat e}_{\left( {\overline \alpha } \right)}},{{\widehat e}_{\left( \alpha \right)}},{{\widehat e}_{\left( {\overline {\overline \alpha } } \right)}}} \right\} is given by the matrix A^=(A^BA)=(δβαΓβα00δβα00Γβαδαβ). \widehat A = \left( {\widehat A_B^A} \right) = \left( {\matrix{ {\delta _\beta ^\alpha } & { - \Gamma _\beta ^\alpha } & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {{\Gamma _\beta }_\alpha } & {\delta _\alpha ^\beta } \cr } } \right).

Since the matrix A^ \widehat A in (17) is non-singular, it has the inverse. Denoting this inverse by (A^)1 {\left( {\widehat A} \right)^{ - 1}} , we have (A^)1=(A^CB)1=(δθβΓθβ00δθβ00Γθβδβθ), {\left( {\widehat A} \right)^{ - 1}} = {\left( {\widehat A_C^B} \right)^{ - 1}} = \left( {\matrix{ {\delta _\theta ^\beta } & {\Gamma _\theta ^\beta } & 0 \cr 0 & {\delta _\theta ^\beta } & 0 \cr 0 & { - {\Gamma _\theta }_\beta } & {\delta _\beta ^\theta } \cr } } \right), where A^(A^)1=(A^BA)(A^CB)1=δCA=I˜ \widehat A{\left( {\widehat A} \right)^{ - 1}} = (\widehat A_B^A){\left( {\widehat A_C^B} \right)^{ - 1}} = \delta _C^A = \widetilde I , where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) , C=(θ¯,θ,θ¯¯) C = \left( {\overline \theta ,\theta ,\overline {\overline \theta } } \right) .

If we take account of (16), we see that the diagonal lift DDF of F11(T(Mn)) F \in \Im _1^1(T({M_n})) has components [10]: DDF=(DDFJI)=(FβαΓεαFβεΓβεFεα00Fβα00ΓβσFασ+ΓασFβσFαβ), ^{^{DD}}F{ = (^{^{DD}}}F_J^I) = \left( {\matrix{ { - F_\beta ^\alpha } & { - \Gamma _\varepsilon ^\alpha F_\beta ^\varepsilon - \Gamma _\beta ^\varepsilon F_\varepsilon ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & {{\Gamma _\beta }_\sigma F_\alpha ^\sigma + {\Gamma _\alpha }_\sigma F_\beta ^\sigma } & { - F_\alpha ^\beta } \cr } } \right), with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) on t* (Mn), where Γεα=yγΓγαε,Γασ=pγΓαγσ. {\Gamma _\varepsilon ^\alpha} = {y^\gamma }{{\Gamma_\gamma^\alpha}_\varepsilon},\quad {\Gamma _{\alpha\sigma} } = {p_\gamma }{{\Gamma_\alpha^\gamma}_\sigma}. which proves (19).

We now see, from (16), that the diagonal lift DDF of F11(T(Mn)) F \in \Im _1^1(T({M_n})) has components of the form DDF=(DDFBA)=(Fβα000Fβα000Fαβ) ^{^{DD}}F{ = (^{^{DD}}}F_B^A) = \left( {\matrix{ { - F_\beta ^\alpha } & 0 & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & 0 & { - F_\alpha ^\beta } \cr } } \right) with respect to the adapted frame {e^(B)} \left\{ {{{\widehat e}_{\left( B \right)}}} \right\} in t* (Mn).

We now obtain from (19) that the diagonal lift DDF of an affinor field F11(T(Mn)) F \in \Im _1^1(T({M_n})) has along βθ (T (Mn)) components of the form [10]: DDF:(Fβα(εVα)Fβε(βVε)Fεα00Fβα00(βθσ)Fασ(αθσ)FβσFαβ), ^{^{DD}}F:\left( {\matrix{ { - F_\beta ^\alpha } & { - \left( {{\nabla _\varepsilon }{V^\alpha }} \right)F_\beta ^\varepsilon - \left( {{\nabla _\beta }{V^\varepsilon }} \right)F_\varepsilon ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & { - \left( {{\nabla _\beta }{\theta _\sigma }} \right)F_\alpha ^\sigma - \left( {{\nabla _\alpha }{\theta _\sigma }} \right)F_\beta ^\sigma } & { - F_\alpha ^\beta } \cr } } \right), with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} .

Then we see from (7) that the horizontal lift HHX of a vector field X01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) has along βθ (T (Mn)) components of the form HHX:(Xβ(βVα)Xα(βθα)Xβ) ^{HH}X:\left( {\matrix{ { - {X^\beta }\left( {{\nabla _\beta }{V^\alpha }} \right)} \hfill \cr {{X^\alpha }} \hfill \cr { - \left( {{\nabla _\beta }{\theta _\alpha }} \right){X^\beta }} \hfill \cr } } \right) with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} [10].

Using (7), (20) and (21), we have along βθ (T (Mn)):

Theorem 3

If F and X are affinor and vector fields on T (Mn), and ω10(Mn) \omega \in \Im _1^0({M_n}) , then with respect to a symetric affine connectionin Mn, we have

(i) DDF (HHX) = HH (FX),

(ii) DDF (vvω) = −vv (ωF) [10].

Theorem 4

If F,G11(Mn) F,G \in \Im _1^1({M_n}) , then with respect to a symetric affine connectionin Mn, we have DDFDDG+DDGDDF=HH(FG+GF). ^{DD}{F^{DD}}G{ + ^{DD}}{G^{DD}}F{ = ^{HH}}\left( {FG + GF} \right).

Proof

If ω10(Mn) \omega \in \Im _1^0({M_n}) and X01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) , then by Theorem 3 and ([11], HHFHHX =HH (FX)), we have (DDFDDG+DDGDDF)HHX=DDF(HHGX)+DDG(HHFX)=HH(FGX)+HH(GFX)=HH((FG+GF)X)=HH(FG+GF)HHX. \matrix{ {{{{(^{DD}}{F^{DD}}G{ + ^{DD}}{G^{DD}}F)}^{HH}}X ={ ^{DD}}F{(^{HH}}GX){ + ^{DD}}G{(^{HH}}FX)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {^{HH}}(FGX){ + ^{HH}}(GFX)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {^{HH}}(\left( {FG + GF} \right)X)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; ={ ^{HH}}{{\left( {FG + GF} \right)}^{HH}}X.} \hfill \cr }

Thus, we have DDFDDG +DD GDDF = HH (FG + GF).

Theorem 5

If F,G11(Mn) F,G \in \Im _1^1({M_n}) , then with respect to a symetric affine connectionin Mn, we have DDFHHG+DDGHHF=HHFDDG+HHGDDF=DD(FG+GF). ^{DD}{F^{HH}}G{ + ^{DD}}{G^{HH}}F{ = ^{HH}}{F^{DD}}G{ + ^{HH}}{G^{DD}}F{ = ^{DD}}\left( {FG + GF} \right).

Proof

If ω10(Mn) \omega \in \Im _1^0({M_n}) and F,G11(T(Mn)) F,G \in \Im _1^1(T({M_n})) , then by Theorem 3 and ([11], HHFvvω =vv (ωF) we have (DDFDDG+DDGDDF)vvω=DDF(vv(ωG))DDG(vv(ωF))=vv(ωGF)vv(ωFG)=vv(ω(GF+FG))=HH(GF+FG)vvω. \matrix{ {{{{(^{DD}}{F^{DD}}G{ + ^{DD}}{G^{DD}}F)}^{vv}}\omega = { - ^{DD}}F{(^{vv}}\left( {\omega \circ G} \right)){ - ^{DD}}G{(^{vv}}\left( {\omega \circ F} \right))} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {^{vv}}\left( {\omega \circ GF} \right){ - ^{vv}}\left( {\omega \circ FG} \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {^{vv}}\left( {\omega \circ (GF + FG} \right))} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {^{HH}}{{(GF + FG)}^{vv}}\omega .} \hfill \cr }

Thus, we have Theorem 5.

Putting F = G in Theorem 4 and Theorem 5, we have HHFDDF=DDFHHF=DD(F2)(DDF)2p=HH(F2p),(DDF)2p+1=DD(F2p+1),(p=1,2,...) \matrix{ {^{HH}{F^{DD}}F{ = ^{DD}}{F^{HH}}F{ = ^{DD}}({F^2})} \hfill \cr {{{{(^{DD}}F)}^{2p}}{ = ^{HH}}({F^{2p}}{{),(}^{DD}}F{)^{2p + 1}}{ = ^{DD}}({F^{2p + 1}}),(p = 1,2,...)} \hfill \cr } for any F11(T(Mn)) F \in \Im _1^1(T({M_n})) .

Theorem 6

The diagonal lift J^ \widehat J of the identity tensor field I of type (1,1) has the components J^:(δβα2Γβα00δβα002Γβαδαβ). \widehat J:\left( {\matrix{ { - \delta _\beta ^\alpha } & {2\Gamma _\beta ^\alpha } & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {2{\Gamma _\beta }_\alpha } & { - \delta _\alpha ^\beta } \cr } } \right).

Proof

Since the matrix J^ \widehat J in (22) is non-singular, it has the inverse. Denoting this inverse by (J^)1 {\left( {\widehat J} \right)^{ - 1}} , we have (J^)1=(J^CB)1=(δθβ2Γθβ00δθβ002Γθβδβθ), {\left( {\widehat J} \right)^{ - 1}} = {\left( {\widehat J_C^B} \right)^{ - 1}} = \left( {\matrix{ { - \delta _\theta ^\beta } & {2\Gamma _\theta ^\beta } & 0 \cr 0 & {\delta _\theta ^\beta } & 0 \cr 0 & {2{\Gamma _\theta }_\beta } & { - \delta _\beta ^\theta } \cr } } \right), where J^(J^)1=(J^BA)(J^CB)1=δCA=I˜ \widehat J{\left( {\widehat J} \right)^{ - 1}} = (\widehat J_B^A){\left( {\widehat J_C^B} \right)^{ - 1}} = \delta _C^A = \widetilde I , where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) , C=(θ¯,θ,θ¯¯) C = \left( {\overline \theta ,\theta ,\overline {\overline \theta } } \right) . In fact, from (22) and (23), we easily see that J^(J^)1=(J^BA)(J^CB)1=(δβα2Γβα00δβα002Γβαδαβ)(δθβ2Γθβ00δθβ002Γθβδβθ)=(δθα2Γθα2Γθα00δθα002Γθα2Γθαδαθ)=(δθα000δθα000δαθ)=δCA=I^. \matrix{ {\widehat J{{\left( {\widehat J} \right)}^{ - 1}} = (\widehat J_B^A){{\left( {\widehat J_C^B} \right)}^{ - 1}}} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\; = \left( {\matrix{ { - \delta _\beta ^\alpha } & {2\Gamma _\beta ^\alpha } & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {2{\Gamma _\beta }_\alpha } & { - \delta _\alpha ^\beta } \cr } } \right)\left( {\matrix{ { - \delta _\theta ^\beta } & {2\Gamma _\theta ^\beta } & 0 \cr 0 & {\delta _\theta ^\beta } & 0 \cr 0 & {2{\Gamma _\theta }_\beta } & { - \delta _\beta ^\theta } \cr } } \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\; = \left( {\matrix{ {\delta _\theta ^\alpha } & {2\Gamma _\theta ^\alpha - 2\Gamma _\theta ^\alpha } & 0 \cr 0 & {\delta _\theta ^\alpha } & 0 \cr 0 & {2{\Gamma _\theta }_\alpha - 2{\Gamma _\theta }_\alpha } & {\delta _\alpha ^\theta } \cr } } \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\; = \left( {\matrix{ {\delta _\theta ^\alpha } & 0 & 0 \cr 0 & {\delta _\theta ^\alpha } & 0 \cr 0 & 0 & {\delta _\alpha ^\theta } \cr } } \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\; = \delta _C^A} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\; = \widehat I.} \hfill \cr }

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