1. bookVolume 6 (2021): Issue 1 (January 2021)
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On Pull-Back Bundle of Tensor Bundles Defined by Projection of The Cotangent Bundle

Published Online: 31 Dec 2020
Page range: 421 - 428
Received: 17 Jul 2019
Accepted: 26 Oct 2019
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Abstract

Using projection (submersion) of the cotangent bundle T*M over a manifold M, we define a semi-tensor (pull-back) bundle tM of type (p,q). The aim of this study is to investigate complete lift of vector fields in a special class of semi-tensor bundle tM of the type (p,q). We also have a new example for good square in this work.

Keywords

MSC 2010

Introduction

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 the tangent bundles and cotangant bundles [1,2,3, 12, 13]. One of these studies is investigating complete lift of vector fields in a special class of semi-tensor (pull-back) bundle tM of type (p,q).

Let Mn be an n-dimensional differentiable manifold of class C, and let (T*(Mn), π1, Mn) be a cotangent bundle over Mn. We use the notation (xi) = xα̅,xα, where the indices i, j,... run from 1 to 2n, the indices α̅,β̅,... from 1 to n and the indices α,β, ... from n + 1 to 2n, xα are coordinates in Mn, xα̅ = pα are fibre coordinates of the cotangent bundle T*(Mn).

Let now (Tqp(Mn),π˜,Mn) \left( {T_q^p({M_n}),\widetilde \pi ,{M_n}} \right) be a tensor bundle [5], [9], [ [10], p.118] with base space Mn, and let T*(Mn) be cotangent bundle determined by a natural projection (submersion) π1 : T*(Mn) → Mn. The semi-tensor bundle (induced, pull-back [6], [7], [9], [11], [13]]) of the tensor bundle (Tqp(Mn),π˜,Mn) \left( {T_q^p({M_n}),\widetilde \pi ,{M_n}} \right) is the bundle (tqp(Mn),π2,T(Mn)) \left( {t_q^p({M_n}),{\pi _2},{T^ * }({M_n})} \right) over cotangent bundle T*(Mn) with a total space tqp(Mn)={((xα¯,xα),xα¯¯)T(Mn)×(Tqp)x(Mn):π1(xα¯,xα)=π˜(xα,xα¯¯)=(xα)}T(Mn)×(Tqp)x(Mn) t_q^p({M_n}) = \left\{ {(\left( {{x^{\overline \alpha }},{x^\alpha }} \right),{x^{\overline {\overline \alpha } }}) \in {T^ * }({M_n}) \times {{\left( {T_q^p} \right)}_x}({M_n}):{\pi _1}\left( {{x^{\overline \alpha }},{x^\alpha }} \right) = \widetilde \pi \left( {{x^\alpha },{x^{\overline {\overline \alpha } }}} \right) = \left( {{x^\alpha }} \right)} \right\} \subset {T^ * }({M_n}) \times {\left( {T_q^p} \right)_x}({M_n}) and with the projection map π2:tqp(Mn)T(Mn) {\pi _2}:t_q^p({M_n}) \to {T^ * }({M_n}) defined by π2(xα¯,xα,xα¯¯)=(xα¯,xα) {\pi _2}({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) = \left( {{x^{\overline \alpha }},{x^\alpha }} \right) , where (Tqp)x(Mn)(x=π1(x˜),x˜=(xα¯,xα)T(Mn)) {\left( {T_q^p} \right)_x}({M_n})\left( {x = {\pi _1}\left( {\widetilde x} \right),\widetilde x = \left( {{x^{\overline \alpha }},{x^\alpha }} \right) \in {T^ * }({M_n})} \right) is the tensor space at a point x of Mn, where xα¯¯=tα1...αqβ1...βp(α¯¯,β¯¯,...=2n+1,...,2n+np+q) {x^{\overline {\overline \alpha } }} = t_{{\alpha _1}...{\alpha _q}}^{{\beta _1}...{\beta _p}}\left( {\overline {\overline \alpha } ,\overline {\overline \beta } ,... = 2n + 1,...,2n + {n^{p + q}}} \right) are fiber coordinates of the tensor bundle Tqp(Mn) T_q^p({M_n}) .

The pull-back (semi-tensor) bundle tqp(Mn) t_q^p({M_n}) of tensor bundle Tqp(Mn) T_q^p({M_n}) has the natural bundle structure over Mn, its bundle projection π:tqp(Mn)Mn \pi :t_q^p({M_n}) \to {M_n} being defined by π:(xα¯,xα,xα¯¯)(xα) \pi :({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^\alpha }) . If we introduce a mapping π2:tqp(Mn)T*(Mn) {\pi _2}:t_q^p({M_n}) \to {T^*}({M_n}) by π2:(xα¯,xα,xα¯¯)(xα¯,xα) {\pi _2}:({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^{\overline \alpha }},{x^\alpha }) , then tqp(Mn) t_q^p({M_n}) has a bundle structure over Mn. Hence π = π1π2.

Now, consider a diagram as AγBαβCπD \matrix{ A & {\mathop \to \limits^\gamma } & B \cr {^\alpha \downarrow } & {} & {{ \downarrow ^\beta }} \cr C & {\mathop \to \limits_\pi } & D \cr }

A good square of vector bundles is a diagram as above verifying

α and β are fibre bundles, but not necessarily vector bundles;

γ and π are vector bundles;

the square is commutative, i.e., πα = βγ;

the local expression

AγBαβCπDUn×Rr×Gs×RtγUn×GsUn×RrπUn(xi,aa,gλ,bσ)γ(xi,gλ)(xi,aa)π(xi) \matrix{ {\matrix{ A & {\mathop \to \limits^\gamma } & B \cr {^\alpha \downarrow } & {} & {{ \downarrow ^\beta }} \cr C & {\mathop \to \limits_\pi } & D \cr } } & {\matrix{ {{U^n} \times {R^r} \times {G^s} \times {R^t}} & {\mathop \to \limits^\gamma } & {{U^n} \times {G^s}} \cr \downarrow & {} & \downarrow \cr {{U^n} \times {R^r}} & {\mathop \to \limits_\pi } & {{U^n}} \cr } } & {\matrix{ {({x^i},{a^a},{g^\lambda },{b^\sigma })} & {\mathop \to \limits^\gamma } & {\left( {{x^i},{g^\lambda }} \right)} \cr \downarrow & {} & \downarrow \cr {({x^i},{a^a})} & {\mathop \to \limits_\pi } & {\left( {{x^i}} \right)} \cr } } \cr } where G is a manifold and superindices denote the dimension of the manifolds [3].

By means of above definition, we have

Theorem 1

Let now π:tqp(Mn)Mn \pi :t_q^p({M_n}) \to {M_n} be a semi-tensor bundle and π1 : T*(Mn) → Mn be a cotangent bundle. Then, the following is a good square: tqp(Mn)π2T(Mn)idπ1tqp(Mn)πMnT(Mn)×(Tqp)x(Mn)π2T(Mn)idπ1T(Mn)×(Tqp)x(Mn)πMn(xα¯,xα,xα¯¯)π2(xα¯,xα)idπ1(xα¯,xα,xα¯¯)π(xα) \matrix{ {\matrix{ {t_q^p({M_n})} & {\mathop \to \limits^{{\pi _2}} } & {{T^ * }({M_n})} \cr {^{id} \downarrow } & {} & {{ \downarrow ^{{\pi _1}}}} \cr {t_q^p({M_n})} & {\mathop \to \limits_\pi } & {{M_n}} \cr } } & {\matrix{ {{T^ * }({M_n}) \times {{\left( {T_q^p} \right)}_x}({M_n})} & {\mathop \to \limits^{{\pi _2}} } & {{T^ * }({M_n})} \cr {^{id} \downarrow } & {} & {{ \downarrow ^{{\pi _1}}}} \cr {{T^ * }({M_n}) \times {{\left( {T_q^p} \right)}_x}({M_n})} & {\mathop \to \limits_\pi } & {{M_n}} \cr } } & {\matrix{ {({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})} & {\mathop \to \limits^{{\pi _2}} } & {\left( {{x^{\overline \alpha }},{x^\alpha }} \right)} \cr {^{id} \downarrow } & {} & {{ \downarrow ^{{\pi _1}}}} \cr {({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})} & {\mathop \to \limits_\pi } & {\left( {{x^\alpha }} \right)} \cr } } \cr }

If (xi')=(xα¯',xα',xα¯¯') ({x^{{i^\prime}}}) = ({x^{{{\overline \alpha }^\prime}}},{x^{{\alpha ^\prime}}},{x^{{{\overline {\overline \alpha } }^\prime}}}) is another system of local adapted coordinates in the semi-tensor bundle tqp(Mn) t_q^p({M_n}) , then we have {xα¯'=xβxα'pβ,xα'=xα'(xβ),xα¯¯'=tα1'...αq'β1'...βp'=Aα1...αpβ1'...βp'Aα1'...αq'β1...βqtβ1...βqα1...αp=A(α)(β')A(α')(β)xβ¯. \left\{ {\matrix{ {{x^{{{\overline \alpha }^\prime}}} = {{\partial {x^\beta }} \over {\partial {x^{{\alpha ^\prime}}}}}{p_\beta },} \hfill \cr {{x^{{\alpha ^\prime}}} = {x^{{\alpha ^\prime}}}\left( {{x^\beta }} \right),} \hfill \cr {{x^{{{\overline {\overline \alpha } }^\prime}}} = t_{\alpha _1^\prime...\alpha _q^\prime}^{\beta _1^\prime...\beta _p^\prime} = A_{{\alpha _1}...{\alpha _p}}^{\beta _1^\prime...\beta _p^\prime}A_{\alpha _1^\prime...\alpha _q^\prime}^{{\beta _1}...{\beta _q}}t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}} = A_{(\alpha )}^{({\beta ^\prime})}A_{({\alpha ^\prime})}^{(\beta )}{x^{\overline \beta }}{\rm{.}}} \hfill \cr } } \right.

The jacobian of (1) has components A¯=(AJI')=(Aα'βpσAββ'Aβ'α'σ00Aβα'00t(σ)(α)βA(α)(β')A(α')(σ)A(α)(β')A(α')(β)), \bar A = \left( {A_J^{{I^\prime}}} \right) = \left( {\matrix{ {A_{{\alpha ^\prime}}^\beta } & {{p_\sigma }A_\beta ^{{\beta ^\prime}}A_{{\beta ^\prime}{\alpha ^\prime}}^\sigma } & 0 \cr 0 & {A_\beta ^{{\alpha ^\prime}}} & 0 \cr 0 & {t_{(\sigma )}^{(\alpha )}{\partial _\beta }A_{(\alpha )}^{({\beta ^\prime})}A_{({\alpha ^\prime})}^{(\sigma )}} & {A_{(\alpha )}^{({\beta ^\prime})}A_{({\alpha ^\prime})}^{(\beta )}} \cr } } \right), where I=(α¯,α,α¯¯) I = (\overline \alpha ,\alpha ,\overline {\overline \alpha } ) , J=(β¯,β,β¯¯) J = (\overline \beta ,\beta ,\overline {\overline \beta } ) , I,J...=1,...,2n+np+q, t(σ)(α)=tσ1...σqα1...αp t_{(\sigma )}^{(\alpha )} = t_{{\sigma _1}...{\sigma _q}}^{{\alpha _1}...{\alpha _p}} , Aβα'=xα'xβ A_\beta ^{{\alpha ^\prime}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}} , Aα'β=xβxα' A_{{\alpha ^\prime}}^\beta = {{\partial {x^\beta }} \over {\partial {x^{{\alpha ^\prime}}}}} , Aβ'α'σ=2xσxβ'xα' A_{{\beta ^\prime}{\alpha ^\prime}}^\sigma = {{{\partial ^2}{x^\sigma }} \over {\partial {x^{{\beta ^\prime}}}\partial {x^{{\alpha ^\prime}}}}} . It is easily verified that the condition DetĀ ≠ 0 is equivalent to the condition: Det(Aα'β)0,Det(Aβα')0,Det(A(α)(β')A(α')(β))0. Det(A_{{\alpha ^\prime}}^\beta ) \ne 0,Det(A_\beta ^{{\alpha ^\prime}}) \ne 0,Det(A_{(\alpha )}^{({\beta ^\prime})}A_{({\alpha ^\prime})}^{(\beta )}) \ne 0.

Also, dimtqp(Mn)=2n+np+q \dim t_q^p({M_n}) = 2n + {n^{p + q}} .

We note that special class of semi-tensor bundle was examined in [4]. The main purpose of this paper is to study semi-tensor (pull-back) bundle tqp(Mn) t_q^p({M_n}) of tensor bundle Tqp(Mn) T_q^p({M_n}) by using projection of the cotangent bundle T*(Mn).

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.

Some lifts of tensor fields and γ−Operator

Let X01(T(Mn)) X \in \Im _0^1({T^ * }({M_n})) , i.e. X = Xαα. The complete lift cX of X to cotangent bundle is defined by cX = Xααpβ (αXβ)α̅ [ [12], p.236]. On putting ccX=(ccXβ¯ccXβccXβ¯¯)=(pε(βXε)Xβλ=1ptβ1...βqα1...ε...αpεXαλμ=1qtβ1...ε...βqα1...αpβμXε), ^{cc}X = \left( {\matrix{ {^{cc}{X^{\overline \beta }}} \hfill \cr {^{cc}{X^\beta }} \hfill \cr {^{cc}{X^{\overline {\overline \beta } }}} \hfill \cr } } \right) = \left( {\matrix{ { - {p_\varepsilon }({\partial _\beta }{X^\varepsilon })} \hfill \cr {{X^\beta }} \hfill \cr {\sum\nolimits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{X^{{\alpha _\lambda }}} - \sum\nolimits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{X^\varepsilon }} \hfill \cr } } \right), from (2), we easily see that ccX =Ā(ccX). The vector field ccX is called the complete lift of cX01(T(Mn)) ^cX \in \Im _0^1({T^ * }({M_n})) to tqp(Mn) t_q^p({M_n}) .

Now, consider Aqp(T(Mn)) A \in \Im _q^p({T^ * }({M_n})) and φ11(Mn) \varphi \in \Im _1^1({M_n}) , then vvA01(tqp(Mn)) ^{vv}A \in \Im _0^1(t_q^p({M_n})) (vertical lift), γφ01(tqp(Mn)) \gamma \varphi \in \Im _0^1(t_q^p({M_n})) and γ˜φ01(tqp(Mn)) \widetilde \gamma \varphi \in \Im _0^1(t_q^p({M_n})) have respectively, components on the semi-tensor bundle tqp(Mn) t_q^p({M_n}) [13] vvA=(vvA)I=(vvAavvAαvvAα¯)=(00Aβ1...βqα1...αp),γφ=(γφ)I=(00λ=1ptβ1...βqα1...ε...αpφεαλ),γ˜φ=(γ˜φ)I=(00μ=1qtβ1...ε...βqα1...αpφβμε) \matrix{ {^{vv}A = {{\left( {^{vv}A} \right)}^I} = \left( {\matrix{ {^{vv}{A^a}} \hfill \cr {^{vv}{A^\alpha }} \hfill \cr {^{vv}{A^{\overline \alpha }}} \hfill \cr } } \right) = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}} \hfill \cr } } \right),} \cr {\gamma \varphi = {{\left( {\gamma \varphi } \right)}^I} = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {\sum\nolimits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}\varphi _\varepsilon ^{{\alpha _\lambda }}} \hfill \cr } } \right),} \cr {\widetilde \gamma \varphi = {{\left( {\widetilde \gamma \varphi } \right)}^I} = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {\sum\nolimits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}\varphi _{{\beta _\mu }}^\varepsilon } \hfill \cr } } \right)} \cr } with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) on tqp(Mn) t_q^p({M_n}) , where Aβ1...βqα1...αp A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}} , φεαλ \varphi _\varepsilon ^{{\alpha _\lambda }} and φβμε \varphi _{{\beta _\mu }}^\varepsilon are local components of A and φ.

On the other hand, vv f the vertical lift of function f00(Mn) f \in \Im _0^0({M_n}) on tqp(Mn) t_q^p({M_n}) is defined by [9]: vvf=vfπ2=fπ1π2=fπ. ^{vv}f{ = ^v}f \circ {\pi _2} = f \circ {\pi _1} \circ {\pi _2} = f \circ \pi .

Theorem 2

For any vector fields X, Y on T*(Mn) and f00(Mn) f \in \Im _0^0({M_n}) , we have

(i) cc(X +Y ) =cc X +cc Y,

(ii) ccXvv f =vv (X f ).

Proof

This immediately follows from (3).

Let X01(T(Mn)) X \in \Im _0^1({T^ * }({M_n})) . Then we get by (3) and (5): ccXvvf=ccXII(vvf)ccXvvf=ccXα¯α¯(vvf)0+ccXαα(vvf)+ccXα¯¯α¯¯(vvf)0=Xαα(vvf)=vv(Xf), \matrix{ {^{cc}{X^{vv}}f ={ ^{cc}}{X^I}{\partial _I}{(^{vv}}f)} \hfill \cr {^{cc}{X^{vv}}f={ ^{cc}}{X^{\overline \alpha }}\underbrace {{\partial _{\overline \alpha }}{(^{vv}}f)}_0{ + ^{cc}}{X^\alpha }{\partial _\alpha }{(^{vv}}f){ + ^{cc}}{X^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{(^{vv}}f)}_0} \hfill \cr {\;\;\;\;\;\;\;\;\;\; = {X^\alpha }{\partial _\alpha }\left( {^{vv}f} \right)} \hfill \cr {\;\;\;\;\;\;\;\;\;\; ={ ^{vv}}(Xf),} \hfill \cr } which gives (ii) of Theorem 2.

Theorem 3

If φ11(Mn) \varphi \in \Im _1^1({M_n}) , f00(Mn) f \in \Im _0^0({M_n}) and Aqp(T(Mn)) A \in \Im _q^p({T^ * }({M_n})) , then

(i) (vvA)vv f = 0,

(ii) (γϕ)(vv f ) = 0,

(iii) (γ̃ϕ)(vv f ) = 0.

Proof

If Aqp(T(Mn)) A \in \Im _q^p({T^ * }({M_n})) , then, by (4) and (5), we find (vvA)vvf=(vvA)II(vvf)=(vvA)α¯0α¯(vvf)+(vvA)α0α(vvf)+(vvA)α¯¯α¯¯(vvf)0=0. \matrix{ {{{\left( {^{vv}A} \right)}^{vv}}f = {{\left( {^{vv}A} \right)}^I}{\partial _I}{(^{vv}}f)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\, =\; \underbrace {{{\left( {^{vv}A} \right)}^{\overline \alpha }}}_0{\partial _{\overline \alpha }}{(^{vv}}f) + \underbrace {{{\left( {^{vv}A} \right)}^\alpha }}_0{\partial _\alpha }{(^{vv}}f) + {{\left( {^{vv}A} \right)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{(^{vv}}f)}_0} \hfill \cr {\;\;\,\;\;\;\;\;\;\;\;\;\; = 0.} \hfill \cr } Thus, we have (i) of Theorem 3.

If φ11(Mn) \varphi \in \Im _1^1({M_n}) , then we have by (4) and (5): (γφ)(vvf)=(γφ)II(vvf)=(γφ)α¯0α¯(vvf)+(γφ)α0α(vvf)+(γφ)α¯¯α¯¯(vvf)0=0. \matrix{ {\left( {\gamma \varphi } \right)\left( {^{vv}f} \right) = {{\left( {\gamma \varphi } \right)}^I}{\partial _I}{(^{vv}}f)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \;\underbrace {{{\left( {\gamma \varphi } \right)}^{\overline \alpha }}}_0{\partial _{\overline \alpha }}{(^{vv}}f) + \underbrace {{{\left( {\gamma \varphi } \right)}^\alpha }}_0{\partial _\alpha }{(^{vv}}f) + {{\left( {\gamma \varphi } \right)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{(^{vv}}f)}_0} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\; = 0.} \hfill \cr } Thus, we have (ii) of Theorem 3.

If φ11(Mn) \varphi \in \Im _1^1({M_n}) , then we have by (4) and (5): (γ˜φ)(vvf)=(γ˜φ)II(vvf)=(γ˜φ)α¯0α¯(vvf)+(γ˜φ)α0α(vvf)+(γ˜φ)α¯¯α¯¯(vvf)0=0. \matrix{ {\left( {\widetilde \gamma \varphi } \right)\left( {^{vv}f} \right) = {{\left( {\widetilde \gamma \varphi } \right)}^I}{\partial _I}{(^{vv}}f)} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\; =\; \underbrace {{{\left( {\widetilde \gamma \varphi } \right)}^{\overline \alpha }}}_0{\partial _{\overline \alpha }}{(^{vv}}f) + \underbrace {{{\left( {\widetilde \gamma \varphi } \right)}^\alpha }}_0{\partial _\alpha }{(^{vv}}f) + {{\left( {\widetilde \gamma \varphi } \right)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{(^{vv}}f)}_0} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = 0.} \hfill \cr } Thus, we have (iii) of Theorem 3.

Theorem 4

Let A,Bqp(T(Mn)) A,B \in \Im _q^p({T^ * }({M_n})) . For the Lie product, we have [vvA,vvB]=0. \left[ {^{vv}A{,^{vv}}B} \right] = 0.

Proof

If A,Bqp(T(Mn)) A,B \in \Im _q^p({T^ * }({M_n})) and ([vvA,vvB]b[vvA,vvB]β[vvA,vvB]β¯) \left( {\matrix{ {{{{[^{vv}}A{,^{vv}}B]}^b}} \hfill \cr {{{{[^{vv}}A{,^{vv}}B]}^\beta }} \hfill \cr {{{{[^{vv}}A{,^{vv}}B]}^{\overline \beta }}} \hfill \cr } } \right) are components of [vvA, vvB]J with respect to the coordinates (xβ¯,xβ,xβ¯¯) ({x^{\overline \beta }},{x^\beta },{x^{\overline {\overline \beta } }}) on tqp(Mn) t_q^p({M_n}) , then we have [vvA,vvB]J=(vvA)II(vvB)J(vvB)II(vvA)J=(vvA)α¯0α¯(vvB)J+(vvA)α0α(vvB)J+(vvA)α¯¯α¯¯(vvB)J(vvB)α¯0α¯(vvA)J(vvB)α0α(vvA)J(vvB)α¯¯α¯¯(vvA)J=Aβ1...βqα1...αpα¯¯(vvB)JBβ1...βqα1...αpα¯¯(vvA)J. \matrix{{{{\left[ {^{vv}A{,^{vv}}B} \right]}^J}} \hfill & = \hfill & {{{{(^{vv}}A)}^I}{\partial _I}{{{(^{vv}}B)}^J} - {{{(^{vv}}B)}^I}{\partial _I}{{{(^{vv}}A)}^J}} \hfill \cr {} \hfill & = \hfill & {\underbrace {{{\left( {^{vv}A} \right)}^{\overline \alpha }}}_0{\partial _{\overline \alpha }}{{{(^{vv}}B)}^J} + \underbrace {{{\left( {^{vv}A} \right)}^\alpha }}_0{\partial _\alpha }{{{(^{vv}}B)}^J}} \hfill \cr {} \hfill & {} \hfill & { + {{\left( {^{vv}A} \right)}^{\overline {\overline \alpha } }}{\partial _{\overline {\overline \alpha } }}{{{(^{vv}}B)}^J} - \underbrace {{{{(^{vv}}B)}^{\overline \alpha }}}_0{\partial _{\overline \alpha }}{{{(^{vv}}A)}^J}} \hfill \cr {} \hfill & {} \hfill & { - \underbrace {{{{(^{vv}}B)}^\alpha }}_0{\partial _\alpha }{{{(^{vv}}A)}^J} - {{{(^{vv}}B)}^{\overline {\overline \alpha } }}{\partial _{\overline {\overline \alpha } }}{{{(^{vv}}A)}^J}} \hfill \cr {} \hfill & = \hfill & {A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}{{\left( {^{vv}B} \right)}^J} - B_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}{{\left( {^{vv}A} \right)}^J}.} \hfill \cr }

Firstly, if J = b, we have [vvA,vvB]β¯=Aβ1...βqα1...αpα¯¯(vvB)β¯0Bβ1...βqα1...αpα¯¯(vvA)β¯0=0, [^{vv}A{,^{vv}}B{]^{\overline \beta }} = A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}\underbrace {{{\left( {^{vv}B} \right)}^{\overline \beta }}}_0 - B_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}\underbrace {{{\left( {^{vv}A} \right)}^{\overline \beta }}}_0 = 0, by virtue of (4). Secondly, if J = β, we have [vvA,vvB]β=Aβ1...βqα1...αpα¯¯(vvB)β0Bβ1...βqα1...αpα¯¯(vvA)β0=0, [^{vv}A{,^{vv}}B{]^\beta } = A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}\underbrace {{{\left( {^{vv}B} \right)}^\beta }}_0 - B_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}\underbrace {{{\left( {^{vv}A} \right)}^\beta }}_0 = 0, by virtue of (4). Thirdly, if J=β¯¯ J = \overline {\overline \beta } , then we have [vvA,vvB]β¯¯=Aβ1...βqα1...αpα¯¯(vvB)β¯¯Bβ1...βqα1...αpα¯¯(vvA)β¯¯=Aβ1...βqα1...αpα¯¯Bθ1...θqβ1...βp0Bβ1...βqα1...αpα¯¯Aθ1...θqβ1...βp0=0 \matrix{ [{^{vv}A{,^{vv}}B{]^{\overline {\overline \beta } }} = A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}{{\left( {^{vv}B} \right)}^{\overline {\overline \beta } }} - B_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{\overline {\overline \alpha } }}{{\left( {^{vv}A} \right)}^{\overline {\overline \beta } }}} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}\underbrace {{\partial _{\overline {\overline \alpha } }}B_{{\theta _1}...{\theta _q}}^{{\beta _1}...{\beta _p}}}_0 - B_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}}\underbrace {{\partial _{\overline {\overline \alpha } }}A_{{\theta _1}...{\theta _q}}^{{\beta _1}...{\beta _p}}}_0} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = 0} \hfill \cr } by virtue of (4). Thus, we have Theorem 4.

Theorem 5

Let X,Y01(T(Mn)) X{\rm{,}}Y \in \Im _0^1({T^ * }({M_n})) . For the Lie product, we have [ccX,ccY]=cc[X,Y](i.e.LccXccY=cc(LXY)). [^{cc}X{,^{cc}}Y{] = ^{cc}}[X,Y](i.e.{L_{^{cc}X}}^{cc}Y{ = ^{cc}}\left( {{L_X}Y} \right)).

Proof

If X,Y01(T(Mn)) X{\rm{,}}Y \in \Im _0^1({T^ * }({M_n})) and ([ccX,ccY]β¯[ccX,ccY]β[ccX,ccY]β¯¯) \left( {\matrix{ {{{{[^{cc}}X{,^{cc}}Y]}^{\overline \beta }}} \hfill \cr {{{{[^{cc}}X{,^{cc}}Y]}^\beta }} \hfill \cr {{{{[^{cc}}X{,^{cc}}Y]}^{\overline {\overline \beta } }}} \hfill \cr } } \right) are components of [ccX, ccY ]J with respect to the coordinates (xβ¯,xβ,xβ¯¯) ({x^{\overline \beta }},{x^\beta },{x^{\overline {\overline \beta } }}) on tqp(Mn) t_q^p({M_n}) , then we have [ccX,ccY]J=(ccX)II(ccY)J(ccY)II(ccX)J. [^{cc}X{,^{cc}}Y{]^J}{ = (^{cc}}X{)^I}{\partial _I}{{(^{cc}}Y)^J} - {{(^{cc}}Y)^I}{\partial _I}{{(^{cc}}X)^J}.

Firstly, if J = b, we have [ccX,ccY]β¯=(ccX)II(ccY)β¯(ccY)II(ccX)β¯=(ccX)α¯α¯(ccY)β¯+(ccX)αα(ccY)β¯+(ccX)α¯¯α¯¯(ccY)β¯0(ccY)α¯α¯(ccX)β¯(ccY)αα(ccX)β¯(ccY)α¯¯α¯¯(ccX)β¯0=pεαXε(βYα)Xααpε(βYε)pεαYε(βXα)+Yααpε(βXε)=pε(βYααXεXααβYεβXααYε+YααβXε)=pε(β(XααYεYααXε)[X,Y]ε)=pε(β[X,Y]ε) \matrix{ {{{{[^{cc}}X{,^{cc}}Y]}^{\overline \beta }}} \hfill & = \hfill & {{{{(^{cc}}X)}^I}{\partial _I}{{{(^{cc}}Y)}^{\overline \beta }} - {{{(^{cc}}Y)}^I}{\partial _I}{{{(^{cc}}X)}^{\overline \beta }}} \hfill \cr {} \hfill & = \hfill & {{{{(^{cc}}X)}^{\overline \alpha }}{\partial _{\overline \alpha }}{{{(^{cc}}Y)}^{\overline \beta }} + {{{(^{cc}}X)}^\alpha }{\partial _\alpha }{{{(^{cc}}Y)}^{\overline \beta }} + {{{(^{cc}}X)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{{{(^{cc}}Y)}^{\overline \beta }}}_0} \hfill \cr {} \hfill & {} \hfill & { - {{{(^{cc}}Y)}^{\overline \alpha }}{\partial _{\overline \alpha }}{{{(^{cc}}X)}^{\overline \beta }} - {{{(^{cc}}Y)}^\alpha }{\partial _\alpha }{{{(^{cc}}X)}^{\overline \beta }} - {{{(^{cc}}Y)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{{{(^{cc}}X)}^{\overline \beta }}}_0} \hfill \cr {} \hfill & = \hfill & {{p_\varepsilon }{\partial _\alpha }{X^\varepsilon }({\partial _\beta }{Y^\alpha }) - {X^\alpha }{\partial _\alpha }{p_\varepsilon }({\partial _\beta }{Y^\varepsilon })} \hfill \cr {} \hfill & {} \hfill & { - {p_\varepsilon }{\partial _\alpha }{Y^\varepsilon }({\partial _\beta }{X^\alpha }) + {Y^\alpha }{\partial _\alpha }{p_\varepsilon }({\partial _\beta }{X^\varepsilon })} \hfill \cr {} \hfill & = \hfill & {{p_\varepsilon }({\partial _\beta }{Y^\alpha }{\partial _\alpha }{X^\varepsilon } - {X^\alpha }{\partial _\alpha }{\partial _\beta }{Y^\varepsilon } - {\partial _\beta }{X^\alpha }{\partial _\alpha }{Y^\varepsilon } + {Y^\alpha }{\partial _\alpha }{\partial _\beta }{X^\varepsilon })} \hfill \cr {} \hfill & = \hfill & { - {p_\varepsilon }({\partial _\beta }\underbrace {({X^\alpha }{\partial _\alpha }{Y^\varepsilon } - {Y^\alpha }{\partial _\alpha }{X^\varepsilon })}_{{{[X,Y]}^\varepsilon }})} \hfill \cr {} \hfill & = \hfill & { - {p_\varepsilon }({\partial _\beta }{{[X,Y]}^\varepsilon })} \hfill \cr } by virtue of (3). Secondly, if J = β, we have [ccX,ccY]β=(ccX)II(ccY)β(ccY)II(ccX)β=(ccX)α¯α¯(ccY)β0+(ccX)αα(ccY)β+(ccX)α¯¯α¯¯(ccY)β0(ccY)α¯α¯(ccX)β0(ccY)αα(ccX)β(ccY)α¯¯α¯¯(ccX)β0=(ccX)αα(ccY)β(ccY)αα(ccX)β=XααYβYααXβ=[X,Y]β \matrix{{{{{[^{cc}}X{,^{cc}}Y]}^\beta }} \hfill & = \hfill & {{{{(^{cc}}X)}^I}{\partial _I}{{{(^{cc}}Y)}^\beta } - {{{(^{cc}}Y)}^I}{\partial _I}{{{(^{cc}}X)}^\beta }} \hfill \cr {} \hfill & = \hfill & {{{{(^{cc}}X)}^{\overline \alpha }}\underbrace {{\partial _{\overline \alpha }}{{{(^{cc}}Y)}^\beta }}_0 + {{{(^{cc}}X)}^\alpha }{\partial _\alpha }{{{(^{cc}}Y)}^\beta } + {{{(^{cc}}X)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{{{(^{cc}}Y)}^\beta }}_0} \hfill \cr {} \hfill & {} \hfill & { - {{{(^{cc}}Y)}^{\overline \alpha }}\underbrace {{\partial _{\overline \alpha }}{{{(^{cc}}X)}^\beta }}_0 - {{{(^{cc}}Y)}^\alpha }{\partial _\alpha }{{{(^{cc}}X)}^\beta } - {{{(^{cc}}Y)}^{\overline {\overline \alpha } }}\underbrace {{\partial _{\overline {\overline \alpha } }}{{{(^{cc}}X)}^\beta }}_0} \hfill \cr {} \hfill & = \hfill & {{{{(^{cc}}X)}^\alpha }{\partial _\alpha }{{{(^{cc}}Y)}^\beta } - {{{(^{cc}}Y)}^\alpha }{\partial _\alpha }{{{(^{cc}}X)}^\beta }} \hfill \cr {} \hfill & = \hfill & {{X^\alpha }{\partial _\alpha }{Y^\beta } - {Y^\alpha }{\partial _\alpha }{X^\beta }} \hfill \cr {} \hfill & = \hfill & {{{[X,Y]}^\beta }} \hfill \cr } by virtue of (3). Thirdly, if J=β¯¯ J = \overline {\overline \beta } , then we have [ccX,ccY]β¯¯=(ccX)II(ccY)β¯¯(ccY)II(ccX)β¯¯=(ccX)α¯α¯(ccY)β¯¯0+(ccX)αα(ccY)β¯¯+(ccX)α¯¯α¯¯(ccY)β¯¯(ccY)α¯α¯(ccX)β¯¯0(ccY)αα(ccX)β¯¯(ccY)α¯¯α¯¯(ccX)β¯¯=Xαα(λ=1ptβ1...βqα1...ε...αpεYβλμ=1qtβ1...ε...βqα1...αpβμYε)+(λ=1ptβ1...βqα1...ε...αpεXαλμ=1qtβ1...γ...βqα1...αpαμXγ)α¯(λ=1ptβ1...βqα1...σ...αpσYβλμ=1qtβ1...γ...βqα1...αpβμYγ)Yαα(λ=1ptβ1...βqα1...ε...αpεXβλμ=1qtβ1...ε...βqα1...αpβμXε)(λ=1ptβ1...βqα1...ε...αpεYαλμ=1qtβ1...ε...βqα1...αpβμYε)α¯(λ=1ptβ1...βqα1...σ...αpσXβλμ=1qtβ1...γ...βqα1...αpβμXγ)=Xαα(λ=1ptβ1...βqα1...ε...αpεYβλ)Xααμ=1qtβ1...ε...βqα1...αp(βμYε)+λ=1ptβ1...βqα1...ε...αpεXαλα¯λ=1ptβ1...βqα1...σ...αpδαλσσYβλYαα(λ=1ptβ1...βqα1...ε...αpεXβλ)+Yααμ=1qtβ1...ε...βqα1...αp(βμXε)λ=1ptβ1...βqα1...ε...αpεYαλα¯λ=1ptβ1...βqα1...σ...αpδαλσσXβλ+μ=1qtβ1...ε...βqα1...αpαμXεα¯μ=1qtβ1...γ...βqα1...αpδγαβμYγμ=1qtβ1...ε...βqα1...αpαμXε(βμYα)μ=1qtβ1...ε...βqα1...αpαμYεα¯μ=1qtβ1...γ...βqα1...αpδγαβμXγμ=1qtβ1...ε...βqα1...αpαμYε(βμXα)=λ=1ptβ1...βqα1...ε...αp(εXσ)(σYβλ)+λ=1ptβ1...βqα1...ε...αpXααεYβλλ=1ptβ1...βqα1...ε...αp(εYσ)(σXβλ)λ=1ptβ1...βqα1...ε...αpYααεXβλ+μ=1qtβ1...ε...βqα1...αp(XααμβμYε+βμYααμXε+YααμβμXεβμXααμYε)μ=1qtβ1...ε...βqα1...αp(βμ(XααμYεYααμXε)[X,Y]ε)=λ=1ptβ1...βqα1...ε...αpε[X,Y]βλμ=1qtβ1...ε...βqα1...αp(βμ[X,Y]ε) \matrix{ [{^{cc}X{,^{cc}}Y{]^{\overline {\overline \beta } }}} \hfill & = \hfill & {{{{(^{cc}}X)}^I}{\partial _I}{{{(^{cc}}Y)}^{\overline {\overline \beta } }} - {{{(^{cc}}Y)}^I}{\partial _I}{{{(^{cc}}X)}^{\overline {\overline \beta } }}} \hfill \cr {} \hfill & = \hfill & {{{{(^{cc}}X)}^{\overline \alpha }}\underbrace {{\partial _{\overline \alpha }}{{{(^{cc}}Y)}^{\overline {\overline \beta } }}}_0 + {{{(^{cc}}X)}^\alpha }{\partial _\alpha }{{{(^{cc}}Y)}^{\overline {\overline \beta } }} + {{{(^{cc}}X)}^{\overline {\overline \alpha } }}{\partial _{\overline {\overline \alpha } }}{{{(^{cc}}Y)}^{\overline {\overline \beta } }}} \hfill \cr {} \hfill & {} \hfill & {\; - {{{(^{cc}}Y)}^{\overline \alpha }}\underbrace {{\partial _{\overline \alpha }}{{{(^{cc}}X)}^{\overline {\overline \beta } }}}_0 - {{{(^{cc}}Y)}^\alpha }{\partial _\alpha }{{{(^{cc}}X)}^{\overline {\overline \beta } }} - {{{(^{cc}}Y)}^{\overline {\overline \alpha } }}{\partial _{\overline {\overline \alpha } }}{{{(^{cc}}X)}^{\overline {\overline \beta } }}} \hfill \cr {} \hfill & = \hfill & {{X^\alpha }{\partial _\alpha }(\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{Y^{{\beta _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{Y^\varepsilon })} \hfill \cr {} \hfill & {} \hfill & { + (\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{X^{{\alpha _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\gamma ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\alpha _\mu }}}{X^\gamma }){\partial _{\overline \alpha }}(\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\sigma ...{\alpha _p}}{\partial _\sigma }{Y^{{\beta _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\gamma ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{Y^\gamma })} \hfill \cr {} \hfill & {} \hfill & { - {Y^\alpha }{\partial _\alpha }(\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{X^{{\beta _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{X^\varepsilon })} \hfill \cr {} \hfill & {} \hfill & { - (\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{Y^{{\alpha _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{Y^\varepsilon }){\partial _{\overline \alpha }}(\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\sigma ...{\alpha _p}}{\partial _\sigma }{X^{{\beta _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\gamma ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{X^\gamma })} \hfill \cr {} \hfill & = \hfill & {{X^\alpha }{\partial _\alpha }(\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{Y^{{\beta _\lambda }}}) - {X^\alpha }{\partial _\alpha }\sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}({\partial _{{\beta _\mu }}}{Y^\varepsilon })} \hfill \cr {} \hfill & {} \hfill & { + \sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{X^{{\alpha _\lambda }}}\underbrace {{\partial _{\overline \alpha }}\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\sigma ...{\alpha _p}}}_{\delta _{{\alpha _\lambda }}^\sigma }{\partial _\sigma }{Y^{{\beta _\lambda }}} - {Y^\alpha }{\partial _\alpha }(\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{X^{{\beta _\lambda }}}) + {Y^\alpha }{\partial _\alpha }\sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}({\partial _{{\beta _\mu }}}{X^\varepsilon })} \hfill \cr {} \hfill & {} \hfill & { - \sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{Y^{{\alpha _\lambda }}}\underbrace {{\partial _{\overline \alpha }}\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\sigma ...{\alpha _p}}}_{\delta _{{\alpha _\lambda }}^\sigma }{\partial _\sigma }{X^{{\beta _\lambda }}}} \hfill \cr {} \hfill & {} \hfill & { + \underbrace {\sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\alpha _\mu }}}{X^\varepsilon }\underbrace {{\partial _{\overline \alpha }}\sum\limits_{\mu = 1}^q t_{{\beta _1}...\gamma ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}}_{\delta _\gamma ^\alpha }{\partial _{{\beta _\mu }}}{Y^\gamma }}_{\sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\alpha _\mu }}}{X^\varepsilon }({\partial _{{\beta _\mu }}}{Y^\alpha })} - \underbrace {\sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\alpha _\mu }}}{Y^\varepsilon }\underbrace {{\partial _{\overline \alpha }}\sum\limits_{\mu = 1}^q t_{{\beta _1}...\gamma ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}}_{\delta _\gamma ^\alpha }{\partial _{{\beta _\mu }}}{X^\gamma }}_{\sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\alpha _\mu }}}{Y^\varepsilon }({\partial _{{\beta _\mu }}}{X^\alpha })}} \hfill \cr {} \hfill & = \hfill & {\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}\left( {{\partial _\varepsilon }{X^\sigma }} \right)\left( {{\partial _\sigma }{Y^{{\beta _\lambda }}}} \right) + \sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{X^\alpha }{\partial _\alpha }{\partial _\varepsilon }{Y^{{\beta _\lambda }}}} \hfill \cr {} \hfill & {} \hfill & { - \sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}\left( {{\partial _\varepsilon }{Y^\sigma }} \right)\left( {{\partial _\sigma }{X^{{\beta _\lambda }}}} \right) - \sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{Y^\alpha }{\partial _\alpha }{\partial _\varepsilon }{X^{{\beta _\lambda }}}} \hfill \cr {} \hfill & {} \hfill & {\underbrace { + \sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}( - {X^\alpha }{\partial _{{\alpha _\mu }}}{\partial _{{\beta _\mu }}}{Y^\varepsilon } + {\partial _{{\beta _\mu }}}{Y^\alpha }{\partial _{{\alpha _\mu }}}{X^\varepsilon } + {Y^\alpha }{\partial _{{\alpha _\mu }}}{\partial _{{\beta _\mu }}}{X^\varepsilon } - {\partial _{{\beta _\mu }}}{X^\alpha }{\partial _{{\alpha _\mu }}}{Y^\varepsilon })}_{ - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}({\partial _{{\beta _\mu }}}\underbrace {({X^\alpha }{\partial _{{\alpha _\mu }}}{Y^\varepsilon } - {Y^\alpha }{\partial _{{\alpha _\mu }}}{X^\varepsilon })}_{{{[X,Y]}^\varepsilon }})}} \hfill \cr {} \hfill & = \hfill & {\sum\limits_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{{[X,Y]}^{{\beta _\lambda }}} - \sum\limits_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}({\partial _{{\beta _\mu }}}{{[X,Y]}^\varepsilon })} \hfill \cr } by virtue of (3). On the other hand, we know that cc[X,Y ] have components cc[X,Y]=(pε(β[X,Y]ε)[X,Y]βλ=1ptβ1...βqα1...ε...αpε[X,Y]βλμ=1qtβ1...ε...βqα1...αpβμ[X,Y]ε) ^{cc}[X,Y] = \left( {\matrix{ { - {p_\varepsilon }({\partial _\beta }{{[X,Y]}^\varepsilon })} \hfill \cr {{{[X,Y]}^\beta }} \hfill \cr {\sum_{\lambda = 1}^p t_{{\beta _1}...{\beta _q}}^{{\alpha _1}...\varepsilon ...{\alpha _p}}{\partial _\varepsilon }{{[X,Y]}^{{\beta _\lambda }}} - \sum_{\mu = 1}^q t_{{\beta _1}...\varepsilon ...{\beta _q}}^{{\alpha _1}...{\alpha _p}}{\partial _{{\beta _\mu }}}{{[X,Y]}^\varepsilon }} \hfill \cr } } \right) with respect to the coordinates (xβ¯,xβ,xβ¯¯) ({x^{\overline \beta }},{x^\beta },{x^{\overline {\overline \beta } }}) on tqp(Mn) t_q^p({M_n}) . Thus Theorem 5 is proved.

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