Journal Details
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Open Access

# On Pull-Back Bundle of Tensor Bundles Defined by Projection of The Cotangent Bundle

###### Accepted: 26 Oct 2019
Journal Details
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English

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.

#### 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 $A→γBα↓↓βC→πD$ \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

$A→γBα↓↓βC→πDUn×Rr×Gs×Rt→γUn×Gs↓↓Un×Rr→πUn(xi,aa,gλ,bσ)→γ(xi,gλ)↓↓(xi,aa)→π(xi)$ \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: $tqp(Mn)→π2T∗(Mn)id↓↓π1tqp(Mn)→πMnT∗(Mn)×(Tqp)x(Mn)→π2T∗(Mn)id↓↓π1T∗(Mn)×(Tqp)x(Mn)→πMn(xα¯,xα,xα¯¯)→π2(xα¯,xα)id↓↓π1(xα¯,xα,xα¯¯)→π(xα)$ \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 ${xα¯'=∂xβ∂xα'pβ,xα'=xα'(xβ),xα¯¯'=tα1'...αq'β1'...βp'=Aα1...αpβ1'...βp'Aα1'...αq'β1...βqtβ1...βqα1...αp=A(α)(β')A(α')(β)xβ¯.$ \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 }}} , $Aα'β=∂xβ∂xα'$ A_{{\alpha ^\prime}}^\beta = {{\partial {x^\beta }} \over {\partial {x^{{\alpha ^\prime}}}}} , $Aβ'α'σ=∂2xσ∂xβ'∂xα'$ 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 $ℑqp(T*(Mn))$ \Im _q^p({T^*}({M_n})) and $ℑqp(Mn)$ \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 $X∈ℑ01(T∗(Mn))$ 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 $ccX=(ccXβ¯ccXβccXβ¯¯)=(−pε(∂βXε)Xβ∑λ=1ptβ1...βqα1...ε...αp∂εXαλ−∑μ=1qtβ1...ε...βqα1...αp∂βμXε),$ ^{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 $cX∈ℑ01(T∗(Mn))$ ^cX \in \Im _0^1({T^ * }({M_n})) to $tqp(Mn)$ t_q^p({M_n}) .

Now, consider $A∈ℑqp(T∗(Mn))$ A \in \Im _q^p({T^ * }({M_n})) and $φ∈ℑ11(Mn)$ \varphi \in \Im _1^1({M_n}) , then $vvA∈ℑ01(tqp(Mn))$ ^{vv}A \in \Im _0^1(t_q^p({M_n})) (vertical lift), $γφ∈ℑ01(tqp(Mn))$ \gamma \varphi \in \Im _0^1(t_q^p({M_n})) and $γ˜φ∈ℑ01(tqp(Mn))$ \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] $vvA=(vvA)I=(vvAavvAαvvAα¯)=(00Aβ1...βqα1...αp),γφ=(γφ)I=(00∑λ=1ptβ1...βqα1...ε...αpφεαλ),γ˜φ=(γ˜φ)I=(00∑μ=1qtβ1...ε...βqα1...αpφβμε)$ \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 $f∈ℑ00(Mn)$ f \in \Im _0^0({M_n}) on $tqp(Mn)$ t_q^p({M_n}) is defined by [9]: $vvf=vf∘π2=f∘π1∘π2=f∘π.$ ^{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 $f∈ℑ00(Mn)$ 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 $X∈ℑ01(T∗(Mn))$ X \in \Im _0^1({T^ * }({M_n})) . Then we get by (3) and (5): $ccXvvf=ccXI∂I(vvf)ccXvvf=ccXα¯∂α¯(vvf)︸0+ccXα∂α(vvf)+ccXα¯¯∂α¯¯(vvf)︸0 =Xα∂α(vvf) =vv(Xf),$ \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 $φ∈ℑ11(Mn)$ \varphi \in \Im _1^1({M_n}) , $f∈ℑ00(Mn)$ f \in \Im _0^0({M_n}) and $A∈ℑqp(T∗(Mn))$ 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 $A∈ℑqp(T∗(Mn))$ A \in \Im _q^p({T^ * }({M_n})) , then, by (4) and (5), we find $(vvA)vvf=(vvA)I∂I(vvf) =(vvA)α¯︸0∂α¯(vvf)+(vvA)α︸0∂α(vvf)+(vvA)α¯¯∂α¯¯(vvf)︸0 =0.$ \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 $φ∈ℑ11(Mn)$ \varphi \in \Im _1^1({M_n}) , then we have by (4) and (5): $(γφ)(vvf)=(γφ)I∂I(vvf) =(γφ)α¯︸0∂α¯(vvf)+(γφ)α︸0∂α(vvf)+(γφ)α¯¯∂α¯¯(vvf)︸0 =0.$ \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 $φ∈ℑ11(Mn)$ \varphi \in \Im _1^1({M_n}) , then we have by (4) and (5): $(γ˜φ)(vvf)=(γ˜φ)I∂I(vvf) =(γ˜φ)α¯︸0∂α¯(vvf)+(γ˜φ)α︸0∂α(vvf)+(γ˜φ)α¯¯∂α¯¯(vvf)︸0 =0.$ \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 $A,B∈ℑqp(T∗(Mn))$ 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 $A,B∈ℑqp(T∗(Mn))$ 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 $[vvA,vvB]J=(vvA)I∂I(vvB)J−(vvB)I∂I(vvA)J=(vvA)α¯︸0∂α¯(vvB)J+(vvA)α︸0∂α(vvB)J+(vvA)α¯¯∂α¯¯(vvB)J−(vvB)α¯︸0∂α¯(vvA)J−(vvB)α︸0∂α(vvA)J−(vvB)α¯¯∂α¯¯(vvA)J=Aβ1...βqα1...αp∂α¯¯(vvB)J−Bβ1...βqα1...αp∂α¯¯(vvA)J.$ \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 $[vvA,vvB]β¯=Aβ1...βqα1...αp∂α¯¯(vvB)β¯︸0−Bβ1...βqα1...αp∂α¯¯(vvA)β¯︸0=0,$ [^{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 $[vvA,vvB]β=Aβ1...βqα1...αp∂α¯¯(vvB)β︸0−Bβ1...βqα1...αp∂α¯¯(vvA)β︸0=0,$ [^{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 $[vvA,vvB]β¯¯=Aβ1...βqα1...αp∂α¯¯(vvB)β¯¯−Bβ1...βqα1...αp∂α¯¯(vvA)β¯¯ =Aβ1...βqα1...αp∂α¯¯Bθ1...θqβ1...βp︸0−Bβ1...βqα1...αp∂α¯¯Aθ1...θqβ1...βp︸0 =0$ \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 $X,Y∈ℑ01(T∗(Mn))$ 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 $X,Y∈ℑ01(T∗(Mn))$ 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 $[ccX,ccY]J=(ccX)I∂I(ccY)J−(ccY)I∂I(ccX)J.$ [^{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 $[ccX,ccY]β¯=(ccX)I∂I(ccY)β¯−(ccY)I∂I(ccX)β¯=(ccX)α¯∂α¯(ccY)β¯+(ccX)α∂α(ccY)β¯+(ccX)α¯¯∂α¯¯(ccY)β¯︸0−(ccY)α¯∂α¯(ccX)β¯−(ccY)α∂α(ccX)β¯−(ccY)α¯¯∂α¯¯(ccX)β¯︸0=pε∂αXε(∂βYα)−Xα∂αpε(∂βYε)−pε∂αYε(∂βXα)+Yα∂αpε(∂βXε)=pε(∂βYα∂αXε−Xα∂α∂βYε−∂βXα∂αYε+Yα∂α∂βXε)=−pε(∂β(Xα∂αYε−Yα∂αXε)︸[X,Y]ε)=−pε(∂β[X,Y]ε)$ \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 $[ccX,ccY]β=(ccX)I∂I(ccY)β−(ccY)I∂I(ccX)β=(ccX)α¯∂α¯(ccY)β︸0+(ccX)α∂α(ccY)β+(ccX)α¯¯∂α¯¯(ccY)β︸0−(ccY)α¯∂α¯(ccX)β︸0−(ccY)α∂α(ccX)β−(ccY)α¯¯∂α¯¯(ccX)β︸0=(ccX)α∂α(ccY)β−(ccY)α∂α(ccX)β=Xα∂αYβ−Yα∂αXβ=[X,Y]β$ \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 $[ccX,ccY]β¯¯=(ccX)I∂I(ccY)β¯¯−(ccY)I∂I(ccX)β¯¯=(ccX)α¯∂α¯(ccY)β¯¯︸0+(ccX)α∂α(ccY)β¯¯+(ccX)α¯¯∂α¯¯(ccY)β¯¯ −(ccY)α¯∂α¯(ccX)β¯¯︸0−(ccY)α∂α(ccX)β¯¯−(ccY)α¯¯∂α¯¯(ccX)β¯¯=Xα∂α(∑λ=1ptβ1...βqα1...ε...αp∂εYβλ−∑μ=1qtβ1...ε...βqα1...αp∂βμYε)+(∑λ=1ptβ1...βqα1...ε...αp∂εXαλ−∑μ=1qtβ1...γ...βqα1...αp∂αμXγ)∂α¯(∑λ=1ptβ1...βqα1...σ...αp∂σYβλ−∑μ=1qtβ1...γ...βqα1...αp∂βμYγ)−Yα∂α(∑λ=1ptβ1...βqα1...ε...αp∂εXβλ−∑μ=1qtβ1...ε...βqα1...αp∂βμXε)−(∑λ=1ptβ1...βqα1...ε...αp∂εYαλ−∑μ=1qtβ1...ε...βqα1...αp∂βμYε)∂α¯(∑λ=1ptβ1...βqα1...σ...αp∂σXβλ−∑μ=1qtβ1...γ...βqα1...αp∂βμXγ)=Xα∂α(∑λ=1ptβ1...βqα1...ε...αp∂εYβλ)−Xα∂α∑μ=1qtβ1...ε...βqα1...αp(∂βμYε)+∑λ=1ptβ1...βqα1...ε...αp∂εXαλ∂α¯∑λ=1ptβ1...βqα1...σ...αp︸δαλσ∂σYβλ−Yα∂α(∑λ=1ptβ1...βqα1...ε...αp∂εXβλ)+Yα∂α∑μ=1qtβ1...ε...βqα1...αp(∂βμXε)−∑λ=1ptβ1...βqα1...ε...αp∂εYαλ∂α¯∑λ=1ptβ1...βqα1...σ...αp︸δαλσ∂σXβλ+∑μ=1qtβ1...ε...βqα1...αp∂αμXε∂α¯∑μ=1qtβ1...γ...βqα1...αp︸δγα∂βμYγ︸∑μ=1qtβ1...ε...βqα1...αp∂αμXε(∂βμYα)−∑μ=1qtβ1...ε...βqα1...αp∂αμYε∂α¯∑μ=1qtβ1...γ...βqα1...αp︸δγα∂βμXγ︸∑μ=1qtβ1...ε...βqα1...αp∂αμYε(∂βμXα)=∑λ=1ptβ1...βqα1...ε...αp(∂εXσ)(∂σYβλ)+∑λ=1ptβ1...βqα1...ε...αpXα∂α∂εYβλ−∑λ=1ptβ1...βqα1...ε...αp(∂εYσ)(∂σXβλ)−∑λ=1ptβ1...βqα1...ε...αpYα∂α∂εXβλ+∑μ=1qtβ1...ε...βqα1...αp(−Xα∂αμ∂βμYε+∂βμYα∂αμXε+Yα∂αμ∂βμXε−∂βμXα∂αμYε)︸−∑μ=1qtβ1...ε...βqα1...αp(∂βμ(Xα∂αμYε−Yα∂αμXε)︸[X,Y]ε)=∑λ=1ptβ1...βqα1...ε...αp∂ε[X,Y]βλ−∑μ=1qtβ1...ε...βqα1...αp(∂βμ[X,Y]ε)$ \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 $cc[X,Y]=(−pε(∂β[X,Y]ε)[X,Y]β∑λ=1ptβ1...βqα1...ε...αp∂ε[X,Y]βλ−∑μ=1qtβ1...ε...βqα1...αp∂βμ[X,Y]ε)$ ^{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.

S. Eker, N. Değirmenci, Seiberg–Witten–like equations without Self–Duality on odd dimensional manifolds, Journal of Partial Differential Equations, 31, No. 4, (2018), 291–303. EkerS. DeğirmenciN. Seiberg–Witten–like equations without Self–Duality on odd dimensional manifolds Journal of Partial Differential Equations 31 4 2018 291 303 Search in Google Scholar

S. Eker, The Bochner vanishing theorems on the conformal killing vector fields, TWMS Journal of Applied and Engineering Mathematics, 9, No. 1, (2019), 114–120. EkerS. The Bochner vanishing theorems on the conformal killing vector fields TWMS Journal of Applied and Engineering Mathematics 9 1 2019 114 120 Search in Google Scholar

F. Etayo, The geometry of good squares of vector bundles, Riv. Mat. Univ. Parma 17, (1991), 131–147. EtayoF. The geometry of good squares of vector bundles Riv. Mat. Univ. Parma 17 1991 131 147 Search in Google Scholar

H. Fattaev, The Lifts of Vector Fields to the Semitensor Bundle of the Type (2, 0), Journal of Qafqaz University, 25, No. 1, (2009), 136–140. FattaevH. The Lifts of Vector Fields to the Semitensor Bundle of the Type (2, 0) Journal of Qafqaz University 25 1 2009 136 140 Search in Google Scholar

A. Gezer, A. A. Salimov, Almost complex structures on the tensor bundles, Arab. J. Sci. Eng. Sect. A Sci. 33, No. 2, (2008), 283–296. GezerA. SalimovA. A. Almost complex structures on the tensor bundles Arab. J. Sci. Eng. Sect. A Sci. 33 2 2008 283 296 Search in Google Scholar

D. Husemoller, Fibre Bundles. Springer, New York, 1994. HusemollerD. Fibre Bundles Springer New York 1994 Search in Google Scholar

H.B. Lawson and M.L. Michelsohn, Spin Geometry. Princeton University Press., Princeton, 1989. LawsonH.B. MichelsohnM.L. Spin Geometry Princeton University Press Princeton 1989 Search in Google Scholar

A.J. Ledger and K. Yano, Almost complex structure on tensor bundles, J. Dif. Geom. 1 (1967), 355–368. LedgerA.J. YanoK. Almost complex structure on tensor bundles J. Dif. Geom. 1 1967 355 368 Search in Google Scholar

M. Polat and N. Cengiz, Some properties of semi-tensor bundle, New Trends in Mathematical Sciences, 6, No. 3, (2018), 147–153. PolatM. CengizN. Some properties of semi-tensor bundle New Trends in Mathematical Sciences 6 3 2018 147 153 Search in Google Scholar

A. Salimov, Tensor Operators and their Applications. Nova Science Publ., New York, 2013. SalimovA. Tensor Operators and their Applications Nova Science Publ. New York 2013 Search in Google Scholar

N. Steenrod, The Topology of Fibre Bundles. Princeton University Press., Princeton, 1951. SteenrodN. The Topology of Fibre Bundles Princeton University Press Princeton 1951 Search in Google Scholar

K. Yano and S. Ishihara, Tangent and Cotangent Bundles. Marcel Dekker, Inc., New York, 1973. YanoK. IshiharaS. Tangent and Cotangent Bundles Marcel Dekker, Inc. New York 1973 Search in Google Scholar

F. Yıldırım, On semi-tensor bundle, International Electronic Journal of Geometry, 11, No. 1, (2018), 93–99. YıldırımF. On semi-tensor bundle International Electronic Journal of Geometry 11 1 2018 93 99 Search in Google Scholar

• #### Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

Recommended articles from Trend MD