On Pull-Back Bundle of Tensor Bundles Defined by Projection of The 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 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 M_n be an n-dimensional differentiable manifold of class C^∞, and let (T^*(M_n), π₁, M_n) be a cotangent bundle over M_n. We use the notation (xⁱ) = x^α̅,x^α, 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, x^α are coordinates in M_n, x^α̅ = p_α are fibre coordinates of the cotangent bundle T^*(M_n).

Let now $(T_{q}^{p} (M_{n}), \tilde{π}, M_{n})$ \left( {T_q^p({M_n}),\widetilde \pi ,{M_n}} \right) be a tensor bundle [5], [9], [ [10], p.118] with base space M_n, and let T^*(M_n) be cotangent bundle determined by a natural projection (submersion) π₁ : T^*(M_n) → M_n. The semi-tensor bundle (induced, pull-back [6], [7], [9], [11], [13]]) of the tensor bundle $(T_{q}^{p} (M_{n}), \tilde{π}, M_{n})$ \left( {T_q^p({M_n}),\widetilde \pi ,{M_n}} \right) is the bundle $(t_{q}^{p} (M_{n}), π_{2}, T^{*} (M_{n}))$ \left( {t_q^p({M_n}),{\pi _2},{T^ * }({M_n})} \right) over cotangent bundle T^*(M_n) with a total space $t_{q}^{p} (M_{n}) = {((x^{\bar{α}}, x^{α}), x^{\bar{\bar{α}}}) \in T^{*} (M_{n}) \times {(T_{q}^{p})}_{x} (M_{n}) : π_{1} (x^{\bar{α}}, x^{α}) = \tilde{π} (x^{α}, x^{\bar{\bar{α}}}) = (x^{α})} \subset T^{*} (M_{n}) \times {(T_{q}^{p})}_{x} (M_{n})$ 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 $π_{2} : t_{q}^{p} (M_{n}) \to T^{*} (M_{n})$ {\pi _2}:t_q^p({M_n}) \to {T^ * }({M_n}) defined by $π_{2} (x^{\bar{α}}, x^{α}, x^{\bar{\bar{α}}}) = (x^{\bar{α}}, x^{α})$ {\pi _2}({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) = \left( {{x^{\overline \alpha }},{x^\alpha }} \right) , where ${(T_{q}^{p})}_{x} (M_{n}) (x = π_{1} (\tilde{x}), \tilde{x} = (x^{\bar{α}}, x^{α}) \in T^{*} (M_{n}))$ {\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 M_n, where $x^{\bar{\bar{α}}} = t_{α_{1} ... α_{q}}^{β_{1} ... β_{p}} (\bar{\bar{α}}, \bar{\bar{β}}, ... = 2 n + 1, ..., 2 n + n^{p + q})$ {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 $T_{q}^{p} (M_{n})$ T_q^p({M_n}) .

The pull-back (semi-tensor) bundle $t_{q}^{p} (M_{n})$ t_q^p({M_n}) of tensor bundle $T_{q}^{p} (M_{n})$ T_q^p({M_n}) has the natural bundle structure over M_n, its bundle projection $π : t_{q}^{p} (M_{n}) \to M_{n}$ \pi :t_q^p({M_n}) \to {M_n} being defined by $π : (x^{\bar{α}}, x^{α}, x^{\bar{\bar{α}}}) \to (x^{α})$ \pi :({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^\alpha }) . If we introduce a mapping $π_{2} : t_{q}^{p} (M_{n}) \to T^{*} (M_{n})$ {\pi _2}:t_q^p({M_n}) \to {T^*}({M_n}) by $π_{2} : (x^{\bar{α}}, x^{α}, x^{\bar{\bar{α}}}) \to (x^{\bar{α}}, x^{α})$ {\pi _2}:({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^{\overline \alpha }},{x^\alpha }) , then $t_{q}^{p} (M_{n})$ t_q^p({M_n}) has a bundle structure over M_n. Hence π = π₁ ○ π₂.

Now, consider a diagram as $\begin{matrix} A & \overset{γ}{\to} & B \\ ^{α} ↓ & ↓^{β} \\ C & \underset{π}{\to} & D \end{matrix}$ \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 i

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

γ and π are vector bundles;

iii

the square is commutative, i.e., π ○ α = β ○ γ;

the local expression

\begin{matrix} \begin{matrix} A & \overset{γ}{\to} & B \\ ^{α} ↓ & ↓^{β} \\ C & \underset{π}{\to} & D \end{matrix} & \begin{matrix} U^{n} \times R^{r} \times G^{s} \times R^{t} & \overset{γ}{\to} & U^{n} \times G^{s} \\ ↓ & ↓ \\ U^{n} \times R^{r} & \underset{π}{\to} & U^{n} \end{matrix} & \begin{matrix} (x^{i}, a^{a}, g^{λ}, b^{σ}) & \overset{γ}{\to} & (x^{i}, g^{λ}) \\ ↓ & ↓ \\ (x^{i}, a^{a}) & \underset{π}{\to} & (x^{i}) \end{matrix} \end{matrix}

\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 $π : t_{q}^{p} (M_{n}) \to M_{n}$ \pi :t_q^p({M_n}) \to {M_n} be a semi-tensor bundle and π₁ : T^*(M_n) → M_n be a cotangent bundle. Then, the following is a good square: $\begin{matrix} \begin{matrix} t_{q}^{p} (M_{n}) & \overset{π_{2}}{\to} & T^{*} (M_{n}) \\ ^{id} ↓ & ↓^{π_{1}} \\ t_{q}^{p} (M_{n}) & \underset{π}{\to} & M_{n} \end{matrix} & \begin{matrix} T^{*} (M_{n}) \times {(T_{q}^{p})}_{x} (M_{n}) & \overset{π_{2}}{\to} & T^{*} (M_{n}) \\ ^{id} ↓ & ↓^{π_{1}} \\ T^{*} (M_{n}) \times {(T_{q}^{p})}_{x} (M_{n}) & \underset{π}{\to} & M_{n} \end{matrix} & \begin{matrix} (x^{\bar{α}}, x^{α}, x^{\bar{\bar{α}}}) & \overset{π_{2}}{\to} & (x^{\bar{α}}, x^{α}) \\ ^{id} ↓ & ↓^{π_{1}} \\ (x^{\bar{α}}, x^{α}, x^{\bar{\bar{α}}}) & \underset{π}{\to} & (x^{α}) \end{matrix} \end{matrix}$ \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 $(x^{i^{'}}) = (x^{{\bar{α}}^{'}}, x^{α^{'}}, x^{{\bar{\bar{α}}}^{'}})$ ({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 $t_{q}^{p} (M_{n})$ t_q^p({M_n}) , then we have (1) ${\begin{array}{l} x^{{\bar{α}}^{'}} = \frac{\partial x^{β}}{\partial x^{α^{'}}} p_{β}, \\ x^{α^{'}} = x^{α^{'}} (x^{β}), \\ x^{{\bar{\bar{α}}}^{'}} = t_{α_{1}^{'} ... α_{q}^{'}}^{β_{1}^{'} ... β_{p}^{'}} = A_{α_{1} ... α_{p}}^{β_{1}^{'} ... β_{p}^{'}} A_{α_{1}^{'} ... α_{q}^{'}}^{β_{1} ... β_{q}} t_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} = A_{(α)}^{(β^{'})} A_{(α^{'})}^{(β)} x^{\bar{β}} . \end{array}$ \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 (2) $\bar{A} = (A_{J}^{I^{'}}) = (\begin{matrix} A_{α^{'}}^{β} & p_{σ} A_{β}^{β^{'}} A_{β^{'} α^{'}}^{σ} & 0 \\ 0 & A_{β}^{α^{'}} & 0 \\ 0 & t_{(σ)}^{(α)} \partial_{β} A_{(α)}^{(β^{'})} A_{(α^{'})}^{(σ)} & A_{(α)}^{(β^{'})} A_{(α^{'})}^{(β)} \end{matrix}),$ \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 $I = (\bar{α}, α, \bar{\bar{α}})$ I = (\overline \alpha ,\alpha ,\overline {\overline \alpha } ) , $J = (\bar{β}, β, \bar{\bar{β}})$ J = (\overline \beta ,\beta ,\overline {\overline \beta } ) , I,J...=1,...,2n+n^p+q, $t_{(σ)}^{(α)} = t_{σ_{1} ... σ_{q}}^{α_{1} ... α_{p}}$ t_{(\sigma )}^{(\alpha )} = t_{{\sigma _1}...{\sigma _q}}^{{\alpha _1}...{\alpha _p}} , $A_{β}^{α^{'}} = \frac{\partial x^{α^{'}}}{\partial x^{β}}$ A_\beta ^{{\alpha ^\prime}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}} , $A_{α^{'}}^{β} = \frac{\partial x^{β}}{\partial x^{α^{'}}}$ A_{{\alpha ^\prime}}^\beta = {{\partial {x^\beta }} \over {\partial {x^{{\alpha ^\prime}}}}} , $A_{β^{'} α^{'}}^{σ} = \frac{\partial^{2} x^{σ}}{\partial x^{β^{'}} \partial x^{α^{'}}}$ 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: $Det (A_{α^{'}}^{β}) \neq 0, Det (A_{β}^{α^{'}}) \neq 0, Det (A_{(α)}^{(β^{'})} A_{(α^{'})}^{(β)}) \neq 0 .$ 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, $dim t_{q}^{p} (M_{n}) = 2 n + n^{p + q}$ \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 $t_{q}^{p} (M_{n})$ t_q^p({M_n}) of tensor bundle $T_{q}^{p} (M_{n})$ T_q^p({M_n}) by using projection of the cotangent bundle T^*(M_n).

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

Some lifts of tensor fields and γ−Operator

Let $X \in ℑ_{0}^{1} (T^{*} (M_{n}))$ X \in \Im _0^1({T^ * }({M_n})) , i.e. X = X^α∂_α. The complete lift ^cX of X to cotangent bundle is defined by ^cX = X^α∂_α − p_β (∂_αX^β)∂_α̅ [ [12], p.236]. On putting (3) $^{cc} X = (\begin{array}{l} ^{cc} X^{\bar{β}} \\ ^{cc} X^{β} \\ ^{cc} X^{\bar{\bar{β}}} \end{array}) = (\begin{array}{l} - p_{ε} (\partial_{β} X^{ε}) \\ X^{β} \\ \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} X^{α_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} X^{ε} \end{array}),$ ^{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 $^{c} X \in ℑ_{0}^{1} (T^{*} (M_{n}))$ ^cX \in \Im _0^1({T^ * }({M_n})) to $t_{q}^{p} (M_{n})$ t_q^p({M_n}) .

Now, consider $A \in ℑ_{q}^{p} (T^{*} (M_{n}))$ A \in \Im _q^p({T^ * }({M_n})) and $φ \in ℑ_{1}^{1} (M_{n})$ \varphi \in \Im _1^1({M_n}) , then $^{vv} A \in ℑ_{0}^{1} (t_{q}^{p} (M_{n}))$ ^{vv}A \in \Im _0^1(t_q^p({M_n})) (vertical lift), $γ φ \in ℑ_{0}^{1} (t_{q}^{p} (M_{n}))$ \gamma \varphi \in \Im _0^1(t_q^p({M_n})) and $\tilde{γ} φ \in ℑ_{0}^{1} (t_{q}^{p} (M_{n}))$ \widetilde \gamma \varphi \in \Im _0^1(t_q^p({M_n})) have respectively, components on the semi-tensor bundle $t_{q}^{p} (M_{n})$ t_q^p({M_n}) [13] (4) $\begin{matrix} ^{vv} A = {(^{vv} A)}^{I} = (\begin{array}{l} ^{vv} A^{a} \\ ^{vv} A^{α} \\ ^{vv} A^{\bar{α}} \end{array}) = (\begin{array}{l} 0 \\ 0 \\ A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \end{array}), \\ γ φ = {(γ φ)}^{I} = (\begin{array}{l} 0 \\ 0 \\ \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} φ_{ε}^{α_{λ}} \end{array}), \\ \tilde{γ} φ = {(\tilde{γ} φ)}^{I} = (\begin{array}{l} 0 \\ 0 \\ \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} φ_{β_{μ}}^{ε} \end{array}) \end{matrix}$ \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 $(x^{\bar{α}}, x^{α}, x^{\bar{\bar{α}}})$ ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) on $t_{q}^{p} (M_{n})$ t_q^p({M_n}) , where $A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}}$ A_{{\beta _1}...{\beta _q}}^{{\alpha _1}...{\alpha _p}} , $φ_{ε}^{α_{λ}}$ \varphi _\varepsilon ^{{\alpha _\lambda }} and $φ_{β_{μ}}^{ε}$ \varphi _{{\beta _\mu }}^\varepsilon are local components of A and φ.

On the other hand, ^vv f the vertical lift of function $f \in ℑ_{0}^{0} (M_{n})$ f \in \Im _0^0({M_n}) on $t_{q}^{p} (M_{n})$ t_q^p({M_n}) is defined by [9]: (5) $^{vv} f =^{v} f \circ π_{2} = f \circ π_{1} \circ π_{2} = f \circ π .$ ^{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^*(M_n) and $f \in ℑ_{0}^{0} (M_{n})$ f \in \Im _0^0({M_n}) , we have

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

(ii) ^ccX^vv f =^vv (X f ).

Proof

(i)

This immediately follows from (3).

(ii)

Let $X \in ℑ_{0}^{1} (T^{*} (M_{n}))$ X \in \Im _0^1({T^ * }({M_n})) . Then we get by (3) and (5): $\begin{array}{l} ^{cc} X^{vv} f =^{cc} X^{I} \partial_{I} (^{vv} f) \\ ^{cc} X^{vv} f =^{cc} X^{\bar{α}} \underset{0}{\underset{︸}{\partial_{\bar{α}} (^{vv} f)}} +^{cc} X^{α} \partial_{α} (^{vv} f) +^{cc} X^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} (^{vv} f)}} \\ = X^{α} \partial_{α} (^{vv} f) \\ =^{vv} (Xf), \end{array}$ \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 $φ \in ℑ_{1}^{1} (M_{n})$ \varphi \in \Im _1^1({M_n}) , $f \in ℑ_{0}^{0} (M_{n})$ f \in \Im _0^0({M_n}) and $A \in ℑ_{q}^{p} (T^{*} (M_{n}))$ 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

(i)

If $A \in ℑ_{q}^{p} (T^{*} (M_{n}))$ A \in \Im _q^p({T^ * }({M_n})) , then, by (4) and (5), we find $\begin{array}{l} {(^{vv} A)}^{vv} f = {(^{vv} A)}^{I} \partial_{I} (^{vv} f) \\ = \underset{0}{\underset{︸}{{(^{vv} A)}^{\bar{α}}}} \partial_{\bar{α}} (^{vv} f) + \underset{0}{\underset{︸}{{(^{vv} A)}^{α}}} \partial_{α} (^{vv} f) + {(^{vv} A)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} (^{vv} f)}} \\ = 0 . \end{array}$ \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.

(ii)

If $φ \in ℑ_{1}^{1} (M_{n})$ \varphi \in \Im _1^1({M_n}) , then we have by (4) and (5): $\begin{array}{l} (γ φ) (^{vv} f) = {(γ φ)}^{I} \partial_{I} (^{vv} f) \\ = \underset{0}{\underset{︸}{{(γ φ)}^{\bar{α}}}} \partial_{\bar{α}} (^{vv} f) + \underset{0}{\underset{︸}{{(γ φ)}^{α}}} \partial_{α} (^{vv} f) + {(γ φ)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} (^{vv} f)}} \\ = 0 . \end{array}$ \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.

(iii)

If $φ \in ℑ_{1}^{1} (M_{n})$ \varphi \in \Im _1^1({M_n}) , then we have by (4) and (5): $\begin{array}{l} (\tilde{γ} φ) (^{vv} f) = {(\tilde{γ} φ)}^{I} \partial_{I} (^{vv} f) \\ = \underset{0}{\underset{︸}{{(\tilde{γ} φ)}^{\bar{α}}}} \partial_{\bar{α}} (^{vv} f) + \underset{0}{\underset{︸}{{(\tilde{γ} φ)}^{α}}} \partial_{α} (^{vv} f) + {(\tilde{γ} φ)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} (^{vv} f)}} \\ = 0 . \end{array}$ \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 $A, B \in ℑ_{q}^{p} (T^{*} (M_{n}))$ A,B \in \Im _q^p({T^ * }({M_n})) . For the Lie product, we have $[^{vv} A,^{vv} B] = 0 .$ \left[ {^{vv}A{,^{vv}}B} \right] = 0.

Proof

If $A, B \in ℑ_{q}^{p} (T^{*} (M_{n}))$ A,B \in \Im _q^p({T^ * }({M_n})) and $(\begin{array}{l} {[^{vv} A,^{vv} B]}^{b} \\ {[^{vv} A,^{vv} B]}^{β} \\ {[^{vv} A,^{vv} B]}^{\bar{β}} \end{array})$ \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 $(x^{\bar{β}}, x^{β}, x^{\bar{\bar{β}}})$ ({x^{\overline \beta }},{x^\beta },{x^{\overline {\overline \beta } }}) on $t_{q}^{p} (M_{n})$ t_q^p({M_n}) , then we have $\begin{array}{l} {[^{vv} A,^{vv} B]}^{J} & = & {(^{vv} A)}^{I} \partial_{I} {(^{vv} B)}^{J} - {(^{vv} B)}^{I} \partial_{I} {(^{vv} A)}^{J} \\ = & \underset{0}{\underset{︸}{{(^{vv} A)}^{\bar{α}}}} \partial_{\bar{α}} {(^{vv} B)}^{J} + \underset{0}{\underset{︸}{{(^{vv} A)}^{α}}} \partial_{α} {(^{vv} B)}^{J} \\ + {(^{vv} A)}^{\bar{\bar{α}}} \partial_{\bar{\bar{α}}} {(^{vv} B)}^{J} - \underset{0}{\underset{︸}{{(^{vv} B)}^{\bar{α}}}} \partial_{\bar{α}} {(^{vv} A)}^{J} \\ - \underset{0}{\underset{︸}{{(^{vv} B)}^{α}}} \partial_{α} {(^{vv} A)}^{J} - {(^{vv} B)}^{\bar{\bar{α}}} \partial_{\bar{\bar{α}}} {(^{vv} A)}^{J} \\ = & A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} {(^{vv} B)}^{J} - B_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} {(^{vv} A)}^{J} . \end{array}$ \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 $[^{vv} A,^{vv} B]^{\bar{β}} = A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} \underset{0}{\underset{︸}{{(^{vv} B)}^{\bar{β}}}} - B_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} \underset{0}{\underset{︸}{{(^{vv} A)}^{\bar{β}}}} = 0,$ [^{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 $[^{vv} A,^{vv} B]^{β} = A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} \underset{0}{\underset{︸}{{(^{vv} B)}^{β}}} - B_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} \underset{0}{\underset{︸}{{(^{vv} A)}^{β}}} = 0,$ [^{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 $J = \bar{\bar{β}}$ J = \overline {\overline \beta } , then we have $\begin{array}{l} [^{vv} A,^{vv} B]^{\bar{\bar{β}}} = A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} {(^{vv} B)}^{\bar{\bar{β}}} - B_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \partial_{\bar{\bar{α}}} {(^{vv} A)}^{\bar{\bar{β}}} \\ = A_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} B_{θ_{1} ... θ_{q}}^{β_{1} ... β_{p}}}} - B_{β_{1} ... β_{q}}^{α_{1} ... α_{p}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} A_{θ_{1} ... θ_{q}}^{β_{1} ... β_{p}}}} \\ = 0 \end{array}$ \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 $X, Y \in ℑ_{0}^{1} (T^{*} (M_{n}))$ X{\rm{,}}Y \in \Im _0^1({T^ * }({M_n})) . For the Lie product, we have $[^{cc} X,^{cc} Y {] =}^{cc} [X, Y] (i . e . L_{^{cc} X}^{cc} Y =^{cc} (L_{X} Y)) .$ [^{cc}X{,^{cc}}Y{] = ^{cc}}[X,Y](i.e.{L_{^{cc}X}}^{cc}Y{ = ^{cc}}\left( {{L_X}Y} \right)).

Proof

If $X, Y \in ℑ_{0}^{1} (T^{*} (M_{n}))$ X{\rm{,}}Y \in \Im _0^1({T^ * }({M_n})) and $(\begin{array}{l} {[^{cc} X,^{cc} Y]}^{\bar{β}} \\ {[^{cc} X,^{cc} Y]}^{β} \\ {[^{cc} X,^{cc} Y]}^{\bar{\bar{β}}} \end{array})$ \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 $(x^{\bar{β}}, x^{β}, x^{\bar{\bar{β}}})$ ({x^{\overline \beta }},{x^\beta },{x^{\overline {\overline \beta } }}) on $t_{q}^{p} (M_{n})$ t_q^p({M_n}) , then we have $[^{cc} X,^{cc} Y]^{J} = (^{cc} X)^{I} \partial_{I} {(^{cc} Y)}^{J} - {(^{cc} Y)}^{I} \partial_{I} {(^{cc} X)}^{J} .$ [^{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 $\begin{array}{l} {[^{cc} X,^{cc} Y]}^{\bar{β}} & = & {(^{cc} X)}^{I} \partial_{I} {(^{cc} Y)}^{\bar{β}} - {(^{cc} Y)}^{I} \partial_{I} {(^{cc} X)}^{\bar{β}} \\ = & {(^{cc} X)}^{\bar{α}} \partial_{\bar{α}} {(^{cc} Y)}^{\bar{β}} + {(^{cc} X)}^{α} \partial_{α} {(^{cc} Y)}^{\bar{β}} + {(^{cc} X)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} {(^{cc} Y)}^{\bar{β}}}} \\ - {(^{cc} Y)}^{\bar{α}} \partial_{\bar{α}} {(^{cc} X)}^{\bar{β}} - {(^{cc} Y)}^{α} \partial_{α} {(^{cc} X)}^{\bar{β}} - {(^{cc} Y)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} {(^{cc} X)}^{\bar{β}}}} \\ = & p_{ε} \partial_{α} X^{ε} (\partial_{β} Y^{α}) - X^{α} \partial_{α} p_{ε} (\partial_{β} Y^{ε}) \\ - p_{ε} \partial_{α} Y^{ε} (\partial_{β} X^{α}) + Y^{α} \partial_{α} p_{ε} (\partial_{β} X^{ε}) \\ = & p_{ε} (\partial_{β} Y^{α} \partial_{α} X^{ε} - X^{α} \partial_{α} \partial_{β} Y^{ε} - \partial_{β} X^{α} \partial_{α} Y^{ε} + Y^{α} \partial_{α} \partial_{β} X^{ε}) \\ = & - p_{ε} (\partial_{β} \underset{{[X, Y]}^{ε}}{\underset{︸}{(X^{α} \partial_{α} Y^{ε} - Y^{α} \partial_{α} X^{ε})}}) \\ = & - p_{ε} (\partial_{β} {[X, Y]}^{ε}) \end{array}$ \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 $\begin{array}{l} {[^{cc} X,^{cc} Y]}^{β} & = & {(^{cc} X)}^{I} \partial_{I} {(^{cc} Y)}^{β} - {(^{cc} Y)}^{I} \partial_{I} {(^{cc} X)}^{β} \\ = & {(^{cc} X)}^{\bar{α}} \underset{0}{\underset{︸}{\partial_{\bar{α}} {(^{cc} Y)}^{β}}} + {(^{cc} X)}^{α} \partial_{α} {(^{cc} Y)}^{β} + {(^{cc} X)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} {(^{cc} Y)}^{β}}} \\ - {(^{cc} Y)}^{\bar{α}} \underset{0}{\underset{︸}{\partial_{\bar{α}} {(^{cc} X)}^{β}}} - {(^{cc} Y)}^{α} \partial_{α} {(^{cc} X)}^{β} - {(^{cc} Y)}^{\bar{\bar{α}}} \underset{0}{\underset{︸}{\partial_{\bar{\bar{α}}} {(^{cc} X)}^{β}}} \\ = & {(^{cc} X)}^{α} \partial_{α} {(^{cc} Y)}^{β} - {(^{cc} Y)}^{α} \partial_{α} {(^{cc} X)}^{β} \\ = & X^{α} \partial_{α} Y^{β} - Y^{α} \partial_{α} X^{β} \\ = & {[X, Y]}^{β} \end{array}$ \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 $J = \bar{\bar{β}}$ J = \overline {\overline \beta } , then we have $\begin{array}{l} [^{cc} X,^{cc} Y]^{\bar{\bar{β}}} & = & {(^{cc} X)}^{I} \partial_{I} {(^{cc} Y)}^{\bar{\bar{β}}} - {(^{cc} Y)}^{I} \partial_{I} {(^{cc} X)}^{\bar{\bar{β}}} \\ = & {(^{cc} X)}^{\bar{α}} \underset{0}{\underset{︸}{\partial_{\bar{α}} {(^{cc} Y)}^{\bar{\bar{β}}}}} + {(^{cc} X)}^{α} \partial_{α} {(^{cc} Y)}^{\bar{\bar{β}}} + {(^{cc} X)}^{\bar{\bar{α}}} \partial_{\bar{\bar{α}}} {(^{cc} Y)}^{\bar{\bar{β}}} \\ - {(^{cc} Y)}^{\bar{α}} \underset{0}{\underset{︸}{\partial_{\bar{α}} {(^{cc} X)}^{\bar{\bar{β}}}}} - {(^{cc} Y)}^{α} \partial_{α} {(^{cc} X)}^{\bar{\bar{β}}} - {(^{cc} Y)}^{\bar{\bar{α}}} \partial_{\bar{\bar{α}}} {(^{cc} X)}^{\bar{\bar{β}}} \\ = & X^{α} \partial_{α} (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} Y^{β_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} Y^{ε}) \\ + (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} X^{α_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... γ ... β_{q}}^{α_{1} ... α_{p}} \partial_{α_{μ}} X^{γ}) \partial_{\bar{α}} (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... σ ... α_{p}} \partial_{σ} Y^{β_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... γ ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} Y^{γ}) \\ - Y^{α} \partial_{α} (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} X^{β_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} X^{ε}) \\ - (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} Y^{α_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} Y^{ε}) \partial_{\bar{α}} (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... σ ... α_{p}} \partial_{σ} X^{β_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... γ ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} X^{γ}) \\ = & X^{α} \partial_{α} (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} Y^{β_{λ}}) - X^{α} \partial_{α} \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} (\partial_{β_{μ}} Y^{ε}) \\ + \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} X^{α_{λ}} \underset{δ_{α_{λ}}^{σ}}{\underset{︸}{\partial_{\bar{α}} \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... σ ... α_{p}}}} \partial_{σ} Y^{β_{λ}} - Y^{α} \partial_{α} (\sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} X^{β_{λ}}) + Y^{α} \partial_{α} \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} (\partial_{β_{μ}} X^{ε}) \\ - \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} Y^{α_{λ}} \underset{δ_{α_{λ}}^{σ}}{\underset{︸}{\partial_{\bar{α}} \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... σ ... α_{p}}}} \partial_{σ} X^{β_{λ}} \\ + \underset{\sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{α_{μ}} X^{ε} (\partial_{β_{μ}} Y^{α})}{\underset{︸}{\sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{α_{μ}} X^{ε} \underset{δ_{γ}^{α}}{\underset{︸}{\partial_{\bar{α}} \sum_{μ = 1}^{q} t_{β_{1} ... γ ... β_{q}}^{α_{1} ... α_{p}}}} \partial_{β_{μ}} Y^{γ}}} - \underset{\sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{α_{μ}} Y^{ε} (\partial_{β_{μ}} X^{α})}{\underset{︸}{\sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{α_{μ}} Y^{ε} \underset{δ_{γ}^{α}}{\underset{︸}{\partial_{\bar{α}} \sum_{μ = 1}^{q} t_{β_{1} ... γ ... β_{q}}^{α_{1} ... α_{p}}}} \partial_{β_{μ}} X^{γ}}} \\ = & \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} (\partial_{ε} X^{σ}) (\partial_{σ} Y^{β_{λ}}) + \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} X^{α} \partial_{α} \partial_{ε} Y^{β_{λ}} \\ - \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} (\partial_{ε} Y^{σ}) (\partial_{σ} X^{β_{λ}}) - \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} Y^{α} \partial_{α} \partial_{ε} X^{β_{λ}} \\ \underset{- \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} (\partial_{β_{μ}} \underset{{[X, Y]}^{ε}}{\underset{︸}{(X^{α} \partial_{α_{μ}} Y^{ε} - Y^{α} \partial_{α_{μ}} X^{ε})}})}{\underset{︸}{+ \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} (- X^{α} \partial_{α_{μ}} \partial_{β_{μ}} Y^{ε} + \partial_{β_{μ}} Y^{α} \partial_{α_{μ}} X^{ε} + Y^{α} \partial_{α_{μ}} \partial_{β_{μ}} X^{ε} - \partial_{β_{μ}} X^{α} \partial_{α_{μ}} Y^{ε})}} \\ = & \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} {[X, Y]}^{β_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} (\partial_{β_{μ}} {[X, Y]}^{ε}) \end{array}$ \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 $^{cc} [X, Y] = (\begin{array}{l} - p_{ε} (\partial_{β} {[X, Y]}^{ε}) \\ {[X, Y]}^{β} \\ \sum_{λ = 1}^{p} t_{β_{1} ... β_{q}}^{α_{1} ... ε ... α_{p}} \partial_{ε} {[X, Y]}^{β_{λ}} - \sum_{μ = 1}^{q} t_{β_{1} ... ε ... β_{q}}^{α_{1} ... α_{p}} \partial_{β_{μ}} {[X, Y]}^{ε} \end{array})$ ^{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 $(x^{\bar{β}}, x^{β}, x^{\bar{\bar{β}}})$ ({x^{\overline \beta }},{x^\beta },{x^{\overline {\overline \beta } }}) on $t_{q}^{p} (M_{n})$ t_q^p({M_n}) . Thus Theorem 5 is proved.

eISSN:: 2444-8656
Sprache:: Englisch

Zeitrahmen der Veröffentlichung:: Volume Open
Fachgebiete der Zeitschrift:: Biologie, andere, Mathematik, Angewandte Mathematik, Allgemeines, Physik

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

Online veröffentlicht: 31. Dez. 2020

Seitenbereich: 421 - 428

Eingereicht: 17. Juli 2019

Akzeptiert: 26. Okt. 2019

DOI: https://doi.org/10.2478/amns.2020.2.00066

SchlüsselwörterVector field, complete lift, pull-back bundle, cotangent bundle, semi-tensor bundle

© 2020 Furkan Yildirim, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Schlüsselwörter
Vector field, complete lift, pull-back bundle, cotangent bundle, semi-tensor bundle