Some Properties of Diagonal Lifts in Semi-Cotangent Bundles

Defining some structure on the tangent bundles and cotangant bundles to obtain subtle information about the topology and geometry of the manifold is the main way for mathematicians. Due to this feature, many authors have been systematically worked on them [1, 2, 9, 13, 14]. One of these studies is analyzing some properties of diagonal lift of tensor fields of type (1,1) in semi-cotangent bundles with the help of adapted frames

Let M_n be an n-dimensional differentiable manifold of class C^∞ and T (M_n) the tangent bundle determined by a natural projection (submersion) π₁ : T (M_n) → M_n. We use the notation (xⁱ) = (x^α̅,x^α), where the indices i, j,... run from 1 to 2n, the indices α,β,... from 1 to n and the indices α̅,β̅ ,... from n + 1 to 2n, x^α are coordinates in M_n, x^α̅ = y^α are fibre coordinates of the tangent bundle T (M_n). If (xi′)=(xα¯′,xα′) ({x^{{i^\prime}}}) = ({x^{{{\overline \alpha }^\prime}}},{x^{{\alpha ^\prime}}}) is another system of local adapted coordinates in the tangent bundle T (M_n), then we have (1) {xα¯′=∂xα′∂xβyβ,xα′=xα′(xβ). \left\{ {\matrix{ {{x^{{{\overline \alpha }^\prime}}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}}{y^\beta },} \hfill & \cr {{x^{{\alpha ^\prime}}} = {x^{{\alpha ^\prime}}}\left( {{x^\beta }} \right).} \hfill & \cr } } \right.

The Jacobian of (1) has components (Aji′)=(∂xi′∂xj)=(Aβα′Aβεα′yε0Aβα′), (A_j^{{i^\prime}}) = \left( {{{\partial {x^{{i^\prime}}}} \over {\partial {x^j}}}} \right) = \left( {\matrix{ {A_\beta ^{{\alpha ^\prime}}} & {A_{\beta \varepsilon }^{{\alpha ^\prime}}{y^\varepsilon }} \cr 0 & {A_\beta ^{{\alpha ^\prime}}} \cr } } \right), where Aβα′=∂xα′∂xβ A_\beta ^{{\alpha ^\prime}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}} , Aβεα′=∂2xα′∂xβ∂xε A_{\beta \varepsilon }^{{\alpha ^\prime}} = {{{\partial ^2}{x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }\partial {x^\varepsilon }}} . Let Tx∗(Mn)(x=π1(x˜),x˜=(xα¯,xα)∈T(Mn)) T_x^ * ({M_n})(x = {\pi _1}(\widetilde x),\widetilde x = ({x^{\overline \alpha }},{x^\alpha }) \in T\left( {{M_n}} \right)) be the cotangent space at a point x of M_n. If p_α are components of p∈Tx∗(Mn) p \in T_x^ * ({M_n}) with respect to the natural coframe {dx^α}, i.e. p = p_i dxⁱ, then by definition the set t^*(M_n) of all points (xI)=(xα¯,xα,xα¯¯) \left( {{x^I}} \right) = ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) , xα¯¯=pα {x^{\overline {\overline \alpha } }} = {p_\alpha } ; I,J,... = 1,...,3n with projection π₂ : t^*(M_n) → T (M_n) (i.e. π2:(xα¯,xα,xα¯¯)→(xα¯,xα)) {\pi _2}:({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^{\overline \alpha }},{x^\alpha })) ) is a semi-cotangent (pull-back [12]) bundle of the cotangent bundle by submersion π₁ : T (M_n) → M_n (For definition of the pull-back bundle, see for example [3], [5], [6], [7]). It is remarkable fact that the semi-cotangent (pull-back) bundle has a degenerate symplectic structure [12] ω=(ωAB)=dp=(00000−δβα0δαβ0). \omega = ({\omega _{AB}}) = dp = \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & { - \delta _\beta ^\alpha } \cr 0 & {\delta _\alpha ^\beta } & 0 \cr } } \right).

It is clear that the pull-back bundle t^*(M_n) of the cotangent bundle T^* (M_n) also has the natural bundle structure over M_n, its bundle projection π: t*(M_n) → M_n being defined by π:(xα¯,xα,xα¯¯)→(xα) \pi :({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^\alpha }) , and hence π = π₁ ○ π₂. Thus (t^*(M_n),π₁ ○ π₂) is the composite bundle [ [14], p.9] or step-like bundle [15].

The main purpose of the present paper is to study complete lift of vector fields and tensor fields of type (1,1) from tangent bundle T (M_n) to semi-cotangent (pull-back) bundle (t^*(M_n),π₂).

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 (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.

To a transformation (1) of local coordinates of T (M_n), there corresponds on t^*(M_n) the coordinate transformation [10]: (2) {xα¯′=∂xα′∂xβyβ,xα′=xα′(xβ),xα¯¯′=∂xβ∂xα′pβ. \left\{ {\matrix{ {{x^{{{\overline \alpha }^\prime}}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}}{y^\beta },} \cr {\;{x^{{\alpha ^\prime}}} = {x^{{\alpha ^\prime}}}\left( {{x^\beta }} \right),} \cr {{x^{{{\overline {\overline \alpha } }^\prime}}} = {{\partial {x^\beta }} \over {\partial {x^{{\alpha ^\prime}}}}}{p_\beta }.} \cr } } \right.

The Jacobian of (2) has components [10]: (3) A¯=(AJI′)=(Aβα′Aβεα′yε00Aβα′00pσAββ′Aβ′α′σAα′β), \overline A = (A_J^{{I^\prime}}) = \left( {\matrix{ {A_\beta ^{{\alpha ^\prime}}} & {A_{\beta \varepsilon }^{{\alpha ^\prime}}{y^\varepsilon }} & 0 \cr 0 & {A_\beta ^{{\alpha ^\prime}}} & 0 \cr 0 & {{p_\sigma }A_\beta ^{{\beta ^\prime}}A_{{\beta ^\prime}{\alpha ^\prime}}^\sigma } & {A_{{\alpha ^\prime}}^\beta } \cr } } \right), where Aβεα′=∂2xα′∂xβ∂xε, Aβ′α′α=∂2xα∂xβ′∂xα′. A_{\beta \varepsilon }^{{\alpha ^\prime}} = {{{\partial ^2}{x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }\partial {x^\varepsilon }}},\quad A_{{\beta ^\prime}{\alpha ^\prime}}^\alpha = {{{\partial ^2}{x^\alpha }} \over {\partial {x^{{\beta ^\prime}}}\partial {x^{{\alpha ^\prime}}}}}.

We denote by ℑqp(T(Mn)) \Im _q^p(T({M_n})) and ℑqp(Mn) \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.

Let θ be a covector field on T (M_n). Then the transformation p → θ_p, θ_p being the value of θ at p ∈ T (M_n), determines a cross-section β_θ of semi-cotangent bundle. Thus if σ : M_n → T^* (M_n) is a cross-section of (T^* (M_n), π̃,M_n), such that π̃ ○ σ = I_(Mn), an associated cross-section β_θ : T (M_n) → t^*(M_n) of semi-cotangent (pull-back) bundle (t^*(M_n),π₂,T (M_n)) of cotangent bundle by using projection (submersion) of the tangent bundle T (M_n) defined by [ [4], p. 217–218], [ [9], p. 301]: βθ(xα¯,xα)=(xα¯,xα,σ∘π1(xα¯,xα))=(xα¯,xα,σ(xα))=(xα¯,xα,θα(xβ)). {\beta _\theta }\left( {{x^{\overline \alpha }},{x^\alpha }} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },\sigma \circ {\pi _1}\left( {{x^{\overline \alpha }},{x^\alpha }} \right)} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },\sigma \left( {{x^\alpha }} \right)} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },{\theta _\alpha }\left( {{x^\beta }} \right)} \right).

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

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

Let X∈ℑ01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) , i.e. X = X^α∂_α. We denote by BX the vector field with local components (5) BX:(B(β¯)AXβ¯)=(δβ¯αXβ¯00)=(Aβ¯αXβ¯00)=(Xα00) BX:\left( {B_{\left( {\overline \beta } \right)}^A{X^{\overline \beta }}} \right) = \left( {\matrix{ {\delta _{\overline \beta }^\alpha {X^{\overline \beta }}} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) = \left( {\matrix{ {A_{\overline \beta }^\alpha {X^{\overline \beta }}} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) = \left( {\matrix{ {{X^\alpha }} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t^*(M_n), which is defined globally along β_θ (T (M_n)). Then a mapping B:ℑ01(T(Mn))→ℑ01(βθ(T(Mn))) B:\Im _0^1(T({M_n})) \to \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) is defined by (5). The mapping B is the differential of β_θ : T (M_n) → t^* (M_n) and so an isomorphism of ℑ01(T(Mn)) \Im _0^1(T({M_n})) onto ℑ01(βθ(T(Mn))) \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) [10].

Since a cross-section is locally expressed by x^α̅ = y^α = const., xα¯¯=pα=const. {x^{\overline {\overline \alpha } }} = {p_\alpha } = const. , x^α = x^α, x^α being considered as parameters. Differentiating (4) by x^α, we have vector fields C_(β) (β = n + 1,...,2n) with components C(β)=∂xA∂xβ=∂βxA=(∂βVα∂βxα∂βθα), {C_{\left( \beta \right)}} = {{\partial {x^A}} \over {\partial {x^\beta }}} = {\partial _\beta }{x^A} = \left( {\matrix{ {{\partial _\beta }{V^\alpha }} \hfill \cr {{\partial _\beta }{x^\alpha }} \hfill \cr {{\partial _\beta }{\theta _\alpha }} \hfill \cr } } \right), which are tangent to the cross-section β_θ (T (M_n)).

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

Let X∈ℑ01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) . Then we denote by CX the vector field with local components (6) CX:(C(β)AXβ)=(Xβ∂βVαXαXβ∂βθα) CX:\left( {C_{\left( \beta \right)}^A{X^\beta }} \right) = \left( {\matrix{ {{X^\beta }{\partial _\beta }{V^\alpha }} \hfill \cr {{X^\alpha }} \hfill \cr {{X^\beta }{\partial _\beta }{\theta _\alpha }} \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in t^* (M_n), which is defined globally along β_θ (T (M_n)). Then a mapping C:ℑ01(T(Mn))→ℑ01(βθ(T(Mn))) C:\Im _0^1(T({M_n})) \to \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) is defined by (6). The mapping C is the differential of β_θ : T (M_n) → t^* (M_n) and so an isomorphism of ℑ01(T(Mn)) \Im _0^1(T({M_n})) onto ℑ01(βθ(T(Mn))) \Im _0^1({\beta _\theta }\left( {T({M_n})} \right)) [10].

Now, consider ω∈ℑ10(Mn) \omega \in \Im _1^0({M_n}) and vector field X∈ℑ01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) , then ^vvω (vertical lift), ^ccX (complete lift) and ^HHX (horizontal lift) have respectively, components on the semi-cotangent bundle t^*(M_n) [11]: (7) vvω=(00ωα), ccX=(yε∂εXαXα−pσ(∂αXσ)), HHX=(−ΓβαXβXαXβΓβα) ^{^{vv}}\omega = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {{\omega _\alpha }} \hfill \cr } } \right),{\quad ^{cc}}X = \left( {\matrix{ {{y^\varepsilon }{\partial _\varepsilon }{X^\alpha }} \hfill \cr {{X^\alpha }} \hfill \cr { - {p_\sigma }({\partial _\alpha }{X^\sigma })} \hfill \cr } } \right),{\quad ^{HH}}X = \left( {\matrix{ { - \Gamma _\beta ^\alpha {X^\beta }} \hfill \cr {{X^\alpha }} \hfill \cr {{X^\beta }{\Gamma _\beta }_\alpha } \hfill \cr } } \right) with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) , where Γβα=VεΓεαβ, Γβα=θεΓβεα. \Gamma _\beta ^\alpha = {V^\varepsilon }{{\Gamma_\varepsilon^\alpha}_\beta},\quad {\Gamma _{\beta\, \alpha}} = {\theta _\varepsilon }{{\Gamma_{\beta \alpha}^\varepsilon}}.

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

Let ω be an 1-form with local components ω_α on M_n, so that ω is a 1-form with local expression ω = ω_αdx^α. We denote by Eω the vector field with local components (8) Eω:(E(β¯¯)Aωβ)=(00ωα), E\omega :\left( {E_{\left( {\overline {\overline \beta } } \right)}^A{\omega _\beta }} \right) = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {{\omega _\alpha }} \hfill \cr } } \right), which is tangent to the fibre. Then a mapping E:ℑ10(Mn)→ℑ01(t∗(Mn)) E:\Im _1^0({M_n}) \to \Im _0^1({t^ * }({M_n})) is defined by (8) and so an isomorphism of ℑ10(Mn) \Im _1^0({M_n}) in to ℑ01(t∗(Mn)) \Im _0^1({t^ * }({M_n})) [10].

We consider in π−¹ (U) 3n local vector fields B_(β̅), C_(β) and E(β¯¯) {E_{\left( {\overline {\overline \beta } } \right)}} along β_θ (T (M_n)), which are respectively represented by B(β¯)=B∂∂xβ¯, C(β)=C∂∂xβ, E(β¯¯)=Edxβ. {B_{\left( {\overline \beta } \right)}} = B{\partial \over {\partial {x^{\overline \beta }}}},\quad {C_{\left( \beta \right)}} = C{\partial \over {\partial {x^\beta }}},\quad {E_{\left( {\overline {\overline \beta } } \right)}} = Ed{x^\beta }.

On the other hand, on putting C(β¯¯)=E(β¯¯) {C_{\left( {\overline {\overline \beta } } \right)}} = {E_{\left( {\overline {\overline \beta } } \right)}} , we write the adapted frame of β_θ (T (M_n)) as {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} . The adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} of β_θ (T (M_n)) is given by the matrix (9) A˜=(A˜BA)=(δβα∂βVα00δβα00∂βθαδαβ). \widetilde A = \left( {\widetilde A_B^A} \right) = \left( {\matrix{ {\delta _\beta ^\alpha } & {{\partial _\beta }{V^\alpha }} & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {{\partial _\beta }{\theta _\alpha }} & {\delta _\alpha ^\beta } \cr } } \right).

Since the matrix à in (9) is non-singular, it has the inverse. Denoting this inverse by (Ã)⁻¹, we have (10) (A˜)−1=(A˜CB)−1=(δθβ−∂θVβ00δθβ00−∂θθβδβθ), {\left( {\widetilde A} \right)^{ - 1}} = {\left( {\widetilde A_C^B} \right)^{ - 1}} = \left( {\matrix{ {\delta _\theta ^\beta } & { - {\partial _\theta }{V^\beta }} & 0 \cr 0 & {\delta _\theta ^\beta } & 0 \cr 0 & { - {\partial _\theta }{\theta _\beta }} & {\delta _\beta ^\theta } \cr } } \right), where A˜(A˜)−1=(A˜BA)(A˜CB)−1=δCA=I˜ \widetilde A{\left( {\widetilde A} \right)^{ - 1}} = (\widetilde A_B^A){\left( {\widetilde A_C^B} \right)^{ - 1}} = \delta _C^A = \widetilde I , where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) , C=(θ¯,θ,θ¯¯) C = \left( {\overline \theta ,\theta ,\overline {\overline \theta } } \right) .

Then we see from Theorem 1 that the complete lift ^ccX of a vector field X∈ℑ01(T(Mn)) X \in \Im _0^1(T({M_n})) has along β_θ (T (M_n)) components of the form ccX:(LVXαXα−LXθα) ^{cc}X:\left( {\matrix{ {{L_V}{X^\alpha }} \hfill \cr {{X^\alpha }} \hfill \cr { - {L_X}{\theta _\alpha }} \hfill \cr } } \right) with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} [10].

BX, CX and Eω also have components: (11) BX=(Xα00), CX=(0Xα0), Eω=(00ωα) BX = \left( {\matrix{ {{X^\alpha }} \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right),\quad CX = \left( {\matrix{ 0 \hfill \cr {{X^\alpha }} \hfill \cr 0 \hfill \cr } } \right),\quad E\omega = \left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {{\omega _\alpha }} \hfill \cr } } \right) respectively, with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} of the cross-section β_θ (T (M_n)) determined by a 1-form θ on T (M_n) [10].

Suppose now that F∈ℑ11(T(Mn)) F \in \Im _1^1(T({M_n})) and F has local components Fβα F_\beta ^\alpha in a neighborhood U of M_n, F=Fβα∂α⊗dxβ F = F_\beta ^\alpha {\partial _\alpha } \otimes d{x^\beta } . Then the semi-cotangent (pull-back) bundle t^* (M_n) of cotangent bundle T^* (M_n) by using projection of the tangent bundle T (M_n) admits the complete lift ^ccF of F with components [11]: (12) ccF=( ccFJI)=(Fβαyε∂εFβα00Fβα00pσ(∂βFασ−∂αFβσ)Fαβ), ^{cc}F = ({\;^{^{cc}}}F_J^I) = \left( {\matrix{ {F_\beta ^\alpha } & {{y^\varepsilon }{\partial _\varepsilon }F_\beta ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & {{p_\sigma }({\partial _\beta }F_\alpha ^\sigma - {\partial _\alpha }F_\beta ^\sigma )} & {F_\alpha ^\beta } \cr } } \right), with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) on t^* (M_n). Then ^ccF has components FBA F_B^A given by (13) ccF=(ccFBA)=(FβαLVFβα00Fβα00φFθFαβ) ^{cc}F{ = (^{^{cc}}}F_B^A) = \left( {\matrix{ {F_\beta ^\alpha } & {{L_V}F_\beta ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & {{\varphi _F}\theta } & {F_\alpha ^\beta } \cr } } \right) with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} of the cross-section β_θ (T (M_n)) determined by a 1-form θ in T (M_n), where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) [10]. Also, the component ccFβα¯¯ ^{^{cc}}F_\beta ^{\overline {\overline \alpha } } of ccFBA ^{^{cc}}F_B^A is defined as Tachibana operator φ_Fθ of F, i.e., ccFβα¯¯=ϕFθ=(∂βFασ−∂αFβσ)θσ−Fβγ∂γθα+Fαγ∂βθγ, {\;^{^{cc}}}F_\beta ^{\overline {\overline \alpha } } = {\phi _F}\theta = ({\partial _\beta }F_\alpha ^\sigma - {\partial _\alpha }F_\beta ^\sigma ){\theta _\sigma } - F_\beta ^\gamma {\partial _\gamma }{\theta _\alpha } + F_\alpha ^\gamma {\partial _\beta }{\theta _\gamma }, and LVFβα {L_V}F_\beta ^\alpha denotes the Lie derivative of Fβα F_\beta ^\alpha with respect to V, i.e., LVFβα=Vγ∂γFβα+Fγα∂βVγ−Fβγ∂γVα. {L_V}F_\beta ^\alpha = {V^\gamma }{\partial _\gamma }F_\beta ^\alpha + F_\gamma ^\alpha {\partial _\beta }{V^\gamma } - F_\beta ^\gamma {\partial _\gamma }{V^\alpha }.

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

Then 3n local vector fields Y_(α), ^HHX_(α) and ^vvθ ^(α) have respectively components of the form (14) Y(α):(δαβ00), HHX(α):(−ΓβαδαβΓβα), vvθ(α):(00δβα) {Y_{\left( \alpha \right)}}:\left( {\matrix{ {\delta _\alpha ^\beta } \hfill \cr 0 \hfill \cr 0 \hfill \cr } } \right),{\quad ^{HH}}{X_{\left( \alpha \right)}}:\left( {\matrix{ { - \Gamma _\beta ^\alpha } \hfill \cr {\delta _\alpha ^\beta } \hfill \cr {{\Gamma _\beta }_\alpha } \hfill \cr } } \right),{\quad ^{vv}}{\theta ^{\left( \alpha \right)}}:\left( {\matrix{ 0 \hfill \cr 0 \hfill \cr {\delta _\beta ^\alpha } \hfill \cr } } \right) with respect to the induced coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) in π⁻¹ (U), where we have used (7). We call the set {Y_(α),^HH X_(α),^vv θ ^(α)} the frame adapted to the symmetric affine connection ∇ in π⁻¹ (U). On putting (15) e^(α¯)=Y(α), e^(α)=HHX(α), e^(α¯¯)=vvθ(α) {\widehat e_{\left( {\overline \alpha } \right)}} = {Y_{\left( \alpha \right)}},\quad {\widehat e_{\left( \alpha \right)}}{ = ^{HH}}{X_{\left( \alpha \right)}},\quad {\widehat e_{\left( {\overline {\overline \alpha } } \right)}}{ = ^{vv}}{\theta ^{\left( \alpha \right)}} we write the adapted frame as (16) {e^(B)}={e^(α¯),e^(α),e^(α¯¯)}. \left\{ {{{\widehat e}_{\left( B \right)}}} \right\} = \left\{ {{{\widehat e}_{\left( {\overline \alpha } \right)}},{{\widehat e}_{\left( \alpha \right)}},{{\widehat e}_{\left( {\overline {\overline \alpha } } \right)}}} \right\}.

The adapted frame {e^(B)}={e^(α¯),e^(α),e^(α¯¯)} \left\{ {{{\widehat e}_{\left( B \right)}}} \right\} = \left\{ {{{\widehat e}_{\left( {\overline \alpha } \right)}},{{\widehat e}_{\left( \alpha \right)}},{{\widehat e}_{\left( {\overline {\overline \alpha } } \right)}}} \right\} is given by the matrix (17) A^=(A^BA)=(δβα−Γβα00δβα00Γβαδαβ). \widehat A = \left( {\widehat A_B^A} \right) = \left( {\matrix{ {\delta _\beta ^\alpha } & { - \Gamma _\beta ^\alpha } & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {{\Gamma _\beta }_\alpha } & {\delta _\alpha ^\beta } \cr } } \right).

Since the matrix A^ \widehat A in (17) is non-singular, it has the inverse. Denoting this inverse by (A^)−1 {\left( {\widehat A} \right)^{ - 1}} , we have (18) (A^)−1=(A^CB)−1=(δθβΓθβ00δθβ00−Γθβδβθ), {\left( {\widehat A} \right)^{ - 1}} = {\left( {\widehat A_C^B} \right)^{ - 1}} = \left( {\matrix{ {\delta _\theta ^\beta } & {\Gamma _\theta ^\beta } & 0 \cr 0 & {\delta _\theta ^\beta } & 0 \cr 0 & { - {\Gamma _\theta }_\beta } & {\delta _\beta ^\theta } \cr } } \right), where A^(A^)−1=(A^BA)(A^CB)−1=δCA=I˜ \widehat A{\left( {\widehat A} \right)^{ - 1}} = (\widehat A_B^A){\left( {\widehat A_C^B} \right)^{ - 1}} = \delta _C^A = \widetilde I , where A=(α¯,α,α¯¯) A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right) , B=(β¯,β,β¯¯) B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right) , C=(θ¯,θ,θ¯¯) C = \left( {\overline \theta ,\theta ,\overline {\overline \theta } } \right) .

If we take account of (16), we see that the diagonal lift ^DDF of F∈ℑ11(T(Mn)) F \in \Im _1^1(T({M_n})) has components [10]: (19) DDF=(DDFJI)=(−Fβα−ΓεαFβε−ΓβεFεα00Fβα00ΓβσFασ+ΓασFβσ−Fαβ), ^{^{DD}}F{ = (^{^{DD}}}F_J^I) = \left( {\matrix{ { - F_\beta ^\alpha } & { - \Gamma _\varepsilon ^\alpha F_\beta ^\varepsilon - \Gamma _\beta ^\varepsilon F_\varepsilon ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & {{\Gamma _\beta }_\sigma F_\alpha ^\sigma + {\Gamma _\alpha }_\sigma F_\beta ^\sigma } & { - F_\alpha ^\beta } \cr } } \right), with respect to the coordinates (xα¯,xα,xα¯¯) ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) on t^* (M_n), where Γεα=yγΓγαε, Γασ=pγΓαγσ. {\Gamma _\varepsilon ^\alpha} = {y^\gamma }{{\Gamma_\gamma^\alpha}_\varepsilon},\quad {\Gamma _{\alpha\sigma} } = {p_\gamma }{{\Gamma_\alpha^\gamma}_\sigma}. which proves (19).

We now see, from (16), that the diagonal lift ^DDF of F∈ℑ11(T(Mn)) F \in \Im _1^1(T({M_n})) has components of the form DDF=(DDFBA)=(−Fβα000Fβα000−Fαβ) ^{^{DD}}F{ = (^{^{DD}}}F_B^A) = \left( {\matrix{ { - F_\beta ^\alpha } & 0 & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & 0 & { - F_\alpha ^\beta } \cr } } \right) with respect to the adapted frame {e^(B)} \left\{ {{{\widehat e}_{\left( B \right)}}} \right\} in t^* (M_n).

We now obtain from (19) that the diagonal lift ^DDF of an affinor field F∈ℑ11(T(Mn)) F \in \Im _1^1(T({M_n})) has along β_θ (T (M_n)) components of the form [10]: (20) DDF:(−Fβα−(∇εVα)Fβε−(∇βVε)Fεα00Fβα00−(∇βθσ)Fασ−(∇αθσ)Fβσ−Fαβ), ^{^{DD}}F:\left( {\matrix{ { - F_\beta ^\alpha } & { - \left( {{\nabla _\varepsilon }{V^\alpha }} \right)F_\beta ^\varepsilon - \left( {{\nabla _\beta }{V^\varepsilon }} \right)F_\varepsilon ^\alpha } & 0 \cr 0 & {F_\beta ^\alpha } & 0 \cr 0 & { - \left( {{\nabla _\beta }{\theta _\sigma }} \right)F_\alpha ^\sigma - \left( {{\nabla _\alpha }{\theta _\sigma }} \right)F_\beta ^\sigma } & { - F_\alpha ^\beta } \cr } } \right), with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} .

Then we see from (7) that the horizontal lift ^HHX of a vector field X∈ℑ01(T(Mn)) X \in \Im _0^1\left( {T({M_n})} \right) has along β_θ (T (M_n)) components of the form (21) HHX:(−Xβ(∇βVα)Xα−(∇βθα)Xβ) ^{HH}X:\left( {\matrix{ { - {X^\beta }\left( {{\nabla _\beta }{V^\alpha }} \right)} \hfill \cr {{X^\alpha }} \hfill \cr { - \left( {{\nabla _\beta }{\theta _\alpha }} \right){X^\beta }} \hfill \cr } } \right) with respect to the adapted frame {B(β¯),C(β),C(β¯¯)} \left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\} [10].

Using (7), (20) and (21), we have along β_θ (T (M_n)):

Thus, we have ^DDF^DDG +^DD G^DDF = ^HH (FG + GF).

Thus, we have Theorem 5.

Putting F = G in Theorem 4 and Theorem 5, we have HHFDDF=DDFHHF=DD(F2)(DDF)2p=HH(F2p),(DDF)2p+1=DD(F2p+1),(p=1,2,...) \matrix{ {^{HH}{F^{DD}}F{ = ^{DD}}{F^{HH}}F{ = ^{DD}}({F^2})} \hfill \cr {{{{(^{DD}}F)}^{2p}}{ = ^{HH}}({F^{2p}}{{),(}^{DD}}F{)^{2p + 1}}{ = ^{DD}}({F^{2p + 1}}),(p = 1,2,...)} \hfill \cr } for any F∈ℑ11(T(Mn)) F \in \Im _1^1(T({M_n})) .