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Lifts of Vector Fields on a Cross-Section in the Semi-Cotangent Bundle
Defining some structure on the tangent bundles and cotangant bundles to obtain subtle information about the topology and geometry of the manifold is the main way for mathematicians. Due to this feature, many authors have been systematically worked on them [1, 2, 9, 13, 14]. One of these studies is analyzing some properties of diagonal lift of tensor fields of type (1,1) in semi-cotangent bundles with the help of adapted frames
Let Mn be an n-dimensional differentiable manifold of class C∞ and T (Mn) the tangent bundle determined by a natural projection (submersion) π1 : T (Mn) → Mn. We use the notation (xi) = (xα̅,xα), where the indices i, j,... run from 1 to 2n, the indices α,β,... from 1 to n and the indices α̅,β̅ ,... from n + 1 to 2n, xα are coordinates in Mn, xα̅ = yα are fibre coordinates of the tangent bundle T (Mn). If
({x^{{i^\prime}}}) = ({x^{{{\overline \alpha }^\prime}}},{x^{{\alpha ^\prime}}})
is another system of local adapted coordinates in the tangent bundle T (Mn), then we have
\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
(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
A_\beta ^{{\alpha ^\prime}} = {{\partial {x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }}}
,
A_{\beta \varepsilon }^{{\alpha ^\prime}} = {{{\partial ^2}{x^{{\alpha ^\prime}}}} \over {\partial {x^\beta }\partial {x^\varepsilon }}}
. Let
T_x^ * ({M_n})(x = {\pi _1}(\widetilde x),\widetilde x = ({x^{\overline \alpha }},{x^\alpha }) \in T\left( {{M_n}} \right))
be the cotangent space at a point x of Mn. If pα are components of
p \in T_x^ * ({M_n})
with respect to the natural coframe {dxα}, i.e. p = pi dxi, then by definition the set t*(Mn) of all points
\left( {{x^I}} \right) = ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})
,
{x^{\overline {\overline \alpha } }} = {p_\alpha }
; I,J,... = 1,...,3n with projection π2 : t*(Mn) → T (Mn) (i.e.
{\pi _2}:({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^{\overline \alpha }},{x^\alpha }))
) is a semi-cotangent (pull-back [12]) bundle of the cotangent bundle by submersion π1 : T (Mn) → Mn (For definition of the pull-back bundle, see for example [3], [5], [6], [7]). It is remarkable fact that the semi-cotangent (pull-back) bundle has a degenerate symplectic structure [12]
\omega = ({\omega _{AB}}) = dp = \left( {\matrix{ 0 & 0 & 0 \cr 0 & 0 & { - \delta _\beta ^\alpha } \cr 0 & {\delta _\alpha ^\beta } & 0 \cr } } \right).
It is clear that the pull-back bundle t*(Mn) of the cotangent bundle T* (Mn) also has the natural bundle structure over Mn, its bundle projection π: t*(Mn) → Mn being defined by
\pi :({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }}) \to ({x^\alpha })
, and hence π = π1 ○ π2. Thus (t*(Mn),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 (Mn) to semi-cotangent (pull-back) bundle (t*(Mn),π2).
We denote by
\Im _q^p(T({M_n}))
and
\Im _q^p({M_n})
the modules over F (T (Mn)) and F (Mn) of all tensor fields of type (p,q) on T (Mn) and Mn respectively, where F (T (Mn)) and F (Mn) denote the rings of real-valued C∞−functions on T (Mn) and Mn, respectively.
To a transformation (1) of local coordinates of T (Mn), there corresponds on t*(Mn) the coordinate transformation [10]:
\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.
We denote by
\Im _q^p(T({M_n}))
and
\Im _q^p({M_n})
the modules over F (T (Mn)) and F (Mn) of all tensor fields of type (p,q) on T (Mn) and Mn, respectively, where F (T (Mn)) and F (Mn) denote the rings of real-valued C∞ −functions on T (Mn) and Mn, respectively.
Let θ be a covector field on T (Mn). Then the transformation p → θp, θp being the value of θ at p ∈ T (Mn), determines a cross-section βθ of semi-cotangent bundle. Thus if σ : Mn → T* (Mn) is a cross-section of (T* (Mn), π̃,Mn), such that π̃ ○ σ = I(Mn), an associated cross-section βθ : T (Mn) → t*(Mn) of semi-cotangent (pull-back) bundle (t*(Mn),π2,T (Mn)) of cotangent bundle by using projection (submersion) of the tangent bundle T (Mn) defined by [ [4], p. 217–218], [ [9], p. 301]:
{\beta _\theta }\left( {{x^{\overline \alpha }},{x^\alpha }} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },\sigma \circ {\pi _1}\left( {{x^{\overline \alpha }},{x^\alpha }} \right)} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },\sigma \left( {{x^\alpha }} \right)} \right) = \left( {{x^{\overline \alpha }},{x^\alpha },{\theta _\alpha }\left( {{x^\beta }} \right)} \right).
If the covector field θ has the local components θα (xβ), the cross-section βθ (T (Mn)) of t*(Mn) is locally expressed by
{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
{x^A} = ({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})
in t* (Mn). xα̅ = yα being considered as parameters. Differentiating (4) by xα̅ = yα, we have vector fields B(β̅) (β̅ = 1,...,n) with components
{B_{\left( {\overline \beta } \right)}} = {{\partial {x^A}} \over {\partial {x^{\overline \beta }}}} = {\partial _{\overline \beta }}{x^A} = \left( {\matrix{ {{\partial _{\overline \beta }}{V^\alpha }} \hfill \cr {{\partial _{\overline \beta }}{x^\alpha }} \hfill \cr {{\partial _{\overline \beta }}{\theta _\alpha }} \hfill \cr } } \right),
which are tangent to the cross-section βθ (T (Mn)) [10].
Let
X \in \Im _0^1\left( {T({M_n})} \right)
, i.e. X = Xα∂α. We denote by BX the vector field with local components
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
({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})
in t*(Mn), which is defined globally along βθ (T (Mn)). Then a mapping
B:\Im _0^1(T({M_n})) \to \Im _0^1({\beta _\theta }\left( {T({M_n})} \right))
is defined by (5). The mapping B is the differential of βθ : T (Mn) → t* (Mn) and so an isomorphism of
\Im _0^1(T({M_n}))
onto
\Im _0^1({\beta _\theta }\left( {T({M_n})} \right))
[10].
Since a cross-section is locally expressed by xα̅ = yα = const.,
{x^{\overline {\overline \alpha } }} = {p_\alpha } = const.
, xα = xα, xα being considered as parameters. Differentiating (4) by xα, we have vector fields C(β) (β = n + 1,...,2n) with components
{C_{\left( \beta \right)}} = {{\partial {x^A}} \over {\partial {x^\beta }}} = {\partial _\beta }{x^A} = \left( {\matrix{ {{\partial _\beta }{V^\alpha }} \hfill \cr {{\partial _\beta }{x^\alpha }} \hfill \cr {{\partial _\beta }{\theta _\alpha }} \hfill \cr } } \right),
which are tangent to the cross-section βθ (T (Mn)).
Let
X \in \Im _0^1\left( {T({M_n})} \right)
. Then we denote by CX the vector field with local components
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
({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})
in t* (Mn), which is defined globally along βθ (T (Mn)). Then a mapping
C:\Im _0^1(T({M_n})) \to \Im _0^1({\beta _\theta }\left( {T({M_n})} \right))
is defined by (6). The mapping C is the differential of βθ : T (Mn) → t* (Mn) and so an isomorphism of
\Im _0^1(T({M_n}))
onto
\Im _0^1({\beta _\theta }\left( {T({M_n})} \right))
[10].
On the other hand, the fibre is locally represented by
{x^{\overline \alpha }} = {y^\alpha } = const.,\quad {x^\alpha } = const.,\quad x\overline {^\alpha } = {p_\alpha } = {p_\alpha },pα being considered as parameters. Thus, on differentiating with respect to pα, we easily see that the vector fields
{E_{\left( {\overline {\overline \beta } } \right)}}{ = ^{vv}}\left( {d{x^\beta }} \right)\;(\overline {\overline \beta } = 2n + 1,...,3n)
with components
{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
\delta _\alpha ^\beta = A_\alpha ^\beta = {{\partial {x^\beta }} \over {\partial {x^\alpha }}}.
Let ω be an 1-form with local components ωα on Mn, so that ω is a 1-form with local expression ω = ωαdxα. We denote by Eω the vector field with local components
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:\Im _1^0({M_n}) \to \Im _0^1({t^ * }({M_n}))
is defined by (8) and so an isomorphism of
\Im _1^0({M_n})
in to
\Im _0^1({t^ * }({M_n}))
[10].
We consider in π−1 (U) 3n local vector fields B(β̅), C(β) and
{E_{\left( {\overline {\overline \beta } } \right)}}
along βθ (T (Mn)), which are respectively represented by
{B_{\left( {\overline \beta } \right)}} = B{\partial \over {\partial {x^{\overline \beta }}}},\quad {C_{\left( \beta \right)}} = C{\partial \over {\partial {x^\beta }}},\quad {E_{\left( {\overline {\overline \beta } } \right)}} = Ed{x^\beta }.
Theorem 1
Let X be a vector field on T (Mn). We have along βθ (T (Mn)) the formula^{cc}X = CX + B\left( {{L_V}X} \right) + E\left( {\; - {L_X}\theta } \right),where LV X denotes the Lie derivative of X with respect to V, and LX θ denotes the Lie derivative of θ with respect to X [10].
On the other hand, on putting
{C_{\left( {\overline {\overline \beta } } \right)}} = {E_{\left( {\overline {\overline \beta } } \right)}}
, we write the adapted frame of βθ (T (Mn)) as
\left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\}
. The adapted frame
\left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\}
of βθ (T (Mn)) is given by the matrix
\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
{\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
\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 = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right)
,
B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right)
,
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 \in \Im _0^1(T({M_n}))
has along βθ (T (Mn)) components of the form
^{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
\left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\}
[10].
Theorem 2
The complete lift ccX of a vector field X in Mn to t* (Mn) is tangent to the cross-section βθ (T (Mn)) determined by a 1 − form θ and vector field V in Mn if and only if{L_X}\theta = 0,{L_V}X = 0,where LV X denotes the Lie derivative of X with respect to V, and LX θ denotes the Lie derivative of θ with respect to X.
BX, CX and Eω also have components:
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
\left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\}
of the cross-section βθ (T (Mn)) determined by a 1-form θ on T (Mn) [10].
Complete Lift of Tensor Fields of Type (1,1) on a Cross-Section in Semi-Cotangent Bundle
Suppose now that
F \in \Im _1^1(T({M_n}))
and F has local components
F_\beta ^\alpha
in a neighborhood U of Mn,
F = F_\beta ^\alpha {\partial _\alpha } \otimes d{x^\beta }
. Then the semi-cotangent (pull-back) bundle t* (Mn) of cotangent bundle T* (Mn) by using projection of the tangent bundle T (Mn) admits the complete lift ccF of F with components [11]:
^{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
({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})
on t* (Mn). Then ccF has components
F_B^A
given by
^{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
\left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\}
of the cross-section βθ (T (Mn)) determined by a 1-form θ in T (Mn), where
A = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right)
,
B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right)
[10]. Also, the component
^{^{cc}}F_\beta ^{\overline {\overline \alpha } }
of
^{^{cc}}F_B^A
is defined as Tachibana operator φFθ of F, i.e.,
{\;^{^{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
{L_V}F_\beta ^\alpha denotes the Lie derivative ofF_\beta ^\alpha with respect to V, i.e.,
{L_V}F_\beta ^\alpha = {V^\gamma }{\partial _\gamma }F_\beta ^\alpha + F_\gamma ^\alpha {\partial _\beta }{V^\gamma } - F_\beta ^\gamma {\partial _\gamma }{V^\alpha }.
Adapted Frames and Diagonal Lifts of Affinor Fields
Let ∇ be a symmetric affine connection in Mn. In each coordinate neighborhood {U,xα} of Mn, we put
{X_{\left( \alpha \right)}} = {\partial \over {\partial {x^\alpha }}},\quad {\theta ^{\left( \alpha \right)}} = d{x^\alpha }.
Then 3n local vector fields Y(α), HHX(α) and vvθ(α) have respectively components of the form
{Y_{\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
({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
{\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
\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\}.
Since the matrix
\widehat A
in (17) is non-singular, it has the inverse. Denoting this inverse by
{\left( {\widehat A} \right)^{ - 1}}
, we have
{\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
\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 = \left( {\overline \alpha ,\alpha ,\overline {\overline \alpha } } \right)
,
B = \left( {\overline \beta ,\beta ,\overline {\overline \beta } } \right)
,
C = \left( {\overline \theta ,\theta ,\overline {\overline \theta } } \right)
.
If we take account of (16), we see that the diagonal lift DDF of
F \in \Im _1^1(T({M_n}))
has components [10]:
^{^{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
({x^{\overline \alpha }},{x^\alpha },{x^{\overline {\overline \alpha } }})
on t* (Mn), where
{\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 \in \Im _1^1(T({M_n}))
has components of the form
^{^{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
\left\{ {{{\widehat e}_{\left( B \right)}}} \right\}
in t* (Mn).
We now obtain from (19) that the diagonal lift DDF of an affinor field
F \in \Im _1^1(T({M_n}))
has along βθ (T (Mn)) components of the form [10]:
^{^{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
\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 \in \Im _0^1\left( {T({M_n})} \right)
has along βθ (T (Mn)) components of the form
^{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
\left\{ {{B_{\left( {\overline \beta } \right)}},{C_{\left( \beta \right)}},{C_{\left( {\overline {\overline \beta } } \right)}}} \right\}
[10].
Using (7), (20) and (21), we have along βθ (T (Mn)):
Theorem 3
If F and X are affinor and vector fields on T (Mn), and\omega \in \Im _1^0({M_n})
, then with respect to a symetric affine connection ∇ in Mn, we have
IfF,G \in \Im _1^1({M_n})
, then with respect to a symetric affine connection ∇ in Mn, we have^{DD}{F^{DD}}G{ + ^{DD}}{G^{DD}}F{ = ^{HH}}\left( {FG + GF} \right).
IfF,G \in \Im _1^1({M_n})
, then with respect to a symetric affine connection ∇ in Mn, we have^{DD}{F^{HH}}G{ + ^{DD}}{G^{HH}}F{ = ^{HH}}{F^{DD}}G{ + ^{HH}}{G^{DD}}F{ = ^{DD}}\left( {FG + GF} \right).
Putting F = G in Theorem 4 and Theorem 5, we have
\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 \in \Im _1^1(T({M_n}))
.
Theorem 6
The diagonal lift\widehat Jof the identity tensor field I of type (1,1) has the components\widehat J:\left( {\matrix{ { - \delta _\beta ^\alpha } & {2\Gamma _\beta ^\alpha } & 0 \cr 0 & {\delta _\beta ^\alpha } & 0 \cr 0 & {2{\Gamma _\beta }_\alpha } & { - \delta _\alpha ^\beta } \cr } } \right).