In this section, we investigate intrinsically a special class of regular connections in Finsler geometry within the pullback formalism. These connections are called P1-connections. It is shown that for these connections, the associated semi-spray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. Some examples of P1-connections are given. All of this section, we use the notions and results of [3].
Definition 1
Let D be a regular connection on π−1(TM) with horizontal map β. Then, the semispray \(S=\beta \overline {\eta }\) will be called the semispray associated with D. Moreover, the nonlinear connection Γ=2β∘ρ−I will be called the nonlinear connection associated with D and will be denoted by ΓD.
Lemma 1
Let D be a regular connection on π−1(TM) whose connection map is K and whose horizontal map is β. If the (h)hv-torsion T of D has the property that \({T}\left (\overline {X},\overline {\eta }\right)=0\), then K=γ−1 on V(TM) and Γ:=β∘ρ−γ∘K is a nonlinear connection on M and coincides with the nonlinear connection associated with D: Γ=ΓD=2β∘ρ−I, and in this case hΓ=hD=β∘ρ and vΓ=vD=γ∘K.
Definition 2
[14] Let (M,L) be a Finsler manifold and g the Finsler metric defined by L. (a) A vector (2) π-form ω is indicatory if and only if \(\omega \left (\overline {X},\overline {\eta }\right)=0=\omega \left (\overline {\eta }, \overline {X}\right)\) and \(g(\omega (\overline {X}, \overline {Y}),\overline {\eta })=0 \). (b) A scaler (2) π-form ω is indicatory if and only if \(\omega (\overline {X},\overline {\eta })=0=\omega (\overline {\eta }, \overline {X})\).
We will study the Finsler connections defined by the set of axiom given in the next definition.
Definition 3
Let (M,L)be a Finsler manifold. A regular connection \(\bar {D}\) on π−1(TM) is called a P1-connection if it satisfies the following conditions: (a)\({\bar {D}}_{\bar {h} X}L=0,\) for all \(X\in \mathfrak {X}({\mathcal {T}} M)\), (b) the (h)hv-torsion \({\bar {T}}\) of \(\bar {D}\) is symmetric and indicatory. (c) the (h)h-torsion tensor \({\bar {Q}}\) of \(\bar {D}\) vanishes, (d) the (v)hv-torsion tensor \(\widehat {\bar {P}}\) of \(\bar {D}\) is symmetric and indicatory.
Examples Cartan and Berwald connections are P1-connection [3]. Also Chern and Hashiguchi connections are P1-connection [4].
Proposition 2
Let \(\bar {D}\) be P1-connection on π−1(TM) whose connection map is \(\bar {K}\) and whose horizontal map is \(\bar {\beta }\). Then, for all \(\,\overline {X},\overline {Y}\in \mathfrak {X}(\pi (M))\), we have:(a)\(\left [\!\gamma \overline {X},\gamma \overline {Y}\right ]= \gamma \left ({\bar {D}}_{\gamma \overline {X}}\overline {Y}- {\bar {D}}_{\gamma \overline {Y}}\overline {X}\right)\)(b)\(\left [\!\gamma \overline {X},{\bar {\beta }} \overline {Y}\right ]=- \gamma \left (\widehat {{\bar {P}}}\left (\overline {Y},\overline {X}\right)+{\bar {D}}_ {{\bar {\beta }} \overline {Y}}\overline {X}\right) +{\bar {\beta }}\left ({\bar {D}}_{\gamma \overline {X}}\overline {Y}-{\bar {T}}\left (\overline {X},\overline {Y}\right)\right)\)(c)\(\left [\!{\bar {\beta }} \overline {X},{\bar {\beta }} \overline {Y}\right ]= \gamma \left (\widehat {{\bar {R}}}(\overline {X},\overline {Y})\right) + \bar {\beta }\left ({\bar {D}}_{\bar {\beta } \overline {X}}\overline {Y}- {\bar {D}}_{\bar {\beta } \overline {Y}}\overline {X}\right)\)
Proof
First we prove that the (v)v-torsion tensor \(\widehat {\bar {S}}\) of a P1-connection vanishes. Since a P1-connection is regular and whose (h)hv-torsion tensor \(\bar {T}\) has the property that \(\,\bar {T}(\overline {X}, \overline {\eta })=0\) (Definition 3(b)), then, by Proposition 2 of [5], the v-curvature tensor \(\bar {S}\) is given by:
$$\begin{array}{@{}rcl@{}} \bar{S}(\overline{X},\overline{Y})\overline{Z} &=& \left(\bar{D}_{\gamma \overline{Y}}\bar{T}\right)\left(\overline{X},\overline{Z}\right)- \left(\bar{D}_{\gamma \overline{X}}\bar{T}\right)\left(\overline{Y},\overline{Z}\right)\\ &&+\bar{T}\left(\overline{X},\bar{T}\left(\overline{Y},\overline{Z}\right)\right) -\bar{T}\left(\overline{Y},\bar{T}\left(\overline{X},\overline{Z}\right)\right)+\bar{T}\left(\widehat{\bar{S}}\left(\overline{X}, \overline{Y}\right),\overline{Z}\right). \end{array} $$
Setting \(\overline {Z}=\overline {\eta }\) into the above relation, noting that \(\,\bar {T}\) is symmetric indicatory, gives
$$ \widehat{\bar{S}}=0. $$
(1)
Now, from (1) and Lemma 2.1 of [5], the result follows. □
Theorem 2
Let (M,L)be a Finsler manifold and \({\bar {D}}\) a P1-connection. Then, the associated nonlinear connection \(\bar {\Gamma }\) has the form:
$$\bar{\Gamma} :=\bar{\beta} o \rho - \gamma o\bar{ K}.$$
Moreover, \(\bar {\Gamma }\) coincides with the Barthel connection of (M,L):\(\bar {\Gamma }=[\!J,G]\).
Proof
Since the (h)hv-torsion of \(\bar {D}\) satisfies \({\bar {T}}\left (\overline {X},\overline {\eta }\right)=0\), then, by Lemma 1, it follows that \(\bar {\Gamma } :=\bar {\beta } o \rho - \gamma o \bar {K} \). Moreover, its projectors are \(\bar {h}=\bar {\beta } o \rho \) and \(\bar {v}=\gamma o \bar { K}\).
We prove that \(\bar {\Gamma }\) enjoys the following properties:
\(\bar {\mathbf {\Gamma }} \)
is conservative:
In fact, if \(E=\frac {1}{2}L^{2}\), then
$$\begin{array}{@{}rcl@{}} d_{\bar{h}} E(X)&=& i_{\bar{h}}dE(X)=L \left(\bar{h}X\cdot L\right) =L\bar{D}_{\bar{h}X}L=0. \ \ \ \ (by \ axiom \ (a)) \end{array} $$
Hence, the result follows.
\(\bar {\mathbf {\Gamma }}\)
is homogenous of degree one:
$$\begin{array}{@{}rcl@{}} {\kern-.5pt}[\!\mathcal{C},\bar{v}](X) &=& \left[\!\mathcal{C},\gamma\,o\,\bar{K}\right](X) = \left[\!\mathcal{C},\gamma\,o\,\bar{K}(X)\right] -\gamma\,o\,\bar{K}([\!\mathcal{C},X])\\ &\,=\,&\left[\!\mathcal{C},\gamma\left(\bar{K} X\right)\right] \!\,-\, \gamma \left(\!\bar{K}\left[\!\mathcal{C},\bar{h}X+\bar{v}X\right]\right)\,=\, \left[\!\mathcal{C},\gamma\left(\bar{K} X\right)\right] \,-\,\! \gamma \left(\bar{K}\left[\mathcal{C},\bar{v}X\right]\right)\,-\,\! \gamma \left(\!\bar{K}\left[\!\mathcal{C},\bar{h}X\right]\right) \end{array} $$
From which, taking into account the fact that \([\mathcal {C},\bar {v}X]\) is vertical and \(\gamma \,o\,\bar {K}=id_{V(TM)}\) and using Proposition 2, we get:
$$\begin{array}{@{}rcl@{}} [\!\mathcal{C},\bar{v}](X) &=& -\gamma (\bar{K}[\mathcal{\!C},\bar{h}X])=-\gamma \,o\,\bar{K}([\!\gamma \overline{\eta},\bar{\beta}\rho X])\\ &=&-\gamma \,o\,\bar{K}\{- \gamma(\widehat{\bar{P}}(\rho X,\overline{\eta})+\bar{D}_ {\bar{\beta} \rho X}\overline{\eta}) +\bar{\beta}(\bar{D}_{\gamma \overline{\eta}}\rho X-\bar{T}(\overline{\eta},\rho X))\}\\ &=&\gamma \{ (\widehat{\bar{P}}(\rho X,\overline{\eta})+\bar{D}_ {\bar{\beta} \rho X}\overline{\eta})\} \bar{\beta}=0)=0. \ \ \ (as\ \widehat{\bar{P}} \ is \ indicatory \ and \ \bar{K}\,o\, \bar{\beta}=0) \end{array} $$
Therefore, \( [\!\mathcal {C},\bar {\Gamma }]= -2[\!\mathcal {C},\bar {v}]=0\), which means that \(\overline {\Gamma }\) is homogenous.
\(\bar {\mathbf {\Gamma }}\)
is torsion-free:
As \(J\bar {v}=0 \) and \(\bar {v}J=J\), we have
$$\begin{array}{@{}rcl@{}} [\!J,\bar{v}](X,Y) &=& [\!JX,\bar{v}Y]+ [\!\bar{v}X,JY]+\bar{v}J[\!X,Y]+J\bar{v}[\!X,Y]\\ && -J[\!\bar{v}X,Y]-J[\!X,\bar{v}Y]-\bar{v}[\!JX,Y]-\bar{v}[\!X,JY]\\ &=&J\left[\!\bar{h}X,\bar{h}Y\right]-\bar{v}\left[\!JX,\bar{h}Y\right]-\bar{v}\left[\!\bar{h}X,JY\right]\\ &=&J\left[\!\bar{\beta} \rho X,\bar{\beta} \rho Y\right]-\bar{v}\left[\!\gamma \rho X,\bar{\beta} \rho Y\right]+\bar{v}\left[\!\gamma \rho Y,\bar{\beta} \rho X\right]. \end{array} $$
Using Proposition 2, we get:
$$\begin{array}{@{}rcl@{}} [\!J,\bar{v}](X,Y)&=&J\left\{\gamma\left(\widehat{\bar{R}}\left(\rho{X},\rho{Y}\right)\right) + \bar{\beta}\left(\bar{D}_{\bar{h}{X}}\rho{Y}- \bar{D}_{\bar{h}{Y}}\rho{X}\right)\right\}\\ &&-\gamma\,o\,\bar{K}\left\{- \gamma\left(\widehat{\bar{P}}\left(\rho{Y},\rho{X}\right)+\bar{D}_ {\bar{h}{Y}}\rho{X}\right) +\bar{\beta}\left(\bar{D}_{J{X}}\rho{Y}-\bar{T}\left(\rho{X},\rho{Y}\right)\right)\right\}\\ &&+\gamma\,o\,\bar{K}\left\{- \gamma\left(\widehat{\bar{P}}\left(\rho{X},\rho{Y}\right)+\bar{D}_ {\bar{h}{X}}\rho{Y}\right) +\bar{\beta}\left(\bar{D}_{J{Y}}\rho{X}-\bar{T}\left(\rho{Y},\rho{X}\right)\right)\right\}. \end{array} $$
Since \( \ \bar {K}\,o\,\bar {\beta }=0, \ \rho \,o\, \gamma =0, \bar {K}\,o\,\gamma =id_{\mathfrak {X}(\pi (M))}, \ \rho \,o\,\bar {\beta }=id_{\mathfrak {X}(\pi (M))}\) and \( \widehat {\bar {P}}\) is symmetric, then the above relation implies that \([\!J,\bar {v}]=0\).
Hence \(t:=\frac {1}{2}\left [\!J,\bar {\Gamma }\right ]=-[\!J,\bar {v}]=0\).
From the above consideration, \(\bar {\Gamma }=\bar {\beta }\circ \rho -\gamma \circ \bar {K}\) is a conservative torsion-free homogenous nonlinear connection. By the uniqueness of the Barthel connection (Theorem 1), it follows that \(\bar {\Gamma }\) coincides with the Barthel connection [ J,G]. □
In view of the above theorem and Definition 1, we obtain the following corollary:
Corollary 1
The semispray associated with a P1-connection is a spray which coincides with the canonical spray.
We reconsider some results in references [3] and [4] with Theorem 2; we get the following corollary:
Corollary 2
Let (M,L) be a Finsler manifold. The nonlinear connection associated with each of Cartan, Berwald, Chern, and Hashiguchi connections coincides with the Barthel connection.
Now, we will give an interesting example of P1-connection called as \(\widetilde {D}\)-connection.
Theorem 3
Let (M,L) be a Finsler manifold. There exists a unique regular connection \(\widetilde {D}\) on π−1(TM) such that: (a)\(\widetilde {D}_{\widetilde {h}X}L=0\), (b)\(\widetilde {D}\) is torsion-free:\({\widetilde {\mathbf {T}}}=0 \), (c) The (v)hv-torsion tensor \(\widehat {\widetilde {P}}\) of \({\widetilde {D}}\) has the form \(\widehat {\widetilde {P}}(\overline {X},\overline {Y})=- LT(\overline {X},\overline {Y})\).
Such a connection is called the \(\widetilde {D}\)-connection associated with the Finsler manifold (M,L).
Proof
It is easy to show first that any regular connection \(\widetilde {D}\) satisfying the conditions (a), (b), and (c) is necessarily a P1-connection. Consequently, in view of Theorem 2, the nonlinear connection \({\Gamma _{\widetilde {D}}}\) associated with \(\widetilde {D}\) coincides with the Barthel connection: \(\Gamma _{\widetilde {D}}=[\!J,G]\). Moreover, \(\widetilde {\beta }=\beta \), \(\widetilde {K}=K\), \(\widetilde {h}=h\), and \(\widetilde {v}=v\), where β, K, h, and v are the horizontal map, connection map, horizontal projector, and the vertical projector of Cartan connection ∇, respectively.
Now, we prove the uniqueness. As \(\widetilde {D}\) is a nonmetric linear connection on π−1(TM) with zero torsion, one can show that:
$${}2g\left(\widetilde{D}_{vX}\rho Y,\rho Z\right) =vX\cdot g\left(\rho Y,\rho Z\right)+ g\left(\rho Y,\rho [\!Z,vX]\right)+g\left(\rho Z,\rho [\!vX,Y]\right)-\left(\widetilde{D}_{vX} g\right)\left(\rho Y,\rho Z\right). $$
Hence, using Theorem 4(a) of [3], the above equation implies that
$$ 2g\left(\widetilde{D}_{vX}\rho Y,\rho Z\right)=2g\left(\nabla_{vX}\rho Y,\rho Z\right) -\left(\widetilde{D}_{vX} g\right)\left(\rho Y,\rho Z\right). $$
(2)
Consequently,
$$\begin{array}{@{}rcl@{}} 2g\left(\widetilde{\mathbf{T}}({vX}, hY),\rho Z\right)&=& 2g\left(\widetilde{D}_{vX}\rho Y-\rho[vX,hY],\rho Z\right)\\ &=&2g\left(\nabla_{vX}\rho Y-\rho[vX,hY],\rho Z\right) -\left(\widetilde{D}_{vX} g\right)\left(\rho Y,\rho Z\right)\\ &=& 2g\left(\mathbf{T}({vX}, hY),\rho Z\right) -\left(\widetilde{D}_{vX} g\right)\left(\rho Y,\rho Z\right). \end{array} $$
From which, taking axiom (b) into account, we get:
$$ \left(\widetilde{D}_{vX} g\right)\left(\rho Y,\rho Z\right)= 2g\left(\mathbf{T}({vX}, hY),\rho Z\right). $$
(3)
Hence, (2) and (3), we obtain
$$ \widetilde{D}_{vX}\rho Y=\nabla_{vX}\rho Y -\mathbf{T}({vX}, hY). $$
(4)
Now, using axiom (c) and noting that K∘J=γ and K∘h=0, we get:
$$\begin{array}{@{}rcl@{}} - LT\left(\rho{X},\rho{Y}\right)=\widehat{\widetilde{P}}(hX,JY)&=&\widetilde{P}(hX,JY)\overline{\eta} \\ &=& -\widetilde{D}_{hX} {\widetilde{D}}_{JY}\overline{\eta} +\widetilde{D}_{JY} \widetilde{D}_{hX}\overline{\eta} +\widetilde{D}_{[hX,JY]}\overline{\eta} \\ &=& - \widetilde{D}_{hX}\rho Y+ K[hX,JY]. \end{array} $$
From which,
$$ \widetilde{D}_{hX}\rho Y= K[hX,JY]+ LT\left(\rho{X},\rho{Y}\right). $$
(5)
Using the definition of the (v)hv-torsion \(\widehat {P}\), (5) may also be written in the form:
$$ \widetilde{D}_{hX}\rho Y=\nabla_ {hX}\rho Y +\widehat{P}\left(\rho {X},\rho{Y}\right)+ LT\left(\rho{X},\rho{Y}\right). $$
(6)
Consequently, from (4) and (6), the full expression of \(\widetilde {D}_{X}\overline {Y}\) is given by:
$$ \widetilde{D}_{X}\overline{Y} = \nabla_{X}\overline{Y} + \widehat{P}\left(\rho X,\overline{Y}\right)+ LT\left(\rho{X},\overline{Y}\right)-T\left(K X, \overline{Y}\right). $$
(7)
Hence \(\widetilde {D}_{X}\overline {Y}\) is uniquely determined by the right-hand side of (7).
To prove the existence, we define \(\widetilde {D}\) by the requirement that (7) holds or, equivalently, (4) and (6) hold. Then, using the properties of Cartan connection ∇ and the results of [3], it is not difficult to show that the connection \(\widetilde {D}\) satisfies the conditions (a), (b), and (c). □
In view of the above theorem, we have the following results :
Corollary 3
The \(\widetilde {D}\)-connection is explicitly expressed in terms of the Cartan connection ∇ in the form:
$$ \widetilde{D}_{X}\overline{Y} = \nabla_{X}\overline{Y} +{\widehat{P}}\left(\rho X,\overline{Y}\right)+ LT\left(\rho{X},\overline{Y}\right) -T\left(K X,\overline{Y}\right). $$
(8)
In particular, we have: (a)\( \widetilde {D}_{\gamma \overline {X}}\overline {Y}=\nabla _{\gamma \overline {X}}\overline {Y}-T\left (\overline {X},\overline {Y}\right)=\rho \left [\!\gamma \overline {X}, \beta \overline {Y}\right ]\). (b)\( \widetilde {D}_{\beta \overline {X}}\overline {Y}=\nabla _{\beta \overline {X}}\overline {Y}+\widehat {P}\left (\overline {X},\overline {Y}\right)+ LT\left ({\overline {X}},\overline {Y}\right)=K\left [\!\beta \overline {X}, \gamma \overline {Y}\right ]+ LT\left ({\overline {X}},\overline {Y}\right).\)
Concerning the metricity properties of \(\widetilde {D}\), we havec
Corollary 4
The \(\widetilde {D}\)-connection has the properties: (a)\(\left (\widetilde {D}_{\gamma \overline {X}}\,g\right)\left (\overline {Y},\overline {Z}\right) =2 T\left (\overline {X}, \overline {Y},\overline {Z}\right)\). (b)\(\left (\widetilde {D}_{\beta \overline {X}}\,g\right)\left (\overline {Y},\overline {Z}\right) =-2LT\left (\overline {X},\overline {Y},\overline {Z}\right) -2 \widehat {P}\left (\overline {X},\overline {Y},\overline {Z}\right)\),
where \(\widehat {P}\left (\overline {X},\overline {Y},\overline {Z}\right):= g\left (\widehat {P}\left (\overline {X},\overline {Y}\right),\overline {Z}\right)\).
Remark 1
In view of the above proposition, one can show that a Finsler manifold (M,L) is Riemannian if and only if \(\widetilde {D}_{\gamma \overline {X}}\,g=0\). Moreover, (M,L) is Landsbergian if and only if \(\left (\widetilde {D}_{\beta \overline {X}}\,g\right)\left (\overline {Y},\overline {Z}\right) =-2L T\left (\overline {X},\overline {Y},\overline {Z}\right)\).
Proposition 3
The Curvature tensors of the \(\widetilde {D}\)-connection are given by: (a)\({{\widetilde {S}}}\left (\overline {X},\overline {Y}\right)\overline {Z}= S\left (\overline {X},\overline {Y}\right)\overline {Z}- \mathfrak {U}_{\overline {X},\overline {Y}}\left \{T\left (\overline {X},T\left (\overline {Y}, \overline {Z}\right)\right)\right \}\). (b)\({{\widetilde {P}}}\left (\overline {X},\overline {Y}\right)\overline {Z}= P\left (\overline {X},\overline {Y}\right)\overline {Z}+ L\,\left (\nabla _{\gamma \overline {Y}}T\right)\left (\overline {X}, \overline {Z}\right)+\left (\nabla _{\gamma \overline {Y}}\widehat {P}\right)\left (\overline {X},\overline {Z}\right)\)\({\qquad \qquad \ \ \ \, }+ \ell \left (\overline {Y}\right)T\left (\overline {X},\overline {Z}\right) + LT\left (T\left (\overline {Y},\overline {X}\right),\overline {Z}\right)+\widehat {P}\left (T\left (\overline {Y},\overline {X}\right),\overline {Z}\right)-\nabla _{\beta \overline {X}}T\left (\overline {Y},\overline {Z}\right)\)\({\qquad \qquad \ \ \ } +T\left (\overline {Y},\nabla _{\beta \overline {X}}\overline {Z}\right)-LT\left (\overline {Y},T\left (\overline {X},\overline {Z}\right)\right)-T\left (\overline {Y},\widehat {P}\left (\overline {X},\overline {Z}\right)\right)\)\({\qquad \qquad \ \ \ } +L\,T \left (\overline {X},T\left (\overline {Y},\overline {Z}\right)\right)+\widehat {P}\left (\overline {X},T\left (\overline {Y},\overline {Z}\right)\right)-T\left (K\left [\gamma \overline {X}, \gamma \overline {Y}\right ],\overline {Z}\right).\)(c)\({{\widetilde {R}}}\left (\overline {X},\overline {Y}\right)\overline {Z}= R\left (\overline {X},\overline {Y}\right)\overline {Z}- T\left (\widehat {R}\left (\overline {X},\overline {Y}\right),\overline {Z}\right)- \mathfrak {U}_{\overline {X},\overline {Y}}\{ L\, \left (\nabla _{\beta \overline {X}}T\right)\left (\overline {Y}, \overline {Z}\right)\)\({\qquad \qquad \ \ \ }+ \left (\nabla _{\beta \overline {X}}\widehat {P}\right)\left (\overline {Y}, \overline {Z}\right) + L^{2}T\left (\overline {X},T\left (\overline {Y},\overline {Z}\right)\right)+L\, \widehat {P}\left (\overline {X},T\left (\overline {Y}, \overline {Z}\right)\right) \)\({\qquad \qquad \ \ \ }+L\, T\left (\overline {X}, \widehat {P}\left (\overline {Y},\overline {Z}\right)\right) +\widehat {P}\left (\overline {X},\widehat {P}\left (\overline {Y},\overline {Z}\right)\right)\},\)
where \(\ell (\overline {X}):=L^{-1}g\left (\overline {X},\overline {\eta }\right)\).
Remark 2
In view of the above Proposition, one can conclude that the v-curvature \(\widetilde {S}\) vanishes and \( \widehat {\widetilde {R}}(\overline {X},\overline {Y})= {\widehat {R}}(\overline {X},\overline {Y})\).
Using Proposition 3, we have the following properties of the hv-curvature and h-curvature of the \(\widetilde {D}\)-connection.
Proposition 4
The hv-curvature tensor \(\widetilde {P}\) of \(\widetilde {D}\)-connection has the properties: (a)\(\widehat {\widetilde {P}}\left (\overline {X},\overline {Y}\right)=- LT\left (\overline {X},\overline {Y}\right)\), (b)\(\widetilde {P}\left (\overline {X},\overline {Y},\overline {Z},\overline {W}\right) +\widetilde {P}\left (\overline {X},\overline {Y},\overline {W},\overline {Z}\right)=2\left (\widetilde {D}_{\beta \overline {X}}T)(\overline {Y},\overline {Z},\overline {W}\right)+ 2 \left (\widetilde {D}_{\gamma \overline {Y}}\widehat {P})(\overline {X},\overline {Z},\overline {W}\right)\)\({\!\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad } 2 \left (\widetilde {D}_{\gamma \overline {Y}}LT)(\overline {X},\overline {Z},\overline {W}\right) -2T\left (\widehat {\widetilde {P}}\left (\overline {X},\overline {Y}\right),\overline {Z},\overline {W}\right)\), (c)\(\widetilde {P}\left (\overline {X},\overline {Y}\right)\overline {Z}= \widetilde {P}\left (\overline {Z},\overline {Y}\right)\overline {X}\), (d)\(\left (\widetilde {D}_{\gamma \overline {X}}\widetilde {P}\right)\left (\overline {Y}, \overline {Z}, \overline {W}\right) =\left (\widetilde {D}_{\gamma \overline {Z}}\widetilde {P}\right)\left (\overline {Y}, \overline {X}, \overline {W}\right)\).
Proposition 5
The h-curvature tensor \(\widetilde {R}\) of \({\widetilde {D}}\)-connection satisfies the following properties: (a)\(\widetilde {R}\left (\overline {X},\overline {Y},\overline {Z},\overline {W}\right)=- \widetilde {R}\left (\overline {Y},\overline {X},\overline {Z},\overline {W}\right)\), (b)\( \widehat {\widetilde {R}}\left (\overline {X}, \overline {Y}\right)= \widehat {R}\left (\overline {X}, \overline {Y}\right)=-K\mathfrak {R}\left (\beta \overline {X},\beta \overline {Y}\right)\), (c)\(\widetilde {R}\left (\overline {X},\overline {Y},\overline {Z},\overline {W}\right)+ \widetilde {R}\left (\overline {X},\overline {Y},\overline {W},\overline {Z}\right)\!=2\mathfrak {A}_{\overline {X},\overline {Y}} \left \{L\left (\widetilde {D}_{\beta \overline {Y}}T\right)\left (\overline {X},\overline {Z},\overline {W}\right)\!+ \left (\widetilde {D}_{\beta \overline {Y}}\widehat {P}\right)\left (\overline {X},\overline {Z},\overline {W}\right)\right \} {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad }-2T\left (\widehat {R}(\overline {X},\overline {Y}),\overline {Z}, \overline {W}\right)\), (d)\(\mathfrak {S}_{\overline {X},\overline {Y},\overline {Z}}\left \{ \widetilde {R}\left (\overline {X},\overline {Y}\right) \overline {Z}\right \}=0\), (e)\(\mathfrak {S}_{\overline {X},\overline {Y},\overline {Z}}\left \{ \left (\widetilde {D}_{\beta \overline {X}}\widetilde {R}\right)\left (\overline {Y},\overline {Z}, \overline {W}\right)+ \widetilde {P}\left (\overline {X}, \widehat {R}\left (\overline {Y},\overline {Z}\right)\right)\overline {W}\right \}=0\), (f)\(\left (\widetilde {D}_{\gamma \overline {X}}\widetilde {R}\right)\left (\overline {Y}, \overline {Z}, \overline {W}\right)=\left (\widetilde {D}_{\beta \overline {Z}}\widetilde {P}\right)\left (\overline {Y}, \overline {X}, \overline {W}\right)-\left (\widetilde {D}_{\beta \overline {Y}}\widetilde {P}\right)\left (\overline {Z}, \overline {X}, \overline {W}\right)\)\({\qquad \qquad \qquad \qquad \,\,\,}-L\widetilde {P}\left (\overline {Z},T\left (\overline {Y},\overline {X}\right)\right)\overline {W} +L\widetilde {P}\left (\overline {Y},T\left (\overline {Z},\overline {X}\right)\right)\overline {W}\),
where \(\mathfrak {R}\) is the curvature of Barthel connection.