Study on generalized BK-5^th recurrent Finsler space

In this paper, we present a novel new class and investigate the connection between the K-projective curvature tensor and other tensors of Finsler space F_n, this space is characterized by the property for Cartan’s 4th curvature tensor $K_{jkh}^i$ satisfies the certain relationship with the given covariant vectors field, we define this space as a generalized BK-5th recurrent space and denote it briefly by GBK-5R_Fn. This paper aims to derive the fifth-order Berwald covariant derivatives of the torsion tensor $H_{kh}^i$ and the deviation tensor $H_h^i$ . Additionally, it demonstrates that the curvature vector H_j, the curvature vector H_k, and the curvature scalar H are all non-vanishing within the considered space. We have identified tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal fifth-order Berwald covariant derivatives with their lower-order counterparts.

1. Introduction

In 1973, Sinha and Singh studied the properties of recurrent tensors in recurrent Finsler space [1]. Several works on recurrent Finsler space were done in the years 1973 and 1987. Verma (1991) discussed the recurrence property of Cartan’s third curvature tensor $R_{jkh}^i$ [2]. Dikshit (1992) discussed the bi-recurrence of Berwald curvature tensor $H_{jkh }^i$ [3]. Qasem (2000) introduced and studied the recurrence conditions of the curvature tensor $U_{jkh}^i$ in the sense of Berwald [4]. Qasem and Abdallah (2016) defined a generalized BR-recurrent Finsler space and obtained the necessary and sufficient conditions for the Berwald curvature tensor and Cartan’s fourth curvature tensorto be generalized recurrent [5]. The generalized BK - recurrent Finsler space was introduced by Qasem and Baleedi (2016) this space whose Cartan’s fourth curvature tensor $K_{jkh}^i$ satisfies a recurrence relation [6]. They showed that the K-Ricci tensor, the curvature vector, and the curvature scalar are non-vanishing in the BK-recurrent Finsler space. Al-Qashbari and Qasem (2017) [7] Studied on generalized BR-Trirecurrent Finsler Space. In 2020, Al-Qashbari [8] introduced some identities for generalized curvature tensors in B-recurrent Finsler space. Bidabad and Sepasi, [9] completed Finsler spaces of constant negative Ricci curvature. Abu-Donia, Shenawy, and Abdehameed [10] studied the W*-Curvature Tensor on Relativistic Space- times. Verstraelen [11] established a new submanifolds theory a contemplation of submanifolds in geometry of submanifolds. Deszcz, Głogowska, and Zafindratafa [12] established some conditions on hypersurfaces. In 2021, Opondo [13] studied the decomposition of Weyl projective curvature tensor in recurrent and bi-recurrent Finsler space. Chen [14] developmental Wintgen inequality and Wintgen ideal submanifolds. Eyasmin [15] studied hypersurfaces in a conformally flat space. Deszcz and Hotloś [16] defined and studied the geodesic mappings in a particular class of roter spaces. Deszcz, M. Głogowska, and M. Hotloś[17] studied the OpozdaVerstraelen affine curvature tensor on hypersurfaces. Decu, Deszcz, and Haesen [18] studied the classification of Roter-type spacetimes. In 2022, Deszcz, Głogowska, Hotloś, and Sawicz [19] studied the particular Roter-type equation on hypersurfaces in space forms. Derdzinski and Terek [20] introduced new examples of compact Weyl-parallel manifolds. Also they [21] studied the topology of compact rank-one ECSmanifolds. In 2023, Al-Qashbari and Al-Maisary [22] studied generalized BW- fourth recurrent in Finsler space. Shaikh, Hul, Datta, and Sakar [23] established new relations between the Kulkarni-Nomizu product of two (0,2) type tensors and the curvature tensors of type (0,4). Ali, Salman, Rahaman, and Pundeer, [24] obtained some properties of M-projective curvature tensor in spacetime.

Delving into the properties of an n-dimensional Finsler space F_n, we assume that its metric function F adheres to the established conditions outlined in the works of Deszcz, M. Głogowska, and M. Hotloś [17].

Positively homogeneous: $\text{F}\left(x,ky\right)=k \text{F}(x,y),k>0.$

Positively: $\text{F}\left(\text{x},\text{y}\right)>0,\text{ y}\ne 0.$

$\left\{ \stackrel{}{{\dot{\partial }}_i} {\dot{\partial }}_j{F}^2\left(x,y\right) \right\} {\xi }^i {\xi }^j, {\dot{\partial }}_i=\frac{\partial }{\partial y^i},\text{is the positive definite}\forall \text{variables }{\xi }^i.$

The corresponding metric tenser denoted by g_ij, the connection coefficients of Cartan represented by ${\text{Γ}}_{jk}^{*i}$ and the connection coefficients of Berwald designated by $G_{jk }^i$ , are all related to the metric function F.

$\begin{array}{l}\text{a) } g_{ij} y^i y^j=F^2,\text{ b) }{g}_{ij} y^j=y_i,\\ \text{c) }{g}_{ij}=\frac{1}{2} {\dot{\partial }}_i{y}_j, \text{ d) }{y}_i y_{ }{}^i=F^2,\\ \text{e) }{g}_{ij }{g}^{ik}={\delta }_j^k=\{ \begin{array}{c}1 if j=k ,\\ 0 if j\ne k . \end{array}\text{f)} {\delta }_{h }^i{g}_{ik}=g_{hk},\\ \text{g}) {\delta }_{k }^i{y}^k=y^i\text{ and h) }{\delta }_{i }^i=n .\text{ (1}\text{.1)}\end{array}$

The torsion tensor $C_{ijk}$ defined by [12]

$C_{ijk}=\frac{1}{2} {\dot{\partial }}_i{g}_{jk}=\frac{1}{4} {\dot{\partial }}_i {\dot{\partial }}_j{\dot{\partial }}_k F^2\text{ (1}\text{.2)}$

and its associate is the torsion tensor $C_{jk }^i$ and it is defined by

$\text{a) }{C}_{ik}^h=g^{hj} C_{ijk} \text{and b)} C_{jk }^i y^k=C_{kj }^i y^k=0.\text{ (1}\text{.3)}$

These tensors satisfy the following conditions

$\begin{array}{l}{\text{a) C}}_{\text{ijk}}{\text{ y}}^{\text{k}}{\text{=C}}_{\text{kij}}{\text{ y}}^{\text{k}}{\text{=C}}_{\text{jki}}{\text{ y}}^{\text{k}}\text{=0 ; }\\ {\text{b) G}}_{\text{jkh }}^{\text{i}}{\text{y}}^{\text{j}}{\text{=G}}_{\text{hjk }}^{\text{i}}{\text{y}}^{\text{j}}{\text{=G}}_{\text{khj }}^{\text{i}}{\text{y}}^{\text{j}}\text{=0 ;}\\ \text{c) }{\delta }_{\text{k}}^{\text{i}}{\text{ C}}_{\text{jin}}{\text{=C}}_{\text{jkn}}\text{;}\\ {\text{d) C}}_{\text{jkr}}{\text{ g}}^{\text{jk}}{\text{=C}}_{\text{r}}\\ \text{and}\\ \text{e) }{\Gamma }_{\text{jkh}}^{\text{*i}}{\text{ y}}^{\text{h}}{\text{= G}}_{\text{jkh }}^{\text{i}}{\text{y}}^{\text{h}}\text{=0 ;}\end{array}$

Where $G_{jkh}^i={\dot{\partial }}_j G_{kh }^i\text{and } {\dot{\partial }}_i=\frac{\partial }{\partial y^h}.\text{ (1}\text{.4)}$

The Berwald covariant derivative ${ℬ}_k{T}_{j }^i$ of an arbitrary tensor field $T_{j }^i$ with respect to $x^k$ is defined as:

${ℬ}_k T_{j }^i={\partial }_k T_{j }^i-\left({\dot{\partial }}_r T_{j }^i\right) G_k^r+T_{j }^r G_{rk}^i-T_{r }^i{G}_{jk }^r.\text{ (1}\text{.5)}$

The Berwald covariant derivatives of the metric function F, the vectors yⁱ, yⁱ and the unit vector lⁱ are all identically zero [11]. In other words,

$\text{a) } {ℬ}_k F=0 ,\text{b) }{ℬ}_{k }{y}^i=0,\text{ c) }{ℬ}_k y_i=0 \text{and d) }{ℬ}_k l^{i }=0. \text{ (1}\text{.6)}$

However, Berwald’s covariant derivative of the metric tensor g_ij is not identically zero, meaning ${ℬ}_{k }{g}_{ij}\ne 0$ . It is expressed as:

${ℬ}_k{g}_{ij}=-2 y^{h }{ℬ}_h C_{ijk}=-2 C_{ijk׀h} y^h.\text{ (1}\text{.7)}$

The covariant differential operator of Berwald with respect to x^h and the partial differential operator with respect to y^k commute, as defined by

$\left({\dot{\partial }}_i{}_{k }{ℬ}_h-{ℬ}_{h }{\dot{\partial }}_k\right) T_{j }^i=T_{j }^r{G}_{khr}^i- T_{r }^i{G}_{khj }^r,\text{ (1}\text{.8)}$

Where $T_{j }^i$ is any arbitrary tensor.

The second Berwald covariant derivative of the vector field Xⁱ, with respect to X^k and X^h is given by:

${ℬ}_k {ℬ}_h X^i={\dot{\partial }}_k{ℬ}_{h }{X}^i-\left( {\dot{\partial }}_{s }{ℬ}_{h }{X}^i\right) G_k^s+\left( {ℬ}_h{X}^r \right) G_{rk}^i-\left( {ℬ}_{r }{X}^i\right) G_{hk}^r. \text{ (1}\text{.9)}$

The tensors $K_{jkh }^i$ and $R_{jkh}^i$ defined by

$\begin{array}{l}\text{a}) K_{jkh}^i={\partial }_k{\Gamma }_{hj}^{*i} +\left({\dot{\partial }}_s{\Gamma }_{jk}^{*i} \right) {\Gamma }_{th}^{*s} y^t+{\Gamma }_{mk}^{*i}{\text{Γ}}_{hj}^{*m}-{\partial }_h{\Gamma }_{kj}^{*i}\\ -\left( {\dot{\partial }}_s{\Gamma }_{jh}^{*i} \right) {\Gamma }_{tk}^{*s} y^t- {\Gamma }_{mh}^{*i}{\Gamma }_{kj}^{*m} \end{array}$

and

$\begin{array}{l}\text{b}) R_{jkh}^i={\partial }_h{\text{Γ}}_{jk}^{*i}+\left({\dot{\partial }}_r{\text{Γ}}_{jk}^{*i}\right) {\text{Γ}}_{sh}^{*r}{y}^s+C_{jm}^i\left({\partial }_k{\text{Γ}}_{sh}^{*m}{y}^s –{\text{Γ}}_{kr}^{*m} {\text{Γ}}_{sh}^{*r} y^s\right)+{\text{Γ}}_{mk}^{*i}{\text{Γ}}_{jh}^{*m}\\ - {\partial }_k{\text{Γ}}_{jh}^{*i}-\left({\dot{\partial }}_r{\text{Γ}}_{jh}^{*i}\right) {\text{Γ}}_{sk}^{*r} y^s-C_{jm}^i\left({\partial }_h{\text{ Γ}}_{sk}^{*m} y^s – {\text{Γ}}_{hr}^{*m} {\text{Γ}}_{sk}^{*r} y^s\right)-{\text{Γ}}_{mh }^{*i}{\text{Γ}}_{jk}^{*m}\end{array}$

The aforementioned tensors, denoted as Cartan’s fourth curvature tensor and Cartan’s third curvature tensor, respectively, exhibit skew-symmetry with respect to their last two lower indices and positive homogeneity of degree zero in their directional arguments. These tensors adhere to the following relations:

$\begin{array}{l}\text{a})R_{jkh}^i y^j=K_{jkh}^i y^j=H_{kh}^i ,\text{b})H_{jkh}^i=K_{jkh}^i+ y^m ({\dot{\partial }}_j K_{mkh}^i),\\ \text{c}) K_{jkh}^i=R_{jkh}^i-C_{js}^i{H}_{kh}^s ,\text{d}) H_{jkh}^i-K_{jkh}^i=P_{jkh}^i+P_{jk}^r{P}_{rh}^i-P_{jhk}^i-P_{jh}^r{P}_{rk}^i,\\ \text{e})R_{ijhk}=K_{ijhk}+C_{ijm} K_{rhk }^s{y}^m\text{ and f})R_{ijkh}=g_{rj}{R}_{ikh}^r.\text{ (1}\text{.11) }\end{array}$

Ricci tensor K_jk, curvature vector K_j and curvature scalar K derived from the curvature tensor $K_{jkh}^i$ are defined as:

$\text{a}) K_{jki}^i=K_{jk},\text{b})K_{jk }{y}^k=K_j\text{ and c}) K_{jk} g^{jk}=K. \text{ (1}\text{.12)}$

Ricci tensor R_jk, the deviation tensor $R_h^i$ and curvature scalar R derived from the curvature tensor $R_{jkh}^i$ are defined as:

$\text{a}) R_{jkr}^r=R_{jk},\text{b}) R_{jkh }^i{g}^{jk}=R_h^i\text{ and c}) R_{jk }{g}^{jk}=R.\text{ (1}\text{.13)}$

The curvature tensor of Berwald $H_{jkh}^i,$ torsion tensor Ricci tensor $H_{kh}^i,$ deviation tensor $H_{jk},$ $H_h^i$ and curvature scalar H is defined as

$\begin{array}{l}\text{a})H_{jkh}^i y^j,=H_{kh}^i\text{b}) H_{kh}^i y^k=H_h^i,\text{c}) H_{k }^i{y}^k=0,\text{d})H_{jk}=H_{jkr}^r\\ \text{e}) H_{kr}^r=H_k,\text{f}) H_{jk }^i{y}_i=0,g)y_{i }{H}_k^i=0,\\ \text{h}) K_{jk }{y}^j=H_k\text{and i})K_j{y}^j=\left(n-1\right)H.\text{ (1}\text{.14)}\end{array}$

A Finsler space in which the Berwald connection parameter $G_{kh}^i$ does not depend on the directional coefficients yⁱ is known as an affinely connected space (Berwald space) [17].

Therefore, an affinely connected space is defined by one of the equivalent conditions

$\text{a}) {ℬ}_k{g}_{ij}=0\text{ and b}){ℬ}_k{g}^{ij}=0\text{ (1}\text{.15)}$

2. A Generalized BK-5th recurrent Finsler space

Let us explore in GBK-RFn for which whose Cartan’s fourth curvature tensor $K_{jkh}^i$ is defined as [9]:

${ℬ}_m{K}_{jkh}^i=a_m{K}_{jkh}^i+b_m\left({\delta }_h^i{g}_{jk}-{\delta }_{k }^i{g}_{jh}\right) ,K_{jkh}^i\ne 0$

Is called generalized BK-recurrent space, where Bm is a covariant derivative of the first order (Berwald’s covariant differential operator) with respect to x^m. Taking the covariant derivative of the fifth order for the above equation in the sense of Berwald with respect to x^l, x^m, xⁿ, x^q and x^s respectively, we obtain

${ℬ}_s{ℬ}_q{ℬ}_n{ℬ}_m{ℬ}_l{K}_{jkh}^i=a_{sqlnm}{K}_{jkh}^i+b_{sqlnm}\left({\delta }_h^i{g}_{jk}-{\delta }_{k }^i{g}_{jh}\right)$

$- c_{sqlnm}\left({\delta }_h^i{C}_{jkn}-{\delta }_{k }^i{C}_{jhn}\right)-d_{sqlnm}\left({\delta }_h^i{C}_{jkl}-{\delta }_{k }^i{C}_{jhl}\right)$

$- e_{sqlnm}\left({\delta }_h^i{C}_{jkq}-{\delta }_{k }^i{C}_{jhq}\right)-2b_{qlnm}{y}^r{ℬ}_r\left({\delta }_h^i{C}_{jks}-{\delta }_{k }^i{C}_{jhs}\right).\text{ (2}\text{.1)}$

Multiplying (2.1) by y^j, using (1.6b), (1.11a), (1.4a), and (1.1b), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_{kh}^i=a_{sqlnm}{H}_{kh}^i+b_{sqlnm}\left({\delta }_h^i{y}_k-{\delta }_k^i{y}_h\right).\text{ (2}\text{.2)}$

Multiplying (2.2) by y^k, using (1.6b), (1.14b), (1.1d), and (1.1g), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_h^i=a_{sqlnm}{H}_h^i+b_{sqlnm}\left({\delta }_h^i{F}^2-y^i{y}_h\right).\text{ (2}\text{.3)}$

In conclusion, we find that

Theorem 2.1: In the GBK-5R_Fn, Berwald's covariant derivatives of the fifth order for the torsion tensor $H_{kh}^i$ and the deviation tensor $H_h^i$ are given by the conditions (2.2) and (2.3), respectively.

Summation over the indices i and h in condition (2.1), using (1.12a), (1.4c), (1.1f) and (1.1h), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{K}_{jk}=a_{sqlnm}{K}_{jk}+b_{sqlnm}\left(n-1\right)g_{jk}$

$- c_{sqlnm}\left(n-1\right)C_{jkn}-d_{sqlnm}\left(n-1\right)C_{jkl}$

$- e_{sqlnm}\left(n-1\right)C_{jkq}-2b_{qlnm}{y}^r{ℬ}_r\left(n-1\right)C_{jks}.\text{ (2}\text{.4)}$

Multiplying (2.4) by yk, using (1.6b), (1.12b), (1.4a) and (1.1b), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{K}_j=a_{sqlnm}{K}_j+b_{sqlnm}\left(n-1\right)y_j. \text{ (2}\text{.5)}$

Multiplying (2.5) by y^j, using (1.6b), (1.14i) and (1.1d), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_{m}H=a_{sqlnm}H+b_{sqlnm}{F}^2.\text{ (2}\text{.6)}$

Multiplying (2.4) by y, using (1.6b), (1.14h), (1.4a) and (1.1b), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_k=a_{sqlnm}{H}_k+b_{sqlnm}\left(n-1\right) y_k.\text{ (2}\text{.7)}$

In conclusion, we find that

Theorem 2.2: In the GBK-5R_Fn, the curvature vector k_j, the curvature vector H_k and the curvature scalar H are all nonzero.

3. Necessary and sufficient condition

Let us explore an GBK-5R_Fn which is characterized by the condition (2.1).

Multiplying (2.4) by g^jk, and let a Berwald space (affinely connected space) and using (1.12c), (1.15b), (1.1e), (1.1h), and (1.4d), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_{m}K=a_{sqlnm}K+n\left(n-1\right)b_{sqlnm}\text{ (3}\text{.1)}$

$- \left(n-1\right)\left[ c_{sqlnm}{c}_n+d_{sqlnm}{c}_l+e_{sqlnm}{c}_q+2b_{qlnm}{y}^r{ℬ}_r{C}_s \right]$

We can express the above equation in different ways as

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_{m}K=a_{sqlnm}K\text{ (3}\text{.2)}$

If and only if

$n{b}_{sqlnm}-\left[c_{sqlnm}{c}_n+d_{sqlnm}{c}_l+e_{sqlnm}{c}_q+2b_{qlnm }{y}^r{ℬ}_r{C}_s\right]=0 \text{ (3}\text{.3)}$

In conclusion, we find that

Theorem 3.1: In the GBK-5R_Fn, (as defined by Berwald space), the fifth-order Berwald covariant derivative of the curvature scalar K is directly proportional to the curvature scalar itself solely under the condition that equation (3.3) is valid.

On account of (2.4), we have

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{K}_{jk}=a_{sqlnm}{K}_{jk}\text{ (3}\text{.4)}$

If and only if

$b_{sqlnm}{g}_{jk}-c_{sqlnm}{C}_{jkn}-d_{sqlnm}{C}_{jkl}-e_{sqlnm}{C}_{jkq}-2b_{qlnm }{y}^r{ℬ}_r{C}_{jks}=0\text{ (3}\text{.5)}$

In conclusion, we find that

Theorem 3.2: In the GBK-5R_Fn, covariant derivative of Berwald on the fifth order for Ricci tensor K_jk is proportional to the tensor itself if and only if (3.5) is valid.

On account of the condition [15]:

${ℬ}_m{H}_{kh}^i={\lambda }_m{H}_{kh }^i+{\mu }_m({\delta }_h^i{y}_k-{\delta }_{k }^i{y}_h)\text{ (3}\text{.6)}$

And in view the condition (2.2), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_{kh}^i={ℬ}_m{H}_{kh}^i\text{ (3}\text{.7)}$

If and only if

${\lambda }_m{H}_{kh}^i+{\mu }_m({\delta }_h^i{y}_k-{\delta }_{k }^i{y}_h)=a_{sqlnm}{H}_{kh}^i+b_{sqlnm}\left({\delta }_h^i{y}_k-{\delta }_k^i{y}_h\right)\text{ (3}\text{.8)}$

In conclusion, we find that

Theorem 3.3: In the GBK-5R_Fn, covariant derivative of Berwald on the first order and fifth order for the torsion tensor both are equal if and only if (3.8) is valid.

On account of the condition [15]:

${ℬ}_m{H}_h^i={\lambda }_m{H}_{h }^i+{\mu }_m({\delta }_h^i{F}^2-y^i{y}_h).\text{ (3}\text{.9)}$

And in view the condition (2.3), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_h^i={ℬ}_m{H}_h^i.\text{ (3}\text{.10)}$

If and only if

${\lambda }_m{H}_{h }^i+{\mu }_m\left({\delta }_h^i{F}^2-y^i{y}_h\right)=a_{sqlnm}{H}_{h }^i+b_{sqlnm}\left({\delta }_h^i{F}^2-y^i{y}_h\right).\text{ (3}\text{.11)}$

In conclusion, we find that

Theorem 3.4: In the GBK-5R_Fn, covariant derivative of Berwald on the first order and fifth order for the deviation tensor $H_h^i$ both are equal if and only if (3.11) is valid.

On account of the condition [8]:

${ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_l W_{kh}^i=a_{lmns}{W}_{kh }^i+b_{lmns}\left({\delta }_h^i{y}_k-{\delta }_{k }^i{y}_h\right) .\text{ (3}\text{.12)}$

And in view the condition (2.2), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_{kh}^i={ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_l{W}_{kh}^i\text{ (3}\text{.13)}$

If and only if

$a_{sqlnm}{H}_{kh }^i+b_{sqlnm}\left({\delta }_h^i{y}_k-{\delta }_k^i{y}_h\right)=a_{lmns}{W}_{kh }^i+b_{lmns}\left({\delta }_h^i{y}_k-{\delta }_{k }^i{y}_h\right).\text{ (3}\text{.14)}$

In conclusion, we find that

Theorem 3.5: In the GBK-5R_Fn, covariant derivative of Berwald on the fifth order for the torsion tensor $H_{kh}^i$ and of the fourth order for projective torsion tensor $W_{kh}^i$ both are equal if and only if (3.14) are valid.

On account of the condition [8]:

${ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_l{W}_h^i=a_{lmns}{W}_{h }^i+b_{lmns}({\delta }_h^i{F}^2-y^i{y}_h).\text{ (3}\text{.15)}$

And in view the condition (2.3), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_h^i={ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_l W_h^i.\text{ (3}\text{.16)}$

If and only if

$a_{sqlnm}{H}_h^i+b_{sqlnm}\left({\delta }_h^i{F}^2-y^i{y}_h\right)=a_{lmns}{W}_{h }^i+b_{lmns}({\delta }_h^i{F}^2-y^i{y}_h)\text{ (3}\text{.17)}$

In conclusion, we find that

Theorem 3.6: In the GBK-5R_Fn, covariant derivative of Berwald on the fifth order for the deviation tensor $H_h^i$ and of the fourth order for projective deviation tensor $W_h^i$ both are equal if and only if (3.17) holds good.

On account of the condition [8]:

${ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_l{W}_k=a_{lmns}{W}_k+\left(n-1\right)b_{lmns }{y}_k\text{ (3}\text{.18)}$

and

${ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_{l}W=a_{lmns}W+\left(n-1\right) b_{lmns}{F}^2\text{ (3}\text{.19)}$

With the condition (2.7) and (2.6), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_k={ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_l{W}_k\text{ (3}\text{.20)}$

If and only if

$a_{sqlnm}{H}_k+b_{sqlnm}\left(n-1\right)y_k=a_{lmns}{W}_k+(n-1)b_{lmns}{y}_k\text{ (3}\text{.21)}$

And

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_{m}H={ℬ}_s{ℬ}_n{ℬ}_m{ℬ}_{l}W\text{ (3}\text{.22)}$

If and only if

$a_{sqlnm}H+b_{sqlnm}{F}^2=a_{lmns}W-\left(n-1\right) b_{lmns}{F}^2\text{ (3}\text{.23)}$

In conclusion, we find that

Theorem 3.7: In the GBK-5R_Fn, covariant derivative of Berwald on the fifth order for curvature vector H_K and curvature scalar H, both are equal to Berwald’s covariant derivative of the fifth order for the curvature vector W_K and curvature scalar W, respectively if and only if (3.21) and (3.23), respectively hold good.

On account of the condition [4]:

${ℬ}_n{ℬ}_m{P}_{kh}^i=a_{mn}{P}_{kh}^i+b_{mn}({\delta }_h^i{y}_k-{\delta }_k^i{y}_h)\text{ (3}\text{.24)}$

And the condition (2.2), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_{kh}^i={ℬ}_n{ℬ}_m{P}_{kh}^i\text{ (3}\text{.25)}$

If and only if

$a_{sqlnm}{H}_{kh }^i+b_{sqlnm}\left({\delta }_h^i{y}_k-{\delta }_k^i{y}_h\right)=a_{mn}{P}_{kh}^i+b_{mn}\left({\delta }_h^i{y}_k-{\delta }_k^i{y}_h\right) \text{ (3}\text{.26)}$

In conclusion, we find that

Theorem 3.8: In the GBK-5R_Fn , covariant derivative of Berwald on the fifth order for torsion tensor $H_{kh}^i$ and of the second order for torsion tensor $P_{kh}^i$ both are equal if and only if (3.26) holds good.

On account of the condition [4]:

${ℬ}_n{ℬ}_m{P}_k=a_{mn}{P}_k+b_{mn}\left(n-1\right)y_k \text{ (3}\text{.27)}$

and in view the condition (2.7), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{H}_k={ℬ}_n{ℬ}_m{P}_k\text{ (3}\text{.28)}$

If and only if

$a_{sqlnm}{H}_k+b_{sqlnm}\left(n-1\right)y_k=a_{mn}{P}_k+b_{mn}(n-1)y_k \text{ (3}\text{.29)}$

In conclusion, we find that

Theorem 3.9: In the GBK-5R_Fn, covariant derivative of Berwald on of the fifth order for the curvature vector H_K and second order for the curvature vector P_k, both are equal if and only if (3.29) holds good.

Using (1.11c) in (2.1), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{jkh}^i=a_{sqlnm}{R}_{jkh }^i+b_{sqlnm}\left({\delta }_h^i{g}_{jk}-{\delta }_{k }^i{g}_{jh}\right)\text{ (3}\text{.30)}$

$- c_{sqlnm}\left({\delta }_h^i{C}_{jkn}-{\delta }_{k }^i{C}_{jhn}\right)-d_{sqlnm}\left({\delta }_h^i{C}_{jkl}-{\delta }_{k }^i{C}_{jhl}\right)$

$- e_{sqlnm}\left({\delta }_h^i{C}_{jkq}-{\delta }_{k }^i{C}_{jhq}\right)-2b_{qlnm}{y}^r{ℬ}_r\left({\delta }_h^i{C}_{jks}-{\delta }_{k }^i{C}_{jhs}\right)$

$+ {ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jt}^i{H}_{kh}^t)-a_{sqlnm}(C_{jt}^i{H}_{kh}^t).$

This exhibits that

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{jkh}^i=a_{sqlnm}{R}_{jkh }^i+b_{sqlnm}\left({\delta }_h^i{g}_{jk}-{\delta }_{k }^i{g}_{jh}\right)$

$- c_{sqlnm}\left({\delta }_h^i{C}_{jkn}-{\delta }_{k }^i{C}_{jhn}\right)-d_{sqlnm}\left({\delta }_h^i{C}_{jkl}-{\delta }_{k }^i{C}_{jhl}\right)$

$- e_{sqlnm}\left({\delta }_h^i{C}_{jkq}-{\delta }_{k }^i{C}_{jhq}\right)-2b_{qlnm}{y}^r{ℬ}_r\left({\delta }_h^i{C}_{jks}-{\delta }_{k }^i{C}_{jhs}\right)\text{ (3}\text{.31)}$

If and only if

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jt}^i{H}_{kh}^t)=a_{sqlnm}(C_{jt}^i{H}_{kh}^t) \text{ (3}\text{.32)}$

In conclusion, we find that

Theorem 3.10: In the GBK-5R_Fn, Cartan’s third curvature tensor $R_{jkh}^i$ is GBK-5R_Fn if and only if (3.32) it holds good.

Multiplying (3.30) by gif, and let the space be a Berwald space and using (1.15a), (1.11f), and (1.1f), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{jfkh}=a_{sqlnm}{R}_{jfkh}+b_{sqlnm}\left(g_{hf}{g}_{jk}-g_{kf}{g}_{jh}\right)$

$- 2b_{qlnm}{y}^r{ℬ}_r\left(g_{hf}{C}_{jks}-g_{kf}{C}_{jhs}\right)-c_{sqlnm}\left(g_{hf}{C}_{jkn}-g_{kf}{C}_{jhn}\right)$

$- d_{sqlnm}\left(g_{hf}{C}_{jkl}-g_{kf}{C}_{jhl}\right)-e_{sqlnm}\left(g_{hf}{C}_{jkq}-g_{kf}{C}_{jhq}\right)$

$+ {ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jft}{H}_{kh}^t)-a_{sqlnm}(C_{jft}{H}_{kh}^t)\text{ (3}\text{.33)}$

This exhibits that

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{jfkh}=a_{sqlnm}{R}_{jfkh}\text{ (3}\text{.34)}$

If and only if

$b_{sqlnm}\left(g_{hf}{g}_{jk}-g_{kf}{g}_{jh}\right)-2b_{qlnm}{y}^r{ℬ}_r\left(g_{hf}{C}_{jks}-g_{kf}{C}_{jhs}\right)\text{ (3}\text{.35)}$

$- c_{sqlnm}\left(g_{hf}{C}_{jkn}-g_{kf}{C}_{jhn}\right)-d_{sqlnm}\left(g_{hf}{C}_{jkl}-g_{kf}{C}_{jhl}\right)$

$- e_{sqlnm}\left(g_{hf}{C}_{jkq}-g_{kf}{C}_{jhq}\right)+{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jft}{H}_{kh}^t)$

$- a_{sqlnm}\left(C_{jft}{H}_{kh}^t\right)=0$

Summation over the indices i and h in condition (3.30), using (1.13a), (1.1h), (1.1f) and (1.14c), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{jk}=a_{sqlnm}{R}_{jk}+b_{sqlnm}\left(n-1\right)g_{jk}-2b_{qlnm}{y}^r{ℬ}_r\left(n-1\right)C_{jks}$

$- c_{sqlnm}\left(n-1\right)C_{jkn}-d_{sqlnm}\left(n-1\right)C_{jkl}-e_{sqlnm}\left(n-1\right)C_{jkq}\text{ (3}\text{.36)}$

$+ {ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jt}^u{H}_{ku}^t)-a_{sqlnm}(C_{jt}^u{H}_{ku}^t)$

This exhibits that

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{jk}=a_{sqlnm}{R}_{jk}\text{ (3}\text{.37)}$

If and only if

$b_{sqlnm}\left(n-1\right)g_{jk}-2b_{qlnm}{y}^r{ℬ}_r\left(n-1\right)C_{jks}-c_{sqlnm}\left(n-1\right)C_{jkn}$

$- d_{sqlnm}\left(n-1\right)C_{jkl}-e_{sqlnm}\left(n-1\right)C_{jkq}+{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jt}^u{H}_{ku}^t)$

$- a_{sqlnm}(C_{jt}^u{H}_{ku}^t)=0 \text{ (3}\text{.38)}$

Multiplying (3.36) by g^jk, and let the space be a Berwald space and using (1.15b), (1.13c), (1.4d) and (1.1e), (1.1h), we obtain

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_{m}R=a_{sqlnm}R+n\left(n-1\right)b_{sqlnm}-2b_{qlnm}{y}^r{ℬ}_r\left(n-1\right)C_s$

$- c_{sqlnm}\left(n-1\right)C_n-d_{sqlnm}\left(n-1\right)C_l-e_{sqlnm}\left(n-1\right)C_q$

$+ [{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jt}^u{H}_{ku}^t)-a_{sqlnm}(C_{jt}^u{H}_{ku}^t)] g^{jk}\text{ (3}\text{.39)}$

This exhibits that

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_{m}R=a_{sqlnm}R\text{ (3}\text{.40)}$

If and only if

$b_{sqlnm}-2b_{qlnm}{y}^r{ℬ}_r{C}_s-c_{sqlnm}{C}_n-d_{sqlnm}{C}_l-e_{sqlnm}{C}_q$

$+ [ {ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m(C_{jt}^u{H}_{ku}^t)-a_{sqlnm}(C_{jt}^u{H}_{ku}^t)] g^{jk}=0 \text{ (3}\text{.41)}$

In conclusion, we find that

Theorem 3.11: In the the GBK-5R_Fn, in the sense of Berwald space the covariant derivative of Berwald on the fifth order for associate curvature tensor R_jfkh of the tensor and the curvature scalar R all are proportional to the tensor itself if and only if (3.35) and (3.41), respectively hold good.

Theorem 3.12: In the GBK-5R_Fn, (in the sense of Berwald space), the covariant derivative of Berwald on the fifth order for the Ricci tensor R_jk is proportional to the tensor itself if and only if (3.38) it holds good.

Using (1.11d) in (2.1), we obtain

${ℬ}_s{ℬ}_q{ℬ}_n{ℬ}_m{ℬ}_l(H_{jkh }^i-P_{jkh}^i-P_{jk}^r{P}_{rh}^i+P_{jhk}^i+P_{jh}^r{P}_{rk}^i)$

$=a_{sqlnm}(H_{jkh }^i-P_{jkh}^i-P_{jk}^r{P}_{rh}^i+P_{jhk}^i+P_{jh}^r{P}_{rk}^i)+b_{sqlnm}\left({\delta }_h^i{g}_{jk}-{\delta }_{k }^i{g}_{jh}\right)$

$- c_{sqlnm}\left({\delta }_h^i{C}_{jkn}-{\delta }_{k }^i{C}_{jhn}\right)-d_{sqlnm}\left({\delta }_h^i{C}_{jkl}-{\delta }_{k }^i{C}_{jhl}\right)$

$- e_{sqlnm}\left({\delta }_h^i{C}_{jkq}-{\delta }_{k }^i{C}_{jhq}\right)-2b_{qlnm}{y}^r{ℬ}_r\left({\delta }_h^i{C}_{jks}-{\delta }_{k }^i{C}_{jhs}\right) \text{ (3}\text{.42)}$

Summation over the indices i and h in condition (3.42), using (1.14d), (1.1h), (1.1f) and (1.4c), we obtain

${ℬ}_s{ℬ}_q{ℬ}_n{ℬ}_m{ℬ}_l(H_{jk}-P_{jkt}^t-P_{jk}^r{P}_{rt}^t+P_{jtk}^t+P_{jt}^r{P}_{rk}^t)$

$=a_{sqlnm}\left(H_{jk}-P_{jkt}^t-P_{jk}^r{P}_{rt}^t+P_{jtk}^t+P_{jt}^r{P}_{rk}^t\right)+b_{sqlnm}\left(n-1\right)g_{jk}$

$- 2b_{qlnm}{y}^r{ℬ}_r\left(n-1\right)C_{jks}-c_{sqlnm}\left(n-1\right)C_{jkn} -d_{sqlnm}\left(n-1\right)C_{jkl}$

$- e_{sqlnm}\left(n-1\right)C_{jkq} \text{ (3}\text{.43)}$

We can express the above equation in different ways as

${ℬ}_s{ℬ}_q{ℬ}_n{ℬ}_m{ℬ}_l{H}_{jk}=a_{sqlnm}{H}_{jk} \text{ (3}\text{.44)}$

If and only if

${ℬ}_s{ℬ}_q{ℬ}_n{ℬ}_m{ℬ}_l\left(P_{jkt}^t+P_{jk}^r{P}_{rt}^t-P_{jtk}^t-P_{jt}^r{P}_{rk}^t\right)$

$+ a_{sqlnm}(-P_{jkt}^t-P_{jk}^r{P}_{rt}^t+P_{jtk}^t+P_{jt}^r{P}_{rk}^t)+b_{sqlnm}\left(n-1\right)g_{jk}$

$- 2b_{qlnm}{y}^r{ℬ}_r\left(n-1\right)C_{jks}-c_{sqlnm}\left(n-1\right)C_{jkn}$

$- d_{sqlnm}\left(n-1\right)C_{jkl}-e_{sqlnm}\left(n-1\right)C_{jkq}=0 \text{ (3}\text{.45)}$

In conclusion, we find that

Theorem 3.13: In the GBK-5R_Fn, in the sense of Berwald space, the covariant derivative of Berwald on the fifth order for the Ricci tensor H_jk is proportional to the tensor itself if and only if (3.45) it holds good.

4. Composition relations between Cartan’s third curvature tensor and conformal Curvature Tensor in GBK-5R_Fn

In this section, we presented the relationship between Cartan’s third curvature tensor R_ijkh and conformal curvature tensor C_ijkh in GBK-5R_Fn

Definition 4.1: A conformal curvature tensor C_ijkh (also known as Weyl conformal curvature tensor) is defined as [2]:

$R_{ijkh}=C_{ijkh}+\frac{1}{2}\left(g_{ik}{R}_{jh}+g_{jh}{R}_{ik}-g_{ih}{R}_{jk}-g_{jk}{R}_{ih}\right)+\frac{R}{6}\left(g_{ih}{g}_{jk}-g_{ik}{g}_{jh}\right)\text{ (4}\text{.1)}$

Taking the covariant derivative of 5th order for (4.1) in the sense of Berwald, we obtain

$\begin{array}{l}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{ijkh}={ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{C}_{ijkh}+\frac{1}{2}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\\ \left( g_{ik}{R}_{jh}+ g_{jh}{R}_{ik}-g_{ih}{R}_{jk}-g_{jk}{R}_{ih}\right)+{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left[\frac{R}{6}\left(g_{ih}{g}_{jk}-g_{ik}{g}_{jh}\right)\right]\text{ (4}\text{.2)}\end{array}$

Using (1.11f) and (1.15 a) in (4.2), we obtain

$\begin{array}{l} g_{rj}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{ikh}^r={ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{C}_{ijkh}+\frac{1}{2}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\\ \left( g_{ik}{R}_{jh}+ g_{jh}{R}_{ik}-g_{ih}{R}_{jk}-g_{jk}{R}_{ih}\right)+{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left[ \frac{R}{6}\left(g_{ih}{g}_{jk}-g_{ik}{g}_{jh}\right) \right] \text{ (4}\text{.3)}\end{array}$

We can express the above equation in different ways as

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{ikh}^r=\frac{1}{ g_{rj}}\left[\begin{array}{l}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{C}_{ijkh}+\frac{1}{2}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left( g_{ik}{R}_{jh}+g_{jh}{R}_{ik}-g_{ih}{R}_{jk}-g_{jk}{R}_{ih}\right)\\ +{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left[ \frac{R}{6}\left(g_{ih}{g}_{jk}-g_{ik}{g}_{jh}\right) \right]\end{array}\right]\text{ (4}\text{.4)}$

In conclusion, we find that

Theorem 4.1: In the GBK-5R_Fn, (in the sense of Berwald space), the covariant derivative of Berwald on the fifth order for Cartan’s third curvature tensor $R_{ikh}^r$ and the conformal curvature tensor C_ijkh, linking together by the relation (4.4).

From (4.3), we have

${ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{C}_{ijkh}=g_{rj}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{C}_{ijkh}=g_{rj}\left({ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{R}_{ikh}^r\right)\text{ (4}\text{.5)}$

If and only if

$\begin{array}{l}\frac{1}{2}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left( g_{ik}{R}_{jh}+ g_{jh}{R}_{ik}-g_{ih}{R}_{jk}-g_{jk}{R}_{ih}\right)\\ + {ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left[ \frac{R}{6}\left(g_{ih}{g}_{jk}-g_{ik}{g}_{jh}\right) \right]=0 \text{ (4}\text{.6)}\end{array}$

In conclusion, we find that

Theorem 4.2: In the GBK-5R_Fn, (in the sense of Berwald space), the covariant derivative of Berwald on the fifth order for the conformal curvature tensor C_ijkh is proportional to the Berwald covariant derivative of the Cartan’s third curvature tensor $R_{ikh}^r$ by (4.5) if and only if (4.6) it holds good.

Using (1. e 1), when $[i\ne h\ne k]$ in (3.31) and using it in (4.4), we obtain

$R_{ikh}^r=\frac{1}{ a_{sqlnm} g_{rj}}\left[\begin{array}{l} {ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m{C}_{ijkh}+\frac{1}{2}{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left(g_{ik}{R}_{jh}+g_{jh}{R}_{ik}- g_{ih}{R}_{jk}-g_{jk}{R}_{ih}\right)\\ +{ℬ}_s{ℬ}_q{ℬ}_l{ℬ}_n{ℬ}_m\left[ \frac{R}{6}\left(g_{ih}{g}_{jk}-g_{ik}{g}_{jh}\right) \right] \end{array}\right] \text{ (4}\text{.7)}$

In conclusion, we find that

Theorem 4.3: In the GBK-5R_Fn, the Cartan’s third curvature tensor $R_{ikh}^r$ and the Berwald covariant derivative of the fifth order for conformal curvature tensor $C_{ijkh},$ linking together by the relation (4.7).

The linking of Cartan’s third curvature tensor with the conformal curvature tensor is a relationship between two tensors in differential geometry. The first tensor is Cartan’s third curvature tensor which measures the local curvature of a metric space. The second tensor is the conformal curvature tensor which measures the local curvature of a transformed metric space. The relationship states that Cartan’s third curvature tensor can be expressed as the product of the conformal curvature tensor and the metric factor.

The relationship linking Cartan’s third curvature tensor with the conformal curvature tensor can be used to study the properties of transformed metric spaces. For example, it can be used to determine whether the metric space is connected or not.

Findings summary

In general relativity, the metric in Finsler space is defined as a function that depends on the velocity vector. A recurrent Finsler space is a Finsler space where the metric is symmetric around the velocity vector. The fifth order is the order of the velocity vector. In this context, (recurrent) refers to the fact that the metric depends on the velocity vector up to the fifth order. A generalized fifth-order recurrent Finsler space can be used to:

- Describe anti-desitter spacetime, where the metric is negative.

- Describe curved spacetime, where the metric depends on the spatial and temporal coordinates.

- Describe multi-dimensional spacetime, where there are more than three spatial dimensions.

Conclusion

A generalized fifth-order recurrent Finsler space is a new geometric structure with great potential. It can be used to describe a variety of geometric structures. Research on these new structures is still ongoing, but there are many potential applications for them. For example: developing new models of the universe, studying the properties of dark matter, and developing new technologies for space travel.