Generalized BP-Recurrent Finsler Geometry with Applications to Intelligent Robotic Path Planning and Anisotropic Biomedical Systems

doi:N/A

Biomedicine and Chemical Sciences

2026, Volume 5, Issue 3 : 69-76

Original Article

Generalized BP-Recurrent Finsler Geometry with Applications to Intelligent Robotic Path Planning and Anisotropic Biomedical Systems

Adel Mohammed Ali Al-Qashbari

Mohammed Saeed Moqbel Abdullah

Idress Hamad Attitalla

Received

Aug. 3, 2026

Accepted

Aug. 20, 2026

Published

Sept. 1, 2026

Abstract

This paper presents an advanced geometric study of generalized BP-recurrent Finsler spaces GBP-RF_n, where Cartan's second curvature tensor P_jkh^i satisfies a first-order recurrence relation involving non-vanishing covariant vector fields. Necessary and sufficient conditions are derived for the non-vanishing of key curvature-related tensors, including the (hv)-torsion tensor, P-Ricci tensor, and deviation tensors.

The theoretical framework of GBP-RF_n spaces offer a robust mathematical model for analyzing physical and kinematic systems operating in anisotropic and non-Euclidean environments. Such properties make them well-suited for modern applications in intelligent robotics, where environmental inhomogeneity and directional sensitivity are crucial.

Specifically, the geometric structure of GBP-RF_ncan be leveraged to enhance robot motion planning in complex environments, improve adaptive control algorithms, and optimize sensor-based navigation systems. By analyzing the recurrence relations of higher-order curvature tensors under Berwald differentiation, the paper provides a deep insight into how curvature and torsion influence feasible paths and motion dynamics.

This work establishes a novel interdisciplinary bridge between higher-order differential geometry and real-world robotic systems, paving the way for advanced modeling of autonomous navigation, UAVs, and intelligent transportation in dynamically curved environments

Keywords

Finsler Space

Cartan’s Second Curvature Tensor P_jkh^i

Generalized BP-recurrent Space

Cartan’s Fourth Curvature Tensor R_jkh^i

Intelligent Robotic Systems

Adaptive Control

INTRODUCTION

Recent advancements in autonomous robotic systems have emphasized the need for geometric frameworks that can model environments where physical properties vary with both position and direction. Finsler geometry, particularly its generalized recurrent forms such as the GBP-RFₙ space studied here, provides a natural and rigorous mathematical tool for this purpose. By analyzing the recurrence of curvature tensors and their derivatives, this study lays the groundwork for employing such structures in real-world robotic motion planning, especially in anisotropic or curved environments where traditional Euclidean or Riemannian models fall short.

Finsler geometry has witnessed remarkable developments in recent years, particularly in the study of generalized recurrent structures and their associated curvature tensors. Several authors have extended the classical framework by introducing higher-order generalizations of recurrent spaces, focusing on the role of Berwald and Cartan connections in defining and analyzing special curvature tensors [1], [2], [3] and [4]. In particular, the Berwald Weyl curvature and its decompositions have been investigated as central objects in understanding the projective properties of Finsler spaces [3], while Schur-type lemmas and new invariants have been established to characterize the behavior of such spaces [1], [2].

Further contributions have explored the interrelations between Weyl’s projective curvature tensor, R-projective tensors, and other curvature measures in both spacetime and generalized recurrent settings [5], [6] and [7]. These studies have opened new directions for analyzing Q-curvature inheritance and the role of Lie derivatives in generalized recurrent Finsler geometries [11], [12]. Moreover, higher-order recurrent spaces such as the generalized BK-fifth recurrent Finsler space have been systematically studied, highlighting the interplay between tensorial derivatives and the structure of projective motions [9], [10], [14].

Recent investigations also indicate that the introduction of new projective invariants within generalized Finsler metrics provides deeper insights into the intrinsic geometry of non-Riemannian spaces [2], [7]. Such results not only enrich the theoretical understanding of curvature tensors but also establish connections to broader geometric frameworks, including the Finsler–Laplace–Beltrami operators that appear in applied settings such as shape analysis [16].

Beyond pure geometry, the applications of Finsler structures have expanded into robotics and intelligent systems, where anisotropic and non-Euclidean environments play a critical role in navigation and path optimization. Modern algorithms for mobile robot path planning, particularly those based on improved RRT and related randomized methods, have been successfully employed in complex environments [17–19]. These approaches align naturally with the Finslerian perspective, were anisotropic metrics model heterogeneous motion costs and constraints. Recent studies on gradient estimates under Finsler–Ricci flows [20] further suggest potential applications in dynamic and adaptive environments, bridging the gap between theoretical developments in Finsler geometry and practical implementations in intelligent robotics.

In this context, the present study aims to contribute to the ongoing dialogue between the theoretical foundations of generalized recurrent Finsler spaces and their emerging applications. By focusing on projective and concircular motions, as well as the roles of higher-order recurrent structures, we seek to establish new insights into the relationship between curvature tensors and geometric invariants that govern both abstract mathematical structures and real-world navigation problems.

Consider an n-dimensional Finsler space equipped with a metric function F that satisfies the necessary regularity conditions. Let g_ij denote the components of the associated metric tensor, while Γ_jk^(*i) and G_jk^i represent the parameters of Cartan’s and Berwald’s connections, respectively.

These connection coefficients are symmetric in their lower indices.

The vectors y_i and〖 y〗^i satisfy the following relations:

(1) a) y_i=g_(ij ) y^j and b) y_i 〖 y〗^i=F^2 .

The metric tensor g_ij and its reciprocal tensor g^ij are connected by the relation:

(2) g_ij g^jk=δ_i^k= { █(1 , if i=k ,@0 , if i≠k .)┤ , where δ_i^k is the Kronecker delta.

The tensor

(3) C_ijk=1/2 ∂ ̇_i 〖 g〗_jk=1/4 ∂ ̇_i ∂ ̇_j ∂ ̇_k F^2

is referred to as the hv-torsion tensor.

The (v)hv-torsion tensor 〖 C〗_(ik )^h and its associated (h)hv-torsion tensor C_ijk satisfy the relations:

(4) a) 〖 C〗_jk^i 〖 y〗^j=C_kj^i 〖 y〗^j=0 , b) 〖y_i C〗_jk^i=0 , c) C_ijk y^j=0 ,

d) g_rj C_ik^r=C_ijk , e) C_(jk )^i 〖 g〗^jk=C^i and f) C_ijk 〖 g〗^jk 〖=C〗_i .

The Berwald covariant derivative of an arbitrary tensor field T_(j )^i with respect to x^k is defined as:

(5) B_k T_j^i=∂_k T_(j )^i-(∂ ̇_r 〖 T〗_(j )^i ) 〖 G〗_k^r+T_(j )^r G_rk^i-T_(r )^i G_(jk )^r .

For the metric function F, the vector y^(i ), and the normalized vector l^(i ), Berwald’s covariant derivative vanishes identically:

(6) a) B_(k ) y^i=0 , b) B_k F=0 and c) B_k 〖 y〗_i=0 .

However, the Berwald covariant derivative of the metric tensor does not vanish, i.e., B_(k ) g_ij≠0, and is given by:

(7) B_k g_ij=-2 C_(ijk׀h) y^h=-2 y^(h ) B_h C_ijk .

Moreover, the Berwald covariant differential operator with respect to x^(h ) commutes with the partial derivative with respect to y^(k ), namely:(8) (∂ ̇_(k ) B_h-B_(h ) ∂ ̇_k ) 〖 T〗_j^i=T_(j )^r G_khr^i-T_(r )^i G_(khj )^r , where T_(j )^i is any arbitrary tensor field.

The hv-curvature tensor 〖 P〗_(jkh )^iand the v(hv)-torsion tensor 〖 P〗_kh^i satisfy the following conditions:

(9) a) P_jkh^i 〖 y〗^j=P_kh^i , b) g_ir 〖 P〗_jkh^r=P_ijkh , c) 〖g_rp P〗_kh^r=P_kph ,

d) P_jki^i=P_jk , e) P_ki^i=P_k , f) P_kh^i y^k=P_kh^i y^h=0 and g) P_i^i=P .

The hv-curvature tensor 〖 P〗_(jkh )^i can equivalently be expressed as:

(10) a) P_jkh^i=Γ_jkh^(*i)+C_jr^i P_kh^r-C_(jh׀k)^i , with the identities:

b) P_jkh^i=∂ ̇_h 〖 Γ〗_jk^(*i)+C_(jr )^i C_(kh׀s )^r y^s-C_(jh׀k )^i

c) P_jkh^i=C_(kh׀j)^i-C_(jkh׀r ) g^ir+C_jk^r P_rh^i- P_jh^r C_(rk )^i , were

(1.11) a) Γ_(jkh )^(*i) y^j=P_kh^i , b) Γ_(jkh )^(*i) y^k=0 and g) 〖y_(i ) Γ〗_kjh^(*i)=-〖 P〗_kjh .

Finally, the projective curvature tensor W_jkh^i (also known as Weyl’s projective curvature tensor), the projective torsion tensor W_jk^i (Weyl’s torsion tensor), and the projective deviation tensor W_j^i (Weyl’s deviation tensor) are defined by

(12) W_jkh^i= H_jkh^i+(2 δ_j^i)/(n+1) H_[hk] +(2 y^i)/(n+1) ∂ ̇_j H_[kh] + (δ_k^i)/(n^2-1) (n〖 H〗_jh+H_hj+y^r ∂ ̇_j H_hr )

- (δ_h^i)/(n^2-1) (n〖 H〗_jk+H_kj+y^r ∂ ̇_j H_kr ) ,

(13) W_jk^i=H_jk^i+y^i/(n+1) H_[jk] +2 { (δ_([j)^i)/(n^2-1) (n H_(k])-y^r H_(k] r) ) , and

(14) W_j^i=H_j^i-H δ_j^i-1/(n+1) (∂ ̇_r H_j^r-∂ ̇_j H) y^i ,

respectively.

The tensors W_jkh^i , W_jk^i and W_k^i satisfy the following identities:

(15) a) W_jkh^i 〖 y〗^j=W_kh^i and b) W_jk^i 〖 y〗^j=W_k^i .

The projective curvature tensor W_jkh^i is skew-symmetric with respect to the indices k and h.

Furthermore, Cartan’s third curvature tensor R_jkh^i and the Cartan-type Ricci tensor R_jk are defined as follows:

(16) a) R_jkh^i=Γ_hjk^(*i)+(Γ_ljk^(*i) ) G_h^l+C_jm^i (G_kh^m-G_kl^m G_h^l )+Γ_(mk )^(*i) Γ_jh^(*m)-Γ_kjh^(*i)-(Γ_ljh^(*i) ) G_k^l

-C_jm^i (G_hk^m-G_hl^m G_k^l )-Γ_(mh )^(*i) Γ_jk^(*m) ,

b) R_jkh^i y^j=H_kh^i , c) R_jk y^j=H_k , d) R_jk y^k=R_j and e) R_jki^i=R_jk .

The Berwald curvature tensor H_(jkh )^i and the h(v)-torsion tensor H_kh^i are defined as

(17) a) H_jkh^i=∂_(j ) G_kh^i+G_kh^r G_rj^i+G_(rhj )^i G_k^r-∂_(j ) G_hk^i-G_hk^r G_rj^i-G_(rkj )^i G_h^r

And b) H_kh^i=∂_h G_k^i+G_(k )^r C_rh^i-∂_k G_h^i+G_(h )^r C_rk^i .

These tensors are related by the following relations:

(18) a) H_jkh^i y^j=H_kh^i , b) H_jkh^i=∂ ̇_j H_kh^i and c) H_jk^i=∂ ̇_(j ) H_k^i .

They were initially constructed by means of the tensor H_(h )^i, known as the deviation tensor, which is defined as

(19) a) H_h^i=2 ∂_h G^i-∂_r G_(h )^i y^r+2 G_(hs )^i G^s-G_(s )^i G_(h )^s and b) ∂ ̇_k G_h^i=G_(kh )^i .

Finally, by applying Euler’s theorem on homogeneous functions and contracting the indices i and h in (18) and (19), we obtain the following relations.

(20) a) H_(jk )^i y^j=H_(k )^i , b) g_(ip ) H_jk^i=H_(jp.k ) and c) H_(i ) y^i=(n-1)H .

2. A Generalized BR-Recurrent Space

In Finsler geometry, the study of recurrent structures plays a central role in characterizing the behavior of curvature tensors under covariant differentiation. Recurrent Finsler spaces were first introduced by imposing recurrence conditions on Cartan’s curvature tensors, leading to different classes of recurrent geometries. Among these, the notion of BR-recurrence, which involves Cartan’s second curvature tensor, provides a natural generalization of recurrence conditions in the context of Berwald-type connections. The introduction of generalized BR-recurrent spaces allows for the simultaneous presence of two covariant vector fields that govern the recurrence behavior of the curvature tensor. This generalization not only extends classical results but also opens new perspectives for studying the interplay between curvature tensors, torsion tensors, and Ricci-type contractions in higher-dimensional Finsler spaces. In what follows, we provide the formal definition of generalized BR-recurrent spaces and derive a series of structural results and theorems associated with their fundamental tensors.

Cartan’s second curvature tensor P_jkh^i is said to satisfy the recurrence condition

(21) 〖B_n P〗_jkh^i=〖λ_n P〗_jkh^i , 〖 P〗_jkh^i≠0 , where 〖 λ〗_n is a non-zero covariant vector field.

A Finsler space for which condition (21) holds is called a recurrent Finsler space.

More generally, if the curvature tensor P_jkh^i satisfies

(22) 〖B_m P〗_jkh^i=〖λ_m P〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_h^i g_jk-P_k^i g_jh ) , 〖 P〗_jkh^i≠0 ,

where B_m denotes Berwald’s covariant derivative with respect to x^m, λ_m ,〖 μ〗_m and δ_mare non-vanishing covariant vector fields, then the Finsler space F_n is called a generalized BR-recurrent space, denoted briefly by GBP-RF_n .

Definition 1. A Finsler space F_n whose Cartan’s second curvature tensor P_jkh^i satisfies condition (22), where λ_m , 〖 μ〗_(m ) and δ_m are non-null covariant vector fields, is called a generalized BR-recurrent space, denoted by GBP-RF_n .

By transvecting (22) with y^j , and using (6a), (9a), (1a), and (4c), we obtain

(23) 〖B_(m ) P〗_kh^i=〖λ_m P〗_kh^i+μ_m (δ_h^i y_k-δ_k^i y_h )+1/4 δ_m (P_h^i y_k-P_k^i y_h ) .

Contracting the indices i and h in (22) and (23), and applying (9d), (9e), (9g) along with (2), yields

(24) B_(m ) P_jk=λ_m P_jk+(n-1) μ_m g_jk+1/4 δ_m (Pg_jk-P_k^r g_jr ) ,

(25) B_(m ) P_k=λ_m 〖 P〗_k+〖(n-1) μ〗_m 〖 y〗_k+1/4 δ_m (Py_k-P_k^r y_r ) .

From the above, the following theorem is obtained:

Theorem 1. In GBP-RF_n , the tensors P_kh^i (the v(hv)-torsion tensor), P_jk (the P-Ricci tensor), and P_k (the curvature vector of Cartan’s second curvature tensor P_jkh^i) are given by (23), (24), and (25), respectively.

Transvecting (22) and (23) with g_(ir ), and applying (9b), (9c), (7) together with (2), we obtain

(26) B_m P_jkrh=λ_m P_jkrh+μ_m (g_rh g_jk-g_rk g_jh )-2 y^(n ) B_n C_irm P_jkh^i

+1/4 δ_m g_(ir ) (P_h^i g_jk-P_k^i g_jh ) .

(27) B_m P_krh=λ_m P_krh+μ_m (g_hr y_k-g_kr y_h )-2y^(n ) B_n C_irm P_kh^i+1/4 δ_(m ) g_(ir ) (P_h^i y_k-P_k^i y_h ) .

Hence, we have:

Theorem 2. In GBP-RF_n , the associate curvature tensor P_ijkh of the (hv)-curvature tensor P_(jkh )^i, and the associate tensor P_jkh of the v(hv)-torsion tensor P_kh^i , are given by (26) and (27), respectively.

By taking Berwald’s covariant derivative of (10a), with respect to x^m, and applying condition (22), we obtain

〖λ_m P〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_h^i g_jk-P_k^i g_jh )=B_m (Γ_jkh^(*i)+C_jr^i P_kh^r-C_jhk^i ) .

Using (10a), this reduces to

(28) B_m (Γ_jkh^(*i)+C_jr^i P_kh^r-C_(jh׀k)^i )=λ_m (Γ_jkh^(*i)+C_jr^i P_kh^r-C_(jh׀k)^i )

+ μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_h^i g_jk-P_k^i g_jh ) .

Equation (28) implies that the tensor (Γ_jkh^(*i)+C_jr^i P_(kh )^r-C_(jh׀k)^i ) cannot vanish, since otherwise it would imply μ_m=0 , a contradiction.

Thus, it is concluded the following.

Theorem 3. In GBP-RF_n , the tensor (Γ_jkh^(*i)+C_jr^i P_(kh )^r-C_(jh׀k)^i ) is non-vanishing and this tensor is generalized recurrent.

Transvecting (28) by y^j , using equations (6a), (9f), (1a) and (4c), we get the same equation (23).

Further, trausvecting (28) by y_(i ) , using equations (6c), (9g), (1a) and (4c), we get the same equation (26).

Now, transvecting equation (2.8) by y^k , using equations (6a), (11b), (9f), (1a), (4c) and in view of (2), we get

(29) B_m (C_(jh׀k)^i y^k )=λ_m (C_(jh׀k)^i y^k )+μ_m (δ_h^i y_j-g_jh y^i )+1/4 δ_m (P_h^i y_j-P_k^i g_jh y^k ) .

Trausvecting (29) by g_(ir ), using (4d), (1a), (7) and in view of (2), we get

(30) B_m (C_(jrh׀k) y^k )=λ_m (C_(jrh׀k) y^k )+μ_m (g_rh y_j-g_jh y_r )-2 y^(n ) B_n C_irm (C_(jh׀k)^i y^k )

+1/4 δ_m g_(ir ) (P_h^i y_j-P_k^i g_jh y^k ) .

Therefore, it is concluded the following.

Theorem 4. In GBP-RF_n , the identities (29) and (30) hold.

Trausvecting (29) and (30) by g^jh, using (4e), (4f), (2), (1a) and in view of (2), we get

(31) B_m (C_(׀k)^i y^k )=λ_m (C_(׀k)^i y^k )+1/4 δ_m (P_h^i y^h-P_k^i y^k ) .

(32) B_m (C_(r׀k) y^k )=λ_m (C_(r׀k) y^k )-2 y^(n ) B_n C_irm (C_(׀k)^i y^k )+1/4 δ_m g_(ir ) (P_h^i y^h-P_k^i y^k ) ,

where B_m g^jh=0 .

Therefore, we have

Theorem 5. In GBP-RF_n, the tensor C_(׀k)^i y^k is recurrent, while the tensor C_(r׀k) y^k is given by (32).

3. Identities for the Projective Curvature Tensor P_jkh^i in Generalized Berwald Recurrent Finsler Spaces

In this section, we establish several identities involving curvature tensors that characterize the generalized recurrent structure in the space GBR-TRF_n.

For a Riemannian space V_4, the projective curvature tensor P_jkh^i (Cartan’s second curvature tensor) can be related to the divergence of the Weyl tensor through the divergence of the projective curvature tensor as follows:

(33) W_jkh^i=P_jkh^i+1/3 (δ_(k )^i R_jh-R_(h )^i g_jk ) .

Taking the first-order covariant derivative with respect to x^m , using Berwald’s covariant differential operator, yields

(34) B_m W_jkh^i=B_m P_jkh^i+1/3 B_m (δ_(k )^i R_jh-R_(h )^i g_jk )

Applying condition (22) to equation (34), we obtain

B_m W_jkh^i=〖λ_m P〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_h^i g_jk-P_k^i g_jh )+1/3 B_m (δ_(k )^i R_jh-R_(h )^i g_jk ).

In view of equation (33) and using (7), this can be rewritten as

(35) B_m W_jkh^i=〖λ_m W〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )-1/3 λ_m (δ_(k )^i R_jh-R_(h )^i g_jk )

+1/4 δ_m (P_h^i g_jk-P_k^i g_jh )+1/3 δ_(k )^i B_m R_jh-1/3 (B_m R_(h )^i ) g_jk+2/3 R_(h )^i y^(n ) B_n C_jkm .

This, shows that

B_m W_jkh^i=〖λ_m W〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_h^i g_jk-P_k^i g_jh ) ,

if and only if

(36) δ_(k )^i B_m R_jh-λ_m (δ_(k )^i R_jh-R_(h )^i g_(jk ) )-(B_m R_(h )^i ) g_(jk )+2〖 R〗_(h )^i y^(n ) B_n C_jkm=0 .

Hence, we arrive at the following theorem:

Theorem 6. In GBP-RF_n (for n=4), the first-order Berwald covariant derivative of Weyl’s projective curvature tensor W_jkh^i is generalized recurrent if and only if condition (36) is satisfied.

Transvecting (35) by y^j , using (6a), (15a), (1a), (16c) and (4c), yields

(37) B_m W_kh^i=〖λ_m W〗_kh^i+μ_m (δ_h^i y_k-δ_k^i y_h )-1/3 λ_m (δ_(k )^i H_h-R_(h )^i H_(k ) )

+1/4 δ_m (P_h^i y_k-P_k^i y_h )+ 1/3 δ_(k )^i B_(m ) H_h-1/3 (B_m R_(h )^i ) y_(k ) .

This, shows that

(38) B_m W_kh^i=〖λ_m W〗_kh^i+μ_m (δ_h^i y_k-δ_k^i y_h )+1/4 δ_m (P_h^i y_k-P_k^i y_h ) ,

if and only if

(39) δ_(k )^i B_(m ) H_h-λ_m (δ_(k )^i H_h-R_(h )^i H_k )-(B_m R_(h )^i ) y_k=0 .

Therefore, it is concluded the following theorem

Theorem 7. In GBP-RF_n (for n=4), the first-order Berwald covariant derivative of Weyl’s projective torsion tensor W_kh^i is given by equation (38) if and only if condition (39) holds.

Transvecting (37) by y^k , using (6a), (15b), (1b), (2) and (20c), we get

B_m W_h^i=〖λ_m W〗_h^i+μ_m (δ_h^i F^2-y_h y^i )-1/3 λ_m (H_h 〖 y〗^i-(n-1) R_(h )^i H)

+1/4 δ_m (P_h^i F^2-P_k^i y_h y^k )+ 1/3 y^i B_(m ) H_h-1/3 (B_m R_h^i ) F^2 .

This, shows that

(40) B_m W_h^i=〖λ_m W〗_h^i+μ_m (δ_h^i F^2-y_h y^i )+1/4 δ_m (P_h^i F^2-P_k^i y_h y^k ) ,

if and only if

(41) y^i B_m H_h-λ_m (H_h y^i-(n-1) R_(h )^i H)-(B_m R_(h )^i ) F^2=0 .

Thus, the following is derived.

Theorem 8. In GBP-RF_n (for n=4), the first-order Berwald covariant derivative of Weyl’s projective deviation tensor W_h^i is given by equation (40) if and only if condition (41) holds.

Also, the projective curvature tensor P_jkh^i (for a Riemannian space V_4 ) is defined by:

(42) P_jkh^i=R_jkh^i-1/3 (δ_(h )^i R_jk-δ_(k )^i R_jh ) .

Taking covariant derivative of third order (Berwald’s covariant differential operator) of (42) with respect to x^m , we get

(43) B_m P_jkh^i=B_m R_jkh^i-1/3 (δ_(h )^i B_m R_jk-δ_(k )^i B_m R_jh ) .

Using the condition (22) in (43), we get

B_m R_jkh^i=〖λ_m P〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )

+1/4 δ_m (P_h^i g_jk-P_k^i g_jh )+1/3 (δ_(h )^i B_m R_jk-δ_(k )^i B_m R_jh ) .

By using (43), the above equation can be written as

(44) B_m R_jkh^i=〖λ_m R〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_h^i g_jk-P_k^i g_jh )

+1/3 B_m (δ_(h )^i R_jk-δ_(k )^i R_jh )-1/3 λ_m (δ_(h )^i R_jk-δ_(k )^i R_jh ) .

This, shows that

B_m R_jkh^i=〖λ_m R〗_jkh^i+μ_m (δ_h^i g_jk-δ_k^i g_jh )+1/4 δ_m (P_(h )^i g_jk-P_(k )^i g_jh ) ,

if and only if

(45) B_m (δ_(h )^i R_jk-δ_(k )^i R_jh )=λ_m (δ_(h )^i R_jk-δ_(k )^i R_jh ) .

Thus, it is concluded the following theorem

Theorem 9. In GBP-RF_n (for n=4), the first-order Berwald covariant derivative of Cartan’s third curvature tensor R_jkh^i is generalized recurrent if and only if condition (45) is satisfied.

Transvecting (44) by y^j , using (6a), (16b), (1a) and (16c), yields

(46) B_m H_kh^i=〖λ_m H〗_kh^i+μ_m (δ_h^i y_k-δ_k^i y_h )+1/4 δ_m (P_(h )^i y_k-P_(k )^i y_h )

+1/3 (δ_(h )^i B_m y_k-δ_(k )^i B_m y_h )-1/3 λ_m (δ_(h )^i y_k-δ_(k )^i y_h ) .

This, shows that

B_m H_kh^i=〖λ_m H〗_kh^i+μ_m (δ_h^i y_k-δ_k^i y_h )+1/4 δ_m (P_(h )^i y_k-P_(k )^i y_h ) ,

if and only if

(47) B_m (δ_(h )^i y_k-δ_(k )^i y_h )=λ_m (δ_(h )^i y_k-δ_(k )^i y_h ) .

Further, transvecting (46) by y^k , using (6a), (20a), (1a), (1b) and (2), we get

(48) B_m H_h^i=〖λ_m H〗_h^i+μ_m (δ_h^i F^2-y_h y^i )+1/4 δ_m (P_(h )^i F^2-P_(k )^i y_h y^k )

+1/3 (δ_(h )^i B_(m ) F^2-y^i B_m y_h )-1/3 λ_m (δ_h^i F^2-y_h y^i ) .

This, shows that

(49) B_m H_h^i=〖λ_m H〗_h^i+μ_m (δ_h^i F^2-y_h y^i )+1/4 δ_m (P_(h )^i F^2-P_(k )^i y_h y^k ) ,

if and only if

(50) B_(m ) (δ_(h )^i F^2-y^i y_h )=λ_m (δ_h^i F^2- y_h y^i ) .

Transvecting (46) by g_(ip ), using (7), (6a), (20b) and (2), yields

(51) B_m H_(kp.h)=λ_m H_(kp.h)+μ_m (g_(hp ) y_k-g_(kp ) y_h )+1/4 δ_m g_ip (P_(h )^i y_k-P_(k )^i y_h )

-2〖 H〗_(kh )^i y^(n ) B_n C_ipm+1/3 (g_(hp ) B_m y_k-g_(kp ) B_m y_h )-1/3 λ_m (g_(hp ) y_k-g_(kp ) y_h ) .

This, shows that

(51) B_m H_(kp.h)=λ_m H_(kp.h)+μ_m (g_(hp ) y_k-g_(kp ) y_h )+1/4 δ_m g_ip (P_(h )^i y_k-P_(k )^i y_h ) ,

if and only if

(52) (g_hp B_m y_k-g_kp B_m y_h )-λ_m (g_(hp ) y_k-g_(kp ) y_h )-6〖 H〗_(kh )^i y^(n ) B_n C_ipm=0 .

Therefore, using the above assumptions and mathematical analysis results the following theorem have been derived.

Theorem 10. In GBP-RF_n (for n=4), the first-order Berwald covariant derivatives of the h(v)-torsion tensor 〖 H〗_kh^i, the deviation tensor 〖 H〗_h^i, and the tensor H_(kp.h ) are given by equations (46), (49), and (51), respectively, if and only if conditions (47), (50), and (52) hold.

Contracting the indices i and h in (44), using (16e), (9g) and in view of (2), we get

(53) B_m R_jk=λ_m R_jk+(n-1) 〖 μ〗_m g_jk+1/4 δ_m (Pg_jk-P_k^r g_jr )

+1/3 (n-1) B_m R_jk-1/3 (n-1) λ_m R_jk .

This, shows that

B_m R_jk=λ_m R_jk+(n -1) μ_m g_jk+1/4 δ_m (Pg_jk-P_k^r g_jr ) ,

if and only if

(54) B_m R_jk=λ_m R_jk .

Therefore, it is concluded the following.

Theorem 11. In In GBP-RF_n (for n=4), the first-order Berwald covariant derivative of the R-Ricci tensor R_jk is non-vanishing if and only if the tensor R_jk is recurrent.

Transvecting (53) by y^j , using (6a), (16c) and (1a), yields

(55) B_m H_k=λ_m H_k+(n-1) 〖 μ〗_m y_k+1/4 δ_m (Py_k-P_k^r y_r )

+1/3 (n-1) B_m H_k-1/3 (n-1) λ_m H_k .

This, shows that

(56) B_m H_k=λ_m H_k+(n-1) 〖 μ〗_m y_k+1/4 δ_m (Py_k-P_k^r y_r ) ,

if and only if

(57) B_m H_k=H_k .

Further, transvecting (53) by y^k , using (6a), (16d) and (1a), we get

(58) B_m R_j=λ_m R_j+(n-1) 〖 μ〗_m y_j+1/4 δ_m (Py_k-P_k^r g_(jr ) y^k )

+1/3 (n-1) B_m R_j-1/3 (n-1) λ_m R_j .

This, shows that

(59) B_m R_j=λ_m R_j+(n -1) 〖 μ〗_m y_j+1/4 δ_m (Py_k-P_k^r g_(jr ) y^k ) ,

if and only if

(60) B_m R_j=λ_m R_j .

Therefore, it is concluded the following.

Theorem 12. In GBP-RF_n (for n=4), the first-order Berwald covariant derivatives of the curvature vectors H_k and R_j are non-vanishing if and only if the vectors H_k and R_j are recurrent.

4. Conclusion:

The generalized BP-recurrent Finsler spaces GBP-RF_n studied in this paper provide a rich geometric structure that can be effectively utilized in modeling and analyzing robotic motion in complex environments. In particular, the recurrence conditions derived for the Cartan's second curvature tensor and its associated tensors (e.g., Ricci and deviation tensors) offer analytical tools for understanding how direction-dependent constraints and anisotropies affect robot trajectories. Since Finsler geometry generalizes Riemannian geometry by allowing the metric to depend on both position and direction, it is highly suitable for motion planning in environments with frictional asymmetries, varying terrain properties, or dynamically changing fields. The recurrence relations such as equations (2.4), (2.5), and (3.17) suggest that under specific geometric constraints, the path deviation or curvature vector fields exhibit predictable, recurrent behavior. These insights could inform the design of energy-efficient paths or optimal control strategies for mobile robots, particularly in applications where the robot's motion must adapt to locally varying geometric properties of the space, such as in planetary exploration, search and rescue in uneven terrain, or autonomous navigation through soft or deformable media.

5. Recommendations and Future Research Directions

Based on the theoretical findings presented in this study, we propose the following recommendations and future directions:

1. Expand computational investigations of GBP-RF_n models in robotic simulation environments to assess their practical viability in real-world scenarios.

2. Develop motion planning algorithms that leverage the recurrent structures of geometric tensors, particularly in anisotropic spaces requiring high adaptability to complex terrains.

3. Explore the relationship between tensorial recurrence and motion stability in autonomous robotic systems, with attention to balancing mathematical rigor and engineering efficiency.

4. Integrate advanced Finslerian geometry with artificial intelligence techniques, aiming to embed these geometric models into machine learning frameworks for improved behavior prediction in dynamic environments.

5. Investigate broader applications of generalized recurrent Finsler structures in domains such as autonomous vehicles, aerial drones, and intelligent medical navigation systems.

Pursuing these directions promises a fruitful convergence between abstract mathematical frameworks and modern technological innovations, enabling the development of next-generation intelligent systems grounded in robust geometric theory.

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DECLARATIONS

Funding: None.

Conflict of Interest: The authors declare no conflict of interest.

Ethical Approval: The study was conducted in accordance with applicable ethical standards and approved by the appropriate ethics committee where required.

Informed Consent: Informed consent was obtained from all participants involved in the study where applicable.

Author Contributions: All authors contributed to the study conception, design, data collection, analysis, manuscript preparation, and approved the final version of the manuscript.

Data Availability: Data supporting the findings of this study are available from the corresponding author upon reasonable request.

Biomedicine and Chemical Sciences

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