Geometric Analysis of Lie and Berwald Derivatives of Inheritance Tensors in Finsler Spaces

Abstract

This paper introduces the concept of P-curvature inheritance in generalized recurrent Finsler spaces and establishes various types of curvature inheritance tensors in such spaces. We prove that the fundamental function of the Finsler space is given by $F=\sqrt{{\displaystyle \frac{y^i{y}^r{\delta }_r^k{g}_{ik}}{n}}}$. Moreover, we infer that the Lie derivative of the curvature scalar R is equal to the Lie derivative of the curvature scalar K, and the Lie derivative of the recurrence vector field ${\mu }_m$ vanishes. Additionally, we establish new mathematical formulas for the scalar function $\alpha \left(x\right)$ and the scalar form of the metric tensor $g_{ij}$ that admit P-curvature inheritance. A tensor $\left({\delta }_l^k{P}_{kh}^i\right)$ and the R-Ricci tensor possess an inheritance property in the generalized $\mathfrak{B}P$-recurrent Finsler space. In the same vein, we obtain conditions under which the Lie derivative and the Berwald covariant derivative of the curvature scalar P commute. Finally, we provide practical examples that illustrate the understanding of the obtained results.

1. Introduction

Finsler geometry, developed by P. Finsler, is a branch of differential geometry that extends Riemannian geometry. It offers a deep understanding of spaces equipped with non-Riemannian metrics, especially concerning curvature and covariant differentiation. Curvature tensors constitute another core aspect of Finsler geometry. They generalize the notion of curvature from Riemannian spaces and are associated with several connections, such as the Cartan, Berwald, and Chern connections, as discussed by Chern and Shen [1]. These tensors are essential for analyzing both intrinsic and extrinsic curvature properties of Finsler manifolds. The relationships among various curvature tensors in Finsler spaces were further investigated by Zafar and Musavvir [2].

The Berwald covariant derivative ${\mathfrak{B}}_k$ is an important concept in Finsler geometry and is considered one of the main types of covariant derivative used to describe how tensors change along directions in a Finsler space. It serves as a foundation for many studies concerning curvature tensors, recurrence conditions, and the interaction between position and direction dependence in Finsler geometry. By applying the Berwald covariant derivative, Abdallah et al. introduced the necessary and sufficient conditions for certain curvature tensors to satisfy the recurrence property in generalized Finsler spaces [3]. In addition, Ghadle et al. [4] studied the generalized $\mathfrak{B}P$-recurrent Finsler space corresponding to the tensor $W_{jkh}^i$ in the Berwald sense.

Regarding the Lie derivative $L_v$, it is a fundamental operator that measures how a tensor field changes along the flow of a smooth vector field. Introduced by Sophus Lie in the 19th century, it serves as a key tool in characterizing the symmetries and invariance properties of geometric objects. Yano and Willmore [5,6] made significant contributions to the development of the Lie derivative, formulating expressions involving partial derivatives of tensor fields and showing that the Lie derivative represents a differential invariant. Several authors have explored these ideas from different perspectives: Pandey [7] examined identities on Lie-recurrent Finsler spaces, while Gouin [8], Gruver, Ohta, Pak, and Kim [9,10,11] discussed the properties and applications of the Lie derivative itself.

In Finsler geometry, the concept of inheritance provides a powerful framework for studying how geometric quantities evolve under the action of a smooth vector field. This idea, originally developed in the context of Riemannian geometry, has been extended to various curvature tensors to investigate whether they remain invariant or change in a proportional manner along specific vector fields. The inheritance property thus describes a form of geometric symmetry or self-similarity within the manifold. The study of P-curvature inheritance offers significant insights into the geometric structure, recurrence properties, and symmetry behavior of Finsler spaces. In particular, P-curvature inheritance describes the behavior of the curvature tensor $P_{jkh}^i$ under the flow of a vector field ${\mu }_m$.

The Lie derivative and inheritance tensors are closely connected through the concepts of symmetry and invariance in differential geometry. The study of Lie derivatives and inheritance tensors is particularly significant, as it provides a means to explore the relationship between vector fields and curvature structures such as $P_{jkh}^i$ and $R_{jkh}^i$. In this context, Opondo and Singh [12,13] investigated the W-curvature inheritance and H-curvature inheritance in recurrent Finsler spaces, respectively. Furthermore, the inheritance property for various tensors in Finsler spaces has been examined in detail by Gatoto, Mishra, and Lodhi [14,15].

Based on what was mentioned previously and through the four main concepts (curvature tensors, inheritance tensors, the Lie derivative, and the Berwald covariant derivative), as well as considering their interrelations and the contributions of previous studies, it has been observed that these concepts require further investigation in a more general context. Accordingly, the present work examines these concepts within a generalized recurrent Finsler space, focusing on the behavior of some curvature tensors and inheritance tensors by employing both the Lie and Berwald derivatives.

The significance of this study stems from examining the four mentioned concepts collectively or individually within the main Finsler space. This approach allows us to explore the relationships among various types of tensors, analyze their behavior, and derive key formulas for them. These results provide a foundation that is highly valuable when extending these concepts to higher-order generalized Finsler spaces.

The main contribution of this paper is the derivation of new formulas in a generalized recurrent Finsler space by jointly applying the Lie and Berwald derivatives. It focuses on the relationships among various tensors, such as the R-Ricci tensor $R_{jk}$ and the P-Ricci tensor $P_{jk}$. In addition, this study derives formulas for the fundamental function F and the scalar function $\alpha \left(x\right)$ of the Finsler space under specific conditions. Furthermore, the paper investigates the P-curvature inheritance along with several other types of curvature inheritance tensors within the main space.

2. Preliminaries

This section provides some important concepts, equations, and lemmas. The metric tensor $g_{ij}$ and its associate $g^{ij}$ are connected by Shen, Y.; Shen, Z.; and Xia [16,17] as follows:

$$\begin{array}{c}\hfill g_{jk}{g}^{kh}={\delta }_j^h=\left\{\begin{array}{c}1\mathrm{if}j=h,\hfill \\ 0\mathrm{if}j\ne h.\hfill \end{array}\right.\end{array}$$

(1)

The Kronecker delta ${\delta }_j^i$ and the vector $y^i$ satisfy the following relations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(a\right)g_{ir}{y}^r=y_i,\left(b\right)y^i{y}_i=F^2,\left(c\right){\delta }_i^i=n,\left(d\right){\delta }_h^i={\dot{\partial }}_h{y}^i,\hfill \\ \left(e\right){\dot{\partial }}_i{y}^i=1,\left(f\right){\delta }_j^i{g}_{ir}=g_{jr},\left(g\right){\delta }_k^i{y}_i=y_k,\hfill \\ \left(h\right)g^{ik}{y}_k=y^i\mathrm{and}\left(i\right){\delta }_k^i{y}^k=y^i.\hfill \end{array}\right.\end{array}$$

(2)

The $\left(h\right)hv$-torsion tensor $C_{ijk}$ and its associate tensor $C_{jk}^i$ are defined by Matsumoto [18] as

$$\begin{array}{c}\hfill \left(a\right)C_{ijk}{y}^i=C_{kij}{y}^i=C_{jki}{y}^i=0\mathrm{and}\left(b\right)C_{jk}^i{y}^j=C_{kj}^i{y}^j=0.\end{array}$$

(3)

The Berwald covariant derivatives of $y^i$ and $y_i$ are given by

$$\begin{array}{c}\hfill \left(a\right){\mathfrak{B}}_k{y}^i=0\mathrm{and}\left(b\right){\mathfrak{B}}_k{y}_i=0.\end{array}$$

(4)

The curvature tensor $R_{jkh}^i$, the $h\left(v\right)$-torsion tensor $H_{kh}^i$, the R-Ricci tensor $R_{jk}$, the K-Ricci tensor $K_{jk}$, and the curvature scalar H satisfy the following relations by Rund [19]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(a\right)R_{jkh}^i{y}^j=H_{kh}^i=K_{jkh}^i{y}^j,\left(b\right)R_{jkh}^i=K_{jkh}^i+C_{js}^i{H}_{kh}^s,\hfill \\ \left(c\right)R_{jk}=K_{jk}+C_{js}^r{H}_{kr}^s,\left(d\right)H_{kh}^i{y}_i=0,\left(e\right)K_{jkr}^r=K_{jk},\hfill \\ \left(f\right)H_{kr}^r=H_k,\left(g\right)H_k{y}^k=(n-1)H,\left(h\right)R_{jk}{g}^{jk}=R,\hfill \\ \left(i\right)R_{jkr}^r=R_{jk},\left(j\right)H_{kh}^i{y}^k=H_h^i,\left(k\right){\dot{\partial }}_r{\dot{\partial }}_j{y}^r{H}_{kh}^i=0,\left(l\right)K_{jk}{g}^{jk}=K.\hfill \end{array}\right.\end{array}$$

(5)

The associate tensor $P_{ijkh}$, the torsion tensor $P_{kh}^i$, the P-Ricci tensor $P_{jk}$, and the curvature vector $P_k$ satisfy the following relations by Rund [19]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\left(a\right)P_{jkh}^i{y}^j=P_{kh}^i,\left(b\right)P_{jkh}^i{g}_{ir}=P_{rjkh},\left(c\right)P_k{y}^k=P,\hfill \\ \left(d\right)P_{jkr}^r=P_{jk}\mathrm{and}\left(e\right)P_{kr}^r=P_k.\hfill \end{array}\right.\end{array}$$

(6)

A Finsler space in which the Berwald connection parameter $G_{kh}^i$ does not depend on the directional coefficients $y^i$ is known as an affinely connected space. Therefore, this space is defined by one of the following equivalent conditions:

$$\begin{array}{c}\hfill \left(a\right){\mathfrak{B}}_k{g}_{ij}=0\mathrm{and}\left(b\right){\mathfrak{B}}_k{g}^{ij}=0.\end{array}$$

(7)

In Finsler geometry, the line element is represented by $(x^i,y^i)$, where $y^i$ denotes the derivative of $x^i$ with respect to the parameter t. The Lie derivative shows how a quantity changes when moved slightly along the direction of a given vector field. Therefore, the change occurs in the positional coordinate x and not in the directional coordinate y. Thus, the Lie derivative of $y^i$ vanishes, which means that $y^i$ is Lie-invariant.

The infinitesimal transformation given by Yano [6] can be explained as

$$x^{-i}=x^i+v^i\left(x\right)\epsilon ,$$

where $\epsilon$ is an infinitesimal constant, while $v^i\left(x\right)\ne 0$ is a contravariant vector field independent of the directional argument and dependent only on the positional coordinates of $x^i$. The infinitesimal method is a tool that leads to the Lie derivative.

This derivative evaluates the rate of change of a vector field or a tensor field along the smooth vector field $v^i\left(x\right)$. The Lie derivative of a general mixed tensor field $T_{jkh}^i(x,\dot{x})$ is expressed in the form given by Al-Qashbari, Baleedi, and Yano [6,20]:

$$\begin{array}{ccc}\hfill L_v{T}_{jkh}^i& =& v^m{\mathfrak{B}}_m{T}_{jkh}^i-T_{jkh}^m{\mathfrak{B}}_m{v}^i+T_{mkh}^i{\mathfrak{B}}_j{v}^m+T_{jmh}^i{\mathfrak{B}}_k{v}^m\hfill \\ & & +T_{jkm}^i{\mathfrak{B}}_h{v}^m+{\dot{\partial }}_m{T}_{jkh}^i{\mathfrak{B}}_r{v}^m{y}^r.\hfill \end{array}$$

(8)

Based on the above and since the vector $y^i$ is Lie-invariant, from Opondo and Verma [12,21], we have

$$\begin{array}{c}\hfill L_v{y}^i=0.\end{array}$$

(9)

Taking the Lie derivative of (2)g and using (1), we obtain

$$\begin{array}{c}\hfill L_v{y}_i=0.\end{array}$$

(10)

By applying the Lie derivative to (2)a and (2)h, respectively, using (9) and (10), we obtain

$$\begin{array}{c}\hfill \left(a\right)L_v{g}_{ij}=0\mathrm{and}\left(b\right)L_v{g}^{ij}=0.\end{array}$$

(11)

The generalized recurrent Finsler space for Cartan’s second curvature tensor $P_{jkh}^i$ in the Berwald sense ($G\mathfrak{B}P$-$R{F}_n$) satisfies the following condition given by Abdallah et al. [3]:

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{P}_{jkh}^i={\lambda }_m{P}_{jkh}^i+{\mu }_m({\delta }_j^i{g}_{kh}-{\delta }_k^i{g}_{jh}),\end{array}$$

(12)

where $P_{jkh}^i\ne 0$ and ${\lambda }_m,{\mu }_m$ are non-zero recurrence vectors. By applying transvection to Equation (12) by $y^j$, using (6a), (4a), (2i), and (2a), we obtain

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{P}_{kh}^i={\lambda }_m{P}_{kh}^i+{\mu }_m(y^i{g}_{kh}-{\delta }_k^i{y}_h).\end{array}$$

(13)

Also, by transvecting (12) by $g_{ir}$ and using (6b), (7a), and (2f), we obtain

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{P}_{rjkh}={\lambda }_m{P}_{rjkh}+{\mu }_m(g_{jr}{g}_{kh}-g_{kr}{g}_{jh})+2P_{jkh}^i{y}^t{\mathfrak{B}}_t{C}_{irm}.\end{array}$$

(14)

By applying contraction on the indices i and h in (13), using (6e), (2a), and (2f), we obtain the following equation:

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{P}_k={\lambda }_m{P}_k.\end{array}$$

By transvecting the above equation by $y^j$ and using (6c) and (4a), we obtain

$$\begin{array}{c}\hfill {\mathfrak{B}}_{m}P={\lambda }_{m}P.\end{array}$$

(15)

The generalized recurrent Finsler space for $K_{jkh}^i$ in Berwald sense ($G\mathfrak{B}K$-$R{F}_n$) satisfies the following condition given by Al-Qashbari and Baleedi [20]:

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{K}_{jkh}^i={\lambda }_m{K}_{jkh}^i+{\mu }_m\left({\delta }_h^i{g}_{jk}-{\delta }_k^i{g}_{jh}\right),K_{jkh}^i\ne 0,\end{array}$$

(16)

where Cartan’s fourth curvature tensor $K_{jkh}^i$ is called K-curvature inheritance and satisfies the relation

$$\begin{array}{c}\hfill L_v{K}_{jkh}^i=\alpha \left(x\right)K_{jkh}^i,\end{array}$$

(17)

where $\alpha \left(x\right)$ is a scalar function with non-zero values.

The space in which $R_{jkh}^i$ satisfies the following condition is called a recurrent Finsler space, as given by Verma [21], where

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{R}_{jkh}^i={\lambda }_m{R}_{jkh}^i.\end{array}$$

By applying contraction to the indices i and h in the above equation and using (5i), we have

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{R}_{jk}={\lambda }_m{R}_{jk}.\end{array}$$

By transvecting the above equation by $g^{jk}$, using (5h) and (7b), we obtain

$$\begin{array}{c}\hfill {\mathfrak{B}}_{m}R={\lambda }_{m}R.\end{array}$$

(18)

The commutation formula of the Lie derivatives and the partial derivative with respect to the directional coordinate $y^l$ for the deviation tensor $T_j^i$ is given by

$$\begin{array}{c}\hfill L_v\left({\dot{\partial }}_l{T}_j^i\right)-{\dot{\partial }}_l\left(L_v{T}_j^i\right)=0.\end{array}$$

(19)

H-curvature inheritance satisfies the following relation given by Singh [13]:

$$\begin{array}{c}\hfill L_v{H}_h^i=\alpha \left(x\right)H_h^i.\end{array}$$

(20)

A scalar function of the tensor $\alpha \left(x\right)T$ is a function whose inputs are n-dimensional tensors but whose output is a scalar value. The scalar function $\alpha \left(x\right)$ of the tensor $T(x,y)$ of different orders is given by Al-Qashbari, Baleedi, and Rund [19,20], where

$$\alpha \left(x\right)T_k=\sum _{k=1}^n{T}_k{y}^k.$$

$$\alpha \left(x\right)T^k=\sum _{k=1}^n{T}^k{y}_k.$$

$$\alpha \left(x\right)T_j^i=\sum _{i=1}^n{T}_i^i.$$

$$\alpha \left(x\right)T_{jk}=\sum _{j,k=1}^n{T}_{jk}{g}^{jk}.$$

$$\alpha \left(x\right)T^{jk}=\sum _{j,k=1}^n{T}^{jk}{g}_{jk}.$$

$$\alpha \left(x\right)T_{kh}^i=\alpha \left(x\right)T_{ki}^i=\alpha \left(x\right)T_k=\sum _{k=1}^n{T}_k{y}^k.$$

$$\alpha \left(x\right)T_{jkh}^i=\alpha \left(x\right)T_{jki}^i=\alpha \left(x\right)T_{jk}=\sum _{j,k=1}^n{T}_{jk}{g}^{jk}.$$

$$\alpha \left(x\right)T_{jkh}^{ir}=\alpha \left(x\right)T_{jkr}^{ir}=\alpha \left(x\right)T_{jk}^i=\alpha \left(x\right)T_{ji}^i=\alpha \left(x\right)T_j=\sum _{j=1}^n{T}_j{y}^j.$$

Lemma 1.

The outer product of two vectors forms a second-rank tensor, as given by Murray [22].

Lemma 2.

The scalar functions of two tensors of the same type are equal, as stated by Murray [22].

This study presents eleven theorems that collectively reveal a fundamental interrelation among their results. The findings are organized into two main parts.

The first part includes Theorems 1–6.

The first and second theorems focus on analyzing the behavior of the vector field ${\mu }_m$ and the fundamental function F, using both the Lie and Berwald derivatives simultaneously in ($G\mathfrak{B}P$-$R{F}_n$). According to Theorem 1, since ${\mu }_m$ vanishes, this result leads to the third and fourth theorems, through which we derive the main condition under which $\left(L_v{P}_{kh}^i\right)$ satisfies the recurrence property, and we establish that the Lie and Berwald derivatives of the curvature scalar P are commutative under specific conditions. We conclude the first part with the fifth and sixth theorems, which focus on the relationships between the tensors discussed in the previous results, such as the R-Ricci tensor $R_{jk}$ and the P-Ricci tensor $P_{jk}$.

The second part includes Theorems 7–11.

Theorem 7 strengthens the concept of P-curvature inheritance, demonstrating that the tensor $\left({\delta }_l^k{P}_{kh}^i\right)$ satisfies an inheritance property governed by the scalar function $\alpha \left(x\right)$. In Theorem 8, we obtain the formula for the scalar function $\alpha \left(x\right)$ by introducing a specific condition associated with the inheritance property. Building upon Theorem 8, which involves the metric tensor $g_{jk}$, Theorem 9 derives a corresponding expression for its associated scalar quantity. Moreover, based on the principal assumption introduced in Theorem 8, Theorem 10 establishes that the Lie derivatives of the curvature scalars R and K are identical. Finally, continuing the logical development of the preceding results and relying on the equality between the curvature scalars R and K established in Theorem 10, Theorem 11 concludes the second part and the entire study by proving that the R-Ricci tensor also satisfies the inheritance property.

3. Main Results

This section presents important theorems using the Lie and Berwald derivatives in $G\mathfrak{B}P$-$R{F}_n$.

Theorem 1.

The Lie derivative of the recurrence vector field ${\mu }_m$ vanishes in $G\mathfrak{B}P$-$R{F}_n$.

Proof. 

Taking the Lie derivative of (6b) and using (8) and (11a), we have

$$\begin{array}{c}\hfill g_{ir}{\mathfrak{B}}_m{P}_{jkh}^i={\mathfrak{B}}_m{P}_{rjkh}.\end{array}$$

(21)

Now, by applying the Lie derivative to (14), using (8), (9), and (11a), we obtain

$$\begin{array}{c}\hfill L_v\left({\mathfrak{B}}_m{P}_{rjkh}\right)=L_v\left({\lambda }_m{P}_{rjkh}\right)+(g_{jr}{g}_{kh}-g_{kr}{g}_{jh})L_v{\mu }_m+2y^t{L}_v{\mu }_m{P}_{jkh}^i{\mathfrak{B}}_t{C}_{irm}.\end{array}$$

Using (21) and (11a) in the above equation, we have

$$\begin{array}{c}\hfill g_{ir}{L}_v\left({\mathfrak{B}}_m{P}_{jkh}^i\right)=L_v\left({\lambda }_m{P}_{rjkh}\right)+(g_{jr}{g}_{kh}-g_{kr}{g}_{jh})L_v{\mu }_m+2y^t{L}_v{\mu }_m{P}_{jkh}^i{\mathfrak{B}}_t{C}_{irm}.\end{array}$$

By applying (1) in (12) and substituting the resulting equation into the above equation, we obtain

$$\begin{array}{c}\hfill g_{ir}{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=L_v\left({\lambda }_m{P}_{rjkh}\right)+(g_{jr}{g}_{kh}-g_{kr}{g}_{jh})L_v{\mu }_m+2y^t{L}_v{\mu }_m{P}_{jkh}^i{\mathfrak{B}}_t{C}_{irm}.\end{array}$$

Also by using (11a) and (6b) in the above equation, we have

$$\begin{array}{c}\hfill (g_{jr}{g}_{kh}-g_{kr}{g}_{jh})L_v{\mu }_m+2y^t{L}_v{\mu }_m{P}_{jkh}^i{\mathfrak{B}}_t{C}_{irm}=0.\end{array}$$

By transvecting the above equation by $y^i$, using (3a), and since the vector $y^i$ and the metric tensor $g_{ij}$ are non-zero, we obtain

$$\begin{array}{c}\hfill L_v{\mu }_m=0.\end{array}$$

(22)

Thus, we have the required results. □

Theorem 2.

In $G\mathfrak{B}P$-$R{F}_n$, the fundamental function of the Finsler space is given by $F=\sqrt{{\displaystyle \frac{y^i{y}^r{\delta }_r^k{g}_{ik}}{n}}}$.

Proof. 

By transvecing (14) by $y^r$, using (3a) and (2a), we have

$$\begin{array}{c}\hfill y^r{\mathfrak{B}}_m{P}_{rjkh}={\lambda }_m{y}^r{P}_{rjkh}+{\mu }_m(y_j{g}_{kh}-y_k{g}_{jh}).\end{array}$$

By applying (6)b to the left-hand side of the above equation, we infer that

$$\begin{array}{c}\hfill y^r{\mathfrak{B}}_m\left(g_{ir}{P}_{jkh}^i\right)={\lambda }_m{y}^r{P}_{rjkh}+{\mu }_m(y_j{g}_{kh}-y_k{g}_{jh}).\end{array}$$

Using (7)a in the above equation, we obtain

$$\begin{array}{c}\hfill y^r{g}_{ir}{\mathfrak{B}}_m{P}_{jkh}^i={\lambda }_m{y}^r{P}_{rjkh}+{\mu }_m(y_j{g}_{kh}-y_k{g}_{jh}).\end{array}$$

After placing (1) in (12) and using the resulting equation in the above equation, we have

$$\begin{array}{c}\hfill y^r{g}_{ir}{\lambda }_m{P}_{jkh}^i={\lambda }_m{y}^r{P}_{rjkh}+{\mu }_m(y_j{g}_{kh}-y_k{g}_{jh}).\end{array}$$

By applying (2)a in the above equation, we have

$$\begin{array}{c}\hfill y_i{\lambda }_m{P}_{jkh}^i={\lambda }_m{y}^r{P}_{rjkh}+{\mu }_m(y_j{g}_{kh}-y_k{g}_{jh}).\end{array}$$

Taking the Lie derivative of the above equation and applying (9), (10), (11a), and (22), we obtain

$$\begin{array}{c}\hfill y_i{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^r{L}_v\left({\lambda }_m{P}_{rjkh}\right).\end{array}$$

By transvecting the above equation by $y^i$ and using (2b), we have

$$\begin{array}{c}\hfill F^2{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^i{y}^r{L}_v\left({\lambda }_m{P}_{rjkh}\right).\end{array}$$

By transvecting the above equation by n and using (2c) on the right-hand side of the resulting equation, we obtain

$$\begin{array}{c}\hfill n{F}^2{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^i{y}^r{\delta }_i^i{L}_v\left({\lambda }_m{P}_{rjkh}\right).\end{array}$$

By applying (1) on the right-hand side of the above equation, we obtain

$$\begin{array}{c}\hfill n{F}^2{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^i{y}^r{g}_{ik}{g}^{ik}{L}_v\left({\lambda }_m{P}_{rjkh}\right).\end{array}$$

Using (6)b in the above equation, we obtain

$$\begin{array}{c}\hfill n{F}^2{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^i{y}^r{g}_{ik}{g}^{ik}{L}_v\left({\lambda }_m{g}_{ir}{P}_{jkh}^i\right).\end{array}$$

Also, by using (11)a in the above equation, we infer that

$$\begin{array}{c}\hfill n{F}^2{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^i{y}^r{g}_{ik}{g}^{ik}{g}_{ir}{L}_v\left({\lambda }_m{P}_{jkh}^i\right).\end{array}$$

By applying (1) in the above equation, we conclude that

$$\begin{array}{c}\hfill n{F}^2{L}_v\left({\lambda }_m{P}_{jkh}^i\right)=y^i{y}^r{\delta }_r^k{g}_{ik}{L}_v\left({\lambda }_m{P}_{jkh}^i\right).\end{array}$$

Thus, we have the required results. □

Theorem 3.

In $G\mathfrak{B}P$-$R{F}_n$, the Lie derivative of the torsion tensor $P_{kh}^i$ behaves as recurrent if

$$\begin{array}{c}\hfill y^j{\mathfrak{B}}_m\left(L_v{P}_{jkh}^i\right)=L_v\left({\mathfrak{B}}_m{P}_{kh}^i\right)-\left(L_v{\lambda }_m\right)P_{kh}^i.\end{array}$$

(23)

Proof. 

Using (1) in (13), we have

$$\begin{array}{c}\hfill {\mathfrak{B}}_m{P}_{kh}^i={\lambda }_m{P}_{kh}^i+{\mu }_m{y}^i{g}_{kh}.\end{array}$$

Taking the Lie derivative of the above equation and applying (22), (9), and (11a), we obtain

$$\begin{array}{c}\hfill {\lambda }_m\left(L_v{P}_{kh}^i\right)=L_v\left({\mathfrak{B}}_m{P}_{kh}^i\right)-\left(L_v{\lambda }_m\right)P_{kh}^i.\end{array}$$

(24)

Now, by applying the Lie derivative of (6a), using (9), we have

$$\begin{array}{c}\hfill y^j{L}_v{P}_{jkh}^i=L_v{P}_{kh}^i.\end{array}$$

Taking the Berwald covariant derivative of the above equation with respect to $x^m$, using (4a), we obtain

$$\begin{array}{c}\hfill {\mathfrak{B}}_m\left(L_v{P}_{kh}^i\right)=y^j{\mathfrak{B}}_m\left(L_v{P}_{jkh}^i\right).\end{array}$$

(25)

In view of (24) and (25), we obtain

$$\begin{array}{c}\hfill {\mathfrak{B}}_m\left(L_v{P}_{kh}^i\right)={\lambda }_m\left(L_v{P}_{kh}^i\right),\end{array}$$

if (23) holds. □

Theorem 4.

In $G\mathfrak{B}P-R{F}_n$, the Lie derivative and the Berwald covariant derivative of the curvature scalar P commute if and only if

$$\begin{array}{c}\hfill {\lambda }_m{L}_{v}P+P{L}_v{\lambda }_m=y^k{\mathfrak{B}}_m\left(L_v{P}_k\right).\end{array}$$

(26)

Proof. 

Taking the Lie derivative of (15), we have

$$\begin{array}{c}\hfill L_v\left({\mathfrak{B}}_{m}P\right)={\lambda }_m{L}_{v}P+P{L}_v{\lambda }_m.\end{array}$$

(27)

Now, by applying the Lie derivative of (6)c, we obtain

$$\begin{array}{c}\hfill L_v\left(P_k{y}^k\right)=L_{v}P.\end{array}$$

Using (9) in the above equation, we obtain

$$\begin{array}{c}\hfill y^k{L}_v{P}_k=L_{v}P.\end{array}$$

Taking the Berwald covariant derivative of the above equation with respect to $x^m$, we have

$$\begin{array}{c}\hfill {\mathfrak{B}}_m\left(y^k{L}_v{P}_k\right)={\mathfrak{B}}_m\left(L_{v}P\right).\end{array}$$

By applying (4)a in the above equation, we have

$$\begin{array}{c}\hfill y^k{\mathfrak{B}}_m\left(L_v{P}_k\right)={\mathfrak{B}}_m\left(L_{v}P\right).\end{array}$$

Otherwise,

$$\begin{array}{c}\hfill {\mathfrak{B}}_m\left(L_{v}P\right)=y^k{\mathfrak{B}}_m\left(L_v{P}_k\right).\end{array}$$

(28)

In view of (27) and (28), we obtain

$$\begin{array}{c}\hfill L_v\left({\mathfrak{B}}_{m}P\right)={\mathfrak{B}}_m\left(L_{v}P\right),\end{array}$$

if (26) holds. □

Theorem 5.

In $G\mathfrak{B}P$-$R{F}_n$, the Berwald covariant derivatives of the two curvature tensors, $K_{jkh}^i$ and $P_{jkh}^i$, and the Lie derivative of the R-Ricci tensor $R_{jk}$ are given by the following relations:

$$\begin{array}{c}\hfill P_{jkh}^i{\mathfrak{B}}_m{K}_{jkh}^i=K_{jkh}^i{\mathfrak{B}}_m{P}_{jkh}^i.\end{array}$$

(29)

and

$$\begin{array}{c}\hfill L_v{R}_{jk}={\displaystyle \frac{1}{n}}{g}_{jk}{v}^m{\mathfrak{B}}_{m}R,\end{array}$$

(30)

respectively.

Proof. 

By transvecting (16) by Cartan’s second curvature tensor $P_{jkh}^i$, we obtain

$$\begin{array}{c}\hfill P_{jkh}^i{\mathfrak{B}}_m{K}_{jkh}^i={\lambda }_m{P}_{jkh}^i{K}_{jkh}^i+{\mu }_m{P}_{jkh}^i\left({\delta }_h^i{g}_{jk}-{\delta }_k^i{g}_{jh}\right).\end{array}$$

Using (1) in the above equation, we have

$$\begin{array}{c}\hfill P_{jkh}^i{\mathfrak{B}}_m{K}_{jkh}^i={\lambda }_m{P}_{jkh}^i{K}_{jkh}^i.\end{array}$$

Using (1) in (12), and then using the resulting equation in the above equation, we obtain (29).

By applying the Lie derivative of (5h) and using (11b), we have

$$\begin{array}{c}\hfill g^{jk}{L}_v{R}_{jk}=L_{v}R.\end{array}$$

By transvecting the above equation by $g_{jk}$, using (1) and (2c), we obtain

$$\begin{array}{c}\hfill n{L}_v{R}_{jk}=g_{jk}{L}_{v}R.\end{array}$$

Using (8) on the right-hand side of the above equation, we obtain (30). Hence, the theorem is proved. □

Theorem 6.

In $G\mathfrak{B}P$-$R{F}_n$, the R-Ricci tensor $R_{jk}$ and the P-Ricci tensor $P_{jk}$ are equal if

$$\begin{array}{c}\hfill P=(n-1)H.\end{array}$$

(31)

Proof. 

As a consequence of the previous theorem, using (1) in (12), and then using the resulting equation and (5b) in the right side of (29), we infer that

$$\begin{array}{c}\hfill P_{jkh}^i{\mathfrak{B}}_m{K}_{jkh}^i=(R_{jkh}^i-C_{js}^i{H}_{kh}^s){\lambda }_m{P}_{jkh}^i.\end{array}$$

By transvecting the above equation by $y^j$ and using (3b), (5a), and (6a), we obtain

$$\begin{array}{c}\hfill P_{kh}^i{\mathfrak{B}}_m{K}_{jkh}^i=H_{kh}^i{\lambda }_m{P}_{jkh}^i.\end{array}$$

Using (1) in (16), and then using the resulting equation in the above equation, we have

$$\begin{array}{c}\hfill P_{kh}^i{K}_{jkh}^i=H_{kh}^i{P}_{jkh}^i.\end{array}$$

(32)

By applying contraction to the indices i and h in (32), using (5e,f) and (6d,e), we obtain

$$\begin{array}{c}\hfill P_k{K}_{jk}=H_k{P}_{jk}.\end{array}$$

Using (5)c in the above equation, we have

$$\begin{array}{c}\hfill P_k(R_{jk}-C_{js}^r{H}_{kr}^s)=H_k{P}_{jk}.\end{array}$$

By transvecting the left side of the above equation by $y^j$, using (3b), we obtain

$$\begin{array}{c}\hfill P_k{R}_{jk}=H_k{P}_{jk}.\end{array}$$

By applying transvection to the above equation by $y^k$, using (5g) and (6c), we infer

$$\begin{array}{c}\hfill R_{jk}=P_{jk}\end{array}$$

(33)

if (31) holds. Hence, the theorem is proved. □

In the next section, we concentrate on an inheritance property; i.e., we obtain the conditions for various tensors that satisfy the inheritance property in $G\mathfrak{B}P-R{F}_n$.

Assume that

$$\begin{array}{c}\hfill P_{kh}^i=H_{kh}^i.\end{array}$$

(34)

Using the above equation in (32), and then taking the Lie derivative of the resulting equation, we have

$$\begin{array}{c}\hfill L_v{K}_{jkh}^i=L_v{P}_{jkh}^i.\end{array}$$

By applying (17) in the above equation, we obtain

$$\begin{array}{c}\hfill L_v{P}_{jkh}^i=\alpha \left(x\right)K_{jkh}^i.\end{array}$$

Since $\alpha \left(x\right)$ is a scalar function, we derive

$$\begin{array}{c}\hfill L_v{P}_{jkh}^i=\alpha \left(x\right)P_{jkh}^i,\end{array}$$

(35)

where $\alpha \left(x\right)$ is a scalar function with non-zero values and $P_{jkh}^i\ne 0$.

Based on that, if the $h\left(v\right)$-torsion tensor $H_{kh}^i$ and the $v\left(hv\right)$-torsion tensor $P_{kh}^i$ are equal, then Cartan’s second curvature tensor $P_{jkh}^i$ is said to exhibit P-curvature inheritance. Hence, we have the following definition:

Definition 1.

In a recurrent Finsler space, if the Lie derivative of $P_{jkh}^i$ with respect to the vector field $v^i\left(x\right)$ is equal to a scalar function multiplied by this curvature tensor, then the infinitesimal transformation (Equation (35)) is called a P-curvature inheritance.

By transvecting (35) by $y^j$ and applying (6a) and (9), we infer that

$$\begin{array}{c}\hfill L_v{P}_{kh}^i=\alpha \left(x\right)P_{kh}^i.\end{array}$$

(36)

By applying contraction to the indices i and h in (35), using (6d), we conclude that

$$\begin{array}{c}\hfill L_v{P}_{jk}=\alpha \left(x\right)P_{jk}.\end{array}$$

(37)

By contracting the indices i and h in (36), using (6e), we obtain

$$\begin{array}{c}\hfill L_v{P}_k=\alpha \left(x\right)P_k.\end{array}$$

(38)

By transvecting (38) by $y^k$, using (6c) and (9), we have

$$\begin{array}{c}\hfill L_{v}P=\alpha \left(x\right)P.\end{array}$$

(39)

Thus, we conclude that Remark 1 is true.

Remark 1.

Various types of P-curvature inheritance are given by (36)–(39).

As a consequence of the previous definition, certain results related to the inheritance property in $G\mathfrak{B}P$-$R{F}_n$ are discussed using the Lie derivative.

Theorem 7.

In $G\mathfrak{B}P$-$R{F}_n$, which admits the P-curvature inheritance, the tensor $\left({\delta }_l^k{P}_{kh}^i\right)$ satisfies an inheritance property.

Proof. 

By applying (19) on the deviation tensor $H_h^i$, and then using (20) in the resulting equation, we have

$$\begin{array}{c}\hfill L_v\left({\dot{\partial }}_l{H}_h^i\right)={\dot{\partial }}_l\alpha \left(x\right)H_h^i.\end{array}$$

(40)

By transvecting (34) by $y^k$, using [(5j)], we obtain

$$\begin{array}{c}\hfill y^k{P}_{kh}^i=H_h^i.\end{array}$$

By applying the above equation in (40), and then using (2d) in the resulting equation, we infer that

$$\begin{array}{c}\hfill L_v\left({\delta }_l^k{P}_{kh}^i\right)=\alpha \left(x\right)\left({\delta }_l^k{P}_{kh}^i\right).\end{array}$$

The above equation refers to the required condition. □

Theorem 8.

In $G\mathfrak{B}P$-$R{F}_n$, which admits the P-curvature inheritance, the scalar function is given by $\alpha \left(x\right)={\displaystyle \frac{R}{n}}$, provided that (31) holds.

Proof. 

Using (33) in (30), we have

$$\begin{array}{c}\hfill L_v{P}_{jk}={\displaystyle \frac{1}{n}}{g}_{jk}{v}^m{\mathfrak{B}}_{m}R.\end{array}$$

By first applying (37) to the preceding equation and then substituting (18) into the result, we derive

$$\begin{array}{c}\hfill \alpha \left(x\right)P_{jk}={\displaystyle \frac{1}{n}}{v}^m{\lambda }_m{g}_{jk}R.\end{array}$$

(41)

Assume that

$$\begin{array}{c}\hfill P_{jk}=v^m{\lambda }_m{g}_{jk}.\end{array}$$

(42)

Using the above equation in (41), we obtain the required result. □

Theorem 9.

In $G\mathfrak{B}P$-$R{F}_n$, which admits the P-curvature inheritance, the scalar value of metric tensor is

$$\begin{array}{c}\hfill g_{jr}=\sum _{r,i=1}^n{T}_{ri}{g}^{ri},\end{array}$$

(43)

where (34) holds and $T_{ri}=y_r{y}_i$ [Lemma 1].

Proof. 

By transvecting (32) by $y_i$ and using (5)d, and then taking the Lie derivative of the resulting equation and applying (10), we have

$$\begin{array}{c}\hfill y_i{L}_v{P}_{kh}^i=0.\end{array}$$

By transvecting the above equation by $y^i$ and using (2b) and (5k) in the above equation, we obtain

$$\begin{array}{c}\hfill F^2{L}_v{P}_{kh}^i=y^i{\dot{\partial }}_r{\dot{\partial }}_j{y}^r{H}_{kh}^i.\end{array}$$

By applying (2e) in the above equation, we infer that

$$\begin{array}{c}\hfill F^2{L}_v{P}_{kh}^i=y^i{\dot{\partial }}_j{H}_{kh}^i.\end{array}$$

Using (34) and (2d) in the above equation, we obtain

$$\begin{array}{c}\hfill L_v{P}_{kh}^i=({\displaystyle \frac{{\delta }_j^i}{F^2}})P_{kh}^i.\end{array}$$

In view of the above equation and (36), we conclude that

$$\begin{array}{c}\hfill {\delta }_j^i=\alpha \left(x\right)F^2.\end{array}$$

By transvecting the above equation by the metric tensor $g_{ir}$ and using (2f), we obtain

$$\begin{array}{c}\hfill g_{jr}=\alpha \left(x\right)g_{ir}{F}^2.\end{array}$$

Using (2a,b) in the above equation, we infer that

$$\begin{array}{c}\hfill g_{jr}=\alpha \left(x\right)y_r{y}_i.\end{array}$$

Since $\alpha \left(x\right)$ is a scalar function, consequently, the expected result is achieved. □

Theorem 10.

In $G\mathfrak{B}P$-$R{F}_n$, which admits the P-curvature inheritance, the Lie derivative of the curvature scalar R is equal to the Lie derivative of the curvature scalar K.

Proof. 

Using (42) in (37), we infer that

$$\begin{array}{c}\hfill L_v{P}_{jk}=\alpha \left(x\right)v^m{\lambda }_m{g}_{jk}.\end{array}$$

By Theorem (8), the above equation can be written as

$$\begin{array}{c}\hfill L_v{P}_{jk}={\displaystyle \frac{R}{n}}{v}^m{\lambda }_m{g}_{jk}.\end{array}$$

By transvecting the above equation by $g^{jk}$, using (1) and (2c), we have

$$\begin{array}{c}\hfill g^{jk}{L}_v{P}_{jk}=v^m{\lambda }_{m}R.\end{array}$$

Using (11b) in the above equation, we obtain

$$\begin{array}{c}\hfill L_v\left(P_{jk}{g}^{jk}\right)=v^m{\lambda }_{m}R.\end{array}$$

(44)

By applying (34) in (32), we have

$$\begin{array}{c}\hfill K_{jkh}^i=P_{jkh}^i.\end{array}$$

By applying contraction to the indices i and h in the above equation, using (5e) and (6d), we obtain

$$\begin{array}{c}\hfill K_{jk}=P_{jk}.\end{array}$$

Taking the Lie derivative of the above equation, and then transvecting the resulting equation by $g^{jk}$, we infer that

$$\begin{array}{c}\hfill g^{jk}{L}_v{K}_{jk}=g^{jk}{L}_v{P}_{jk}.\end{array}$$

By applying (11)b to the previous equation and using the result in (44), we derive

$$\begin{array}{c}\hfill L_v\left(K_{jk}{g}^{jk}\right)=v^m{\lambda }_{m}R.\end{array}$$

Using (18) in the above equation, we obtain

$$\begin{array}{c}\hfill L_v\left(K_{jk}{g}^{jk}\right)=v^m{\mathfrak{B}}_{m}R.\end{array}$$

By applying (8) to the curvature scalar R, and then using the resulting equation in the above equation, we conclude that

$$\begin{array}{c}\hfill L_v\left(K_{jk}{g}^{jk}\right)=L_{v}R.\end{array}$$

Using (5l) in the above equation, we have

$$\begin{array}{c}\hfill L_{v}K=L_{v}R.\end{array}$$

(45)

Hence, we obtain the required result. □

Theorem 11.

In $G\mathfrak{B}P$-$R{F}_n$, which admits the P-curvature inheritance, the R-Ricci tensor satisfies the inheritance property.

Proof. 

Using (5l,h) in (45), and then using (11b) in the resulting equation, we obtain

$$\begin{array}{c}\hfill L_v{K}_{jk}=L_v{R}_{jk}.\end{array}$$

(46)

By applying contraction to the indices i and h in (17) and using (5e), we have

$$\begin{array}{c}\hfill L_v{K}_{jk}=\alpha \left(x\right)K_{jk},\end{array}$$

Using the above equation in (46), we obtain

$$\begin{array}{c}\hfill L_v{R}_{jk}=\alpha \left(x\right)K_{jk},\end{array}$$

Since $\alpha \left(x\right)$ is a scalar function, according to [Lemma 2], we obtain

$$\begin{array}{c}\hfill L_v{R}_{jk}=\alpha \left(x\right)R_{jk}.\end{array}$$

The above equation refers to the required result. □

4. Applications

The results obtained in this paper regarding P-curvature inheritance in generalized recurrent Finsler spaces have potential applications in both theoretical physics and advanced differential geometry. Furthermore, the use of Berwald’s covariant derivative in this context facilitates the description of how tensorial quantities evolve in spaces where standard Riemannian tools are insufficient. This becomes useful especially in modeling phenomena in anisotropic cosmology, modified gravity theories, or media with directionally dependent internal structures, such as in optical or relativistic Finsler media.

The newly established formulas for the scalar function $\alpha \left(x\right)$ and the scalar metric tensor $g_{ij}$ can be employed in modeling space-time manifolds with directional dependence, for example, in cosmology or relativistic mechanics involving anisotropic media. The inheritance properties of curvature tensors help simplify complex tensorial equations and enable deep analysis of conserved quantities and field invariants in such geometric models. For example, we can examine the P-curvature tensor within a Finsler space and demonstrate that, under the action of $L_v$, it fulfills an inheritance condition such as

$$\begin{array}{c}\hfill L_v{P}_{ijk}=\alpha P_{ijk},\end{array}$$

where $\alpha$ is a scalar function. This means that the P-curvature is inherited under the flow of v, and it is a property often connected to symmetries or conserved quantities. Examples derived from the results are presented below.

• Example 1. According to Theorem 8 and Theorem 9, if the P-Ricci tensor is given by $P_{jk}=y_k{y}_i$, then the value of the Lie derivative of the curvature scalar R is equal to n.
• Example 2. According to Theorem 6 and Theorem 8, if the scalar vectors are $v^m=a$ and ${\lambda }_m=b$, then the scalar function is given by $\alpha \left(x\right)=c$, where the scalar value is $c=ab$.
• Example 3. According to Theorem 5 and Theorem 6, the value of the curvature scalar R is ${\displaystyle \frac{n}{2}}$ if the P-Ricci tensor is $P_{jk}={\displaystyle \frac{1}{2}}{g}_{jk}$.
• Example 4. According to Theorem 9, the value of the scalar function for a quadratic fundamental function in Finsler space is $\alpha \left(x\right)F^2=1$.

5. Conclusions

In this paper, we inferred certain relations between the curvature tensor $P_{jkh}^i$ and other tensors by using the Lie derivative and Berwald’s covariant derivative. Several tensors were shown to satisfy the inheritance property in a generalized recurrent Finsler space through the Lie derivative. Furthermore, we derived new mathematical expressions for the scalar function $\alpha \left(x\right)$ and the scalar component of the metric tensor $g_{ij}$ in the main space admitted by the P-curvature inheritance. The Lie of the curvature scalar R was also shown to be equal to the Lie derivative of the curvature scalar K in $G\mathfrak{B}P$-$R{F}_n$.

This study examined specific concepts related to various tensor types within the main space discussed. Based on these findings, future research will expand the scope of this study to include generalized Finsler spaces of higher orders.