On an R-Randers 𝑚th-Root Space

Abstract

We consider an n-dimensional Finsler space 𝐹^𝑛(𝑛>2) with the metric 𝐿(𝑥,𝑦)=𝐹(𝑥,𝑦)+𝛼(𝑥,𝑦), where 𝐹 is an 𝑚th-root metric and 𝛼 is a Riemannian metric. We call such space as an R-Randers 𝑚th-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers 𝑚th-root space. Some other properties of such space have also been discussed.

1. Introduction

The theory of an mth-root metric was introduced by Shimada [1] in 1979. By introducing the regularity of the metric, various fundamental quantities of a Finsler metric could be found. In particular, the Cartan connection of a Finsler space with mth-root metric was introduced from the theoretical standpoint. Matsumoto and Okubo [2] studied Berwald connection of a Finsler space with mth-root metric and gave main scalars in two dimensional cases and defined higher-order Christoffel symbols. The mth-root metric is used in many problems of theoretical physics [3]. Pandey et al. [4] studied three-dimensional Finsler space with mth-root metric. To discuss general relativity with the electromagnetic field, Randers [5] introduced a metric of the form 𝐿(𝑥,𝑦)=𝛼(𝑥,𝑦)+𝛽(𝑥,𝑦), where 𝛼 is a square root metric and 𝛽 is a differential one form. In his honor, this metric is called Randers metric, and it has been extensively studied by several geometers and physicists [6–8].

Munteanu and Purcaru [9] first defined complex Finsler spaces by reducing the scalar 𝜆∈𝑅 in the homogeneity condition of fundamental function, that is, 𝐹(𝑧,𝜆𝜂,𝑧,𝜆𝜂)=|𝜆|𝐹(𝑧,𝜂,𝑧,𝜂),∀𝜆∈𝑅, and named such space as an R-complex Finsler space. Aldea and Purcaru [10] introduced the concept of R-complex Finsler spaces with (𝛼,𝛽)-metrics. In 2011, Purcaru [11] also discussed the notion of R-complex Finsler space with Kropina metric. Lungu and Nimineţ [12] studied a special Finsler space with the metric 𝐿 of the form 𝐿(𝑥,𝑦)=𝐹(𝑥,𝑦)+𝛼(𝑥,𝑦),∀𝑦∈𝑇𝑀, where 𝐹 is a quartic root metric and 𝛼 is a square root metric. They regarded this space as an R-Randers quartic space and obtained many results related to it. The aim of the present paper is to study a more general space with the metric 𝐿(𝑥,𝑦)=𝐹(𝑥,𝑦)+𝛼(𝑥,𝑦), where 𝐹 is an mth-root metric and 𝛼 is a Riemannian metric. We call the space endowed with this metric as an R-Randers mth-root space.

The paper is organized as follows. Section 2 deals with some preliminary concepts required for the discussion of the following sections. It includes the notion of an R-Randers mth-root space. In Section 3, we derive certain identities satisfied in an R-Randers mth-root space. We obtain the fundamental metric tensor 𝑔_𝑖𝑗, its inverse 𝑔^𝑖𝑗, and the Cartan tensor 𝐶_𝑖𝑗𝑘 for an R-Randers mth-root space. In Section 4, we obtain the spray coefficients of an R-Randers mth-root space. It includes the equations of the geodesics. Section 5 discusses the nonlinear connection in an R-Randers mth-root space.

2. Preliminaries

Let 𝐹^𝑛=(𝑀,𝐿(𝑥,𝑦))(𝑛>2) be an n-dimensional Finsler space. The mth-root metric on 𝑀 is defined as 𝐿^𝑚=𝑎_{𝑖1𝑖2⋯𝑖𝑚}(𝑥)𝑦^𝑖1𝑦^𝑖2⋯𝑦^𝑖𝑚, where 𝑎_{𝑖1𝑖2⋯𝑖𝑚}(𝑥) are components of an mth-order covariant symmetric tensor.

In case of 𝑚=2, the metric 𝐿 is Riemannian, and in the cases 𝑚=3 and 𝑚=4 these metrics are called cubic and quartic, respectively.

The covariant symmetric metric tensor 𝑔_𝑖𝑗 of 𝐹^𝑛(𝑀,𝐿(𝑥,𝑦)) is defined by 𝑔_𝑖𝑗1=∶2𝜕̇_𝑖𝜕̇_𝑗𝐿²,𝜕̇_𝑖≡𝜕𝜕𝑥̇^𝑖. (1) This tensor is positively homogeneous of degree zero in 𝑥̇^𝑖. From the metric tensor 𝑔_𝑖𝑗, we construct the Cartan tensor 𝐶_𝑖𝑗𝑘 by 𝐶_𝑖𝑗𝑘1=∶2𝜕̇_𝑘𝑔_𝑖𝑗. (2) The tensor 𝐶_𝑖𝑗𝑘 is symmetric in its lower indices and is positively homogeneous of degree −1 in 𝑥̇^𝑖. Due to its homogeneous and symmetric properties, it satisfies the following: 𝐶_𝑖𝑗𝑘𝑥̇^𝑖=𝐶_𝑘𝑖𝑗𝑥̇^𝑖=𝐶_𝑗𝑘𝑖𝑥̇^𝑖=0. (3) The angular metric tensor of a Finsler space is written as ℎ_𝑖𝑗=𝑔_𝑖𝑗−𝑙_𝑖𝑙_𝑗, (4) where 𝑙_𝑖=𝜕̇_𝑖𝐿.

The geodesic of a Finsler space is given by 𝑑²𝑥^𝑖𝑑𝑡²+2𝐺^𝑖=0, (5) where 𝐺^𝑖 is the geodesic spray coefficients given by 2𝐺^𝑖=12𝑔^𝑖𝑗𝑦^𝑘𝜕̇_𝑗𝜕_𝑘𝐿²−𝜕_𝑗𝐿²,𝜕_𝑗≡𝜕𝜕𝑥^𝑗. (6) The nonlinear connection of a Finsler space is defined as 𝑁^𝑖_𝑗=𝜕̇_𝑗𝐺^𝑖. (7) In the present paper, we study the space whose fundamental function is given by 𝐿𝑥,𝑦=𝐹𝑥,𝑦+𝛼𝑥,𝑦, (8) where 𝐹^𝑚=𝑎_{𝑖1𝑖2⋯𝑖𝑚}(𝑥)𝑦^𝑖1𝑦^𝑖2⋯𝑦^𝑖𝑚 (9) is  an  mth-root  metric and 𝛼²=𝑏_𝑖1𝑖2(𝑥)𝑦^𝑖1𝑦^𝑖2 (10) is a Riemannian metric. We call this space an R-Randers mth-root space.

3. Fundamental Metric Tensor and Cartan Tensor

In this section, we find the fundamental metric tensor 𝑔_𝑖𝑗, its inverse 𝑔^𝑖𝑗, angular metric tensor ℎ_𝑖𝑗, and the Cartan tensor 𝐶_𝑖𝑗𝑘 for an R-Randers mth-root space.

Differentiating (9) partially with respect to 𝑥̇^𝑖, we get 𝐹^𝑚−1𝜕̇_𝑖𝐹=𝑎_𝑖, (11) where 𝑎_𝑖(𝑥,𝑦)=𝑎_{𝑖𝑖2⋯𝑖𝑚}(𝑥)𝑦^𝑖2𝑦^𝑖3⋯𝑦^𝑖𝑚.

Differentiating (11) partially with respect to 𝑥̇^𝑗, we find (𝑚−1)𝐹^𝑚−2𝜕̇_𝑗𝐹𝜕̇_𝑖𝐹+𝐹^𝑚−1𝜕̇_𝑗𝜕̇_𝑖𝐹=(𝑚−1)𝑎_𝑖𝑗, (12) where 𝑎_𝑖𝑗(𝑥,𝑦)=𝑎_{𝑖𝑗𝑖3⋯𝑖𝑚}(𝑥)𝑦^𝑖3𝑦^𝑖4⋯𝑦^𝑖𝑚.

Take 𝜌_{−(𝑚−1)}=(𝑚−1)𝐹^{−(𝑚−1)} and 𝜌_{−(2𝑚−1)}=(𝑚−1)𝐹^{−(2𝑚−1)}, where the subscripts denote the degree of homogeneity of the corresponding entities with respect to 𝑥̇^𝑖. In view of this, (11) and (12) give 𝜕̇_𝑖𝜌𝐹=^{−(𝑚−1)}()𝑎𝑚−1^𝑖,𝜕̇_𝑗𝜕̇_𝑖𝐹=𝜌_{−(𝑚−1)}𝑎_𝑖𝑗−𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗, (13)(14) where 𝜕̇_𝑗𝜌_{−(𝑚−1)}=−(𝑚−1)𝜌_{−(2𝑚−1)}𝑎_𝑗. (15) Now, differentiating (10) partially with respect to 𝑥̇^𝑖, we get 𝜕̇_𝑖𝛼=𝜇₋₁𝑏_𝑖, (16) where 𝑏_𝑖(𝑥,𝑦)=𝑏_𝑖𝑖2(𝑥)𝑦^𝑖2 and 𝜇₋₁=𝛼⁻¹.

Further differentiating (16) partially with respect to 𝑥̇^𝑗, we find 𝜕̇_𝑗𝜕̇_𝑖𝛼=𝜇₋₁𝑏_𝑖𝑗+𝜇₋₃𝑏_𝑖𝑏_𝑗, (17) where 𝜇₋₃=−𝛼⁻³.

Differentiating 𝜇₋₁=𝛼⁻¹ partially with respect to 𝑥̇^𝑗, we get 𝜕̇_𝑗𝜇₋₁=𝜇₋₃𝑏_𝑗. (18) Thus, we have the following.

Proposition 1. In an R-Randers mth-root space, the following identities hold: 𝜕̇_𝑖𝜌𝐹=^{−(𝑚−1)}()𝑎𝑚−1^𝑖,𝜕̇_𝑗𝜕̇_𝑖𝐹=𝜌_{−(𝑚−1)}𝑎_𝑖𝑗−𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗,𝜕̇_𝑖𝛼=𝜇₋₁𝑏_𝑖,𝜕̇_𝑗𝜕̇_𝑖𝛼=𝜇₋₁𝑏_𝑖𝑗+𝜇₋₃𝑏_𝑖𝑏_𝑗,𝜕̇_𝑗𝜌_{−(𝑚−1)}=−(𝑚−1)𝜌_{−(2𝑚−1)}𝑎_𝑗,𝜕̇_𝑗𝜇₋₁=𝜇₋₃𝑏_𝑗. (19)

Differentiating (8) partially with respect to 𝑥̇^𝑖 and using (13) and (16), we have 𝜕̇_𝑖𝜌𝐿=^{−(𝑚−1)}()𝑎𝑚−1^𝑖+𝜇₋₁𝑏_𝑖. (20) Further, differentiating (20) partially with respect to 𝑥̇^𝑗 and using (14) and (17), we get 𝜕̇_𝑗𝜕̇_𝑖𝐿=𝜌_{−(𝑚−1)}𝑎_𝑖𝑗−𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗+𝜇₋₁𝑏_𝑖𝑗+𝜇₋₃𝑏_𝑖𝑏_𝑗. (21) From (1), we have 𝑔_𝑖𝑗=𝜕̇_𝑖𝐿𝜕̇_𝑗𝐿+𝐿𝜕̇_𝑖𝜕̇_𝑗𝐿. (22) Using (20) and (21), (22) yields 𝑔_𝑖𝑗=𝜌_{−(𝑚−1)}()𝑎𝑚−1^𝑖+𝜇₋₁𝑏_𝑖𝜌_{−(𝑚−1)}()𝑎𝑚−1^𝑗+𝜇₋₁𝑏_𝑗+𝐿𝜌_{−(𝑚−1)}𝑎_𝑖𝑗−𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗+𝜇₋₁𝑏_𝑖𝑗+𝜇₋₃𝑏_𝑖𝑏_𝑗. (23) Let us take 𝑑_𝑖𝑗=𝜌_{−(𝑚−1)}𝑎_𝑖𝑗+𝜇₋₁𝑏_𝑖𝑗 and 𝑐_𝑖=𝑞₀𝑎_𝑖+𝑞₋₁𝑏_𝑖, where 𝑞₀ and 𝑞₋₁ satisfy 𝑞²₀=𝜌²_{−(𝑚−1)}(𝑚−1)²−𝐿𝜌_{−(2𝑚−1)},𝑞²₋₁=𝜇²₋₁+𝐿𝜇₋₃,𝑞₀𝑞₋₁=𝜌_{−(𝑚−1)}()𝜇𝑚−1⁻¹. (24) The matrix (𝑑_𝑖𝑗) is nonsingular, in general, for 𝑎_𝑖𝑗's are functions of 𝑥 and 𝑦 whereas 𝑏_𝑖𝑗's are functions of 𝑥 only.

Equation (23) takes the form 𝑔_𝑖𝑗=𝐿𝑑_𝑖𝑗+𝑐_𝑖𝑐_𝑗. (25) Thus, we have the following.

Theorem 2. The fundamental metric tensor of an R-Randers mth-root space is given by (25).

Theorem 3. In an R-Randers mth-root space, the inverse 𝑔^𝑖𝑗 of the fundamental metric tensor 𝑔_𝑖𝑗 is given by 𝑔^𝑖𝑗=1𝐿𝑑^𝑖𝑗−1𝐿+𝑐²𝑐^𝑖𝑐^𝑗, (26) where 𝑐^𝑖=𝑑^𝑖𝑗𝑐_𝑗 and 𝑐²=𝑐_𝑖𝑐^𝑖.

Proof. Let (𝑑^𝑖𝑗) be the inverse of nonsingular matrix (𝑑_𝑖𝑗). Suppose that (𝑔^𝑖𝑗) is given by (26).
Now, 𝑔_𝑖𝑗𝑔^𝑗𝑘=𝐿𝑑_𝑖𝑗+𝑐_𝑖𝑐_𝑗1𝐿𝑑^𝑗𝑘−1𝐿+𝑐²𝑐^𝑗𝑐^𝑘=𝑑_𝑖𝑗𝑑^𝑗𝑘−𝑑_𝑖𝑗𝐿+𝑐²𝑐^𝑗𝑐^𝑘+1𝐿𝑐_𝑖𝑐_𝑗𝑑^𝑗𝑘−1𝐿𝐿+𝑐²𝑐_𝑖𝑐_𝑗𝑐^𝑗𝑐^𝑘=𝛿^𝑖_𝑘−𝑐_𝑖𝑐^𝑘𝐿+𝑐²+1𝐿𝑐_𝑖𝑐^𝑘−1𝐿𝐿+𝑐²𝑐²𝑐_𝑖𝑐^𝑘=𝛿^𝑖_𝑘. (27)
Therefore, (𝑔^𝑖𝑗) given by (26) is the inverse of the matrix (𝑔_𝑖𝑗). This also shows that (𝑔_𝑖𝑗) is nondegenerate.

Using (20) and (23) in (4), we get the angular metric tensor of an R-Randers mth-root space: ℎ_𝑖𝑗=𝐿𝜌_{−(𝑚−1)}𝑎_𝑖𝑗−𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗+𝜇₋₁𝑏_𝑖𝑗+𝜇₋₃𝑏_𝑖𝑏_𝑗. (28) Thus, we have the following.

Theorem 4. In an R-Randers mth-root space, the angular metric tensor ℎ_𝑖𝑗 is given by (28).

Differentiating 𝜌_{−(2𝑚−1)}=(𝑚−1)𝐹^{−(2𝑚−1)} partially with respect to 𝑥̇^𝑘, we get 𝜕̇_𝑘𝜌_{−(2𝑚−1)}=−(2𝑚−1)𝜌_{−(3𝑚−1)}𝑎_𝑘, (29) where 𝜌_{−(3𝑚−1)}=(𝑚−1)𝐹^{−(3𝑚−1)}.

Also, we have 𝜕̇_𝑘𝜇₋₃=3𝜇₋₅𝑏_𝑘, (30) where 𝜇₋₅=𝛼⁻⁵.

Thus, we have the following.

Proposition 5. In an R-Randers mth-root space, the following holds good: 𝜕̇_𝑘𝜌_{−(2𝑚−1)}=−(2𝑚−1)𝜌_{−(3𝑚−1)}𝑎_𝑘,𝜕̇_𝑘𝜇₋₃=3𝜇₋₅𝑏_𝑘. (31)

Differentiating (23) partially with respect to 𝑥̇^𝑘, we get 𝜕̇_𝑘𝑔_𝑖𝑗=2𝜌_{−(𝑚−1)}𝜕̇_𝑘𝜌_{−(𝑚−1)}𝑎_𝑖𝑎_𝑗+𝜌_{−(𝑚−1)}²𝑎_𝑗𝜕̇_𝑘𝑎_𝑖+𝑎_𝑖𝜕̇_𝑘𝑎_𝑗+𝜕̇_𝑘𝜌_{−(𝑚−1)}𝜇₋₁+𝜌_{−(𝑚−1)}𝜕̇_𝑘𝜇₋₁𝑏_𝑖𝑎_𝑗+𝑏_𝑗𝑎_𝑖+𝜌_{−(𝑚−1)}𝜇₋₁𝑎_𝑗𝜕̇_𝑘𝑏_𝑖+𝑏_𝑖𝜕̇_𝑘𝑎_𝑗+𝑎_𝑖𝜕̇_𝑘𝑏_𝑗+𝑏_𝑗𝜕̇_𝑘𝑎_𝑖+2𝜇₋₁𝜕̇_𝑘𝜇₋₁𝑏_𝑖𝑏_𝑗+𝜇²₋₁𝑏_𝑗𝜕̇_𝑘𝑏_𝑖+𝑏_𝑖𝜕̇_𝑘𝑏_𝑗+𝜕̇_𝑘𝐿𝜌_{−(𝑚−1)}𝑎_𝑖𝑗−𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗+𝜇₋₁𝑏_𝑖𝑗+𝜇₋₃𝑏_𝑖𝑏_𝑗+𝐿𝜕̇_𝑘𝜌_{−(𝑚−1)}𝑎_𝑖𝑗+𝜕̇_𝑘𝑎_𝑖𝑗𝜌_{−(𝑚−1)}−𝜕̇_𝑘𝜌_{−(2𝑚−1)}𝑎_𝑖𝑎_𝑗−𝜌_{−(2𝑚−1)}×𝜕̇_𝑘𝑎_𝑖𝑎_𝑗+𝜕̇_𝑘𝑎_𝑗𝑎_𝑖+𝜕̇_𝑘𝜇₋₁𝑏_𝑖𝑗+𝜕̇_𝑘𝑏_𝑖𝑗𝜇₋₁+𝜕̇_𝑘𝜇₋₃𝑏_𝑖𝑏_𝑗+𝜇₋₃𝜕̇_𝑘𝑏_𝑖𝑏_𝑗+𝜕̇_𝑘𝑏_𝑗𝑏_𝑖. (32)

Partial differentiation of 𝑎_𝑖 and 𝑏_𝑖, with respect to 𝑥̇^𝑘, yields 𝜕̇_𝑘𝑎_𝑖=(𝑚−1)𝑎_𝑖𝑘,𝜕̇_𝑘𝑏_𝑖=𝑏_𝑖𝑘. (33) Further, differentiation of 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 with respect to 𝑥̇^𝑘 gives 𝜕̇_𝑘𝑎_𝑖𝑗=(𝑚−2)𝑎_𝑖𝑗𝑘,𝜕̇_𝑘𝑏_𝑖𝑗=0, (34) where 𝑎_𝑖𝑗𝑘(𝑥,𝑦)=𝑎_{𝑖𝑗𝑘𝑖4⋯𝑖𝑚}(𝑥)𝑦^𝑖4⋯𝑦^𝑖𝑚.

Thus, we have the following.

Proposition 6. In an R-Randers mth-root space the following holds good: 𝜕̇_𝑘𝑎_𝑖=(𝑚−1)𝑎_𝑖𝑘,𝜕̇_𝑘𝑎_𝑖𝑗=(𝑚−2)𝑎_𝑖𝑗𝑘,𝜕̇_𝑘𝑏_𝑖=𝑏_𝑖𝑘,𝜕̇_𝑘𝑏_𝑖𝑗=0. (35)

If we use (2), (15), (18), (20), (29), (30), (33), and (34) in (32), on simplification it follows that 2𝐶_𝑖𝑗𝑘=𝜌_{−(2𝑚−2)}𝑎_𝑘𝑎_𝑖𝑗+𝜉_{−(2𝑚−2)}𝑎_𝑗𝑎_𝑖𝑘+𝑎_𝑖𝑎_𝑗𝑘+𝜇₋₂𝑏_𝑖𝑏_𝑗𝑘+𝑏_𝑗𝑏_𝑖𝑘+𝑏_𝑘𝑏_𝑖𝑗+𝜇₋₄𝑏_𝑖𝑏_𝑗𝑏_𝑘+𝜌_{−(3𝑚−2)}𝑎_𝑖𝑎_𝑗𝑎_𝑘+𝜌_{−(𝑚−2)}𝑎_𝑖𝑗𝑘+𝜌_{−(𝑚−1)}𝜇₋₁×1𝑎𝑚−1^𝑘𝑏_𝑖𝑗+𝑏_𝑘𝑎_𝑖𝑗+𝑎_𝑗𝑏_𝑖𝑘+(𝑚−1)𝑏_𝑖𝑎_𝑗𝑘+𝑏_𝑗𝑎_𝑖𝑘+𝑎_𝑖𝑏_𝑗𝑘−(𝑚−1)𝜌_{−(2𝑚−1)}𝜇₋₁𝑏_𝑖𝑎_𝑗+𝑏_𝑗𝑎_𝑖𝑎_𝑘+𝜌_{−(𝑚−1)}𝜇₋₃𝑏_𝑖𝑎_𝑗+𝑏_𝑗𝑎_𝑖𝑏_𝑘, (36) where 𝜌_{−(2𝑚−2)}=𝜌²_{−(𝑚−1)}()𝜌𝑚−1−𝐿𝑚−1_{−(2𝑚−1)},𝜉_{−(2𝑚−2)}=(𝑚−1)𝜌²_{−(𝑚−1)}−𝐿𝜌_{−(2𝑚−1)},𝜇₋₂=𝜇²₋₁+𝐿𝜇₋₃,𝜇₋₄=3𝐿𝜇₋₅+2𝜇₋₁𝜇₋₃,𝜌_{−(𝑚−2)}=𝐿(𝑚−2)𝜌_{−(𝑚−1)},𝜌_{−(3𝑚−2)}=(2𝑚−1)𝐿𝜌_{−(3𝑚−1)}−2(𝑚−1)𝜌_{−(𝑚−1)}𝜌_{−(2𝑚−1)}. (37) Thus, we have the following.

Theorem 7. In an R-Randers mth-root space, the Cartan tensor 𝐶_𝑖𝑗𝑘 is given by (36).

4. Spray and Equation of Geodesics

In this section, we discuss the spray of an R-Randers mth-root space and obtain its local coefficients. We also obtain the equation of geodesics in such space.

If we differentiate (9) partially with respect to 𝑥^𝑗, we get 𝜕_𝑗𝜌𝐹=^{−(𝑚−1)}𝑚()𝐴𝑚−1^𝑗, (38) where 𝐴_𝑗=𝜕_𝑗𝑎_{𝑖1⋯𝑖𝑚}𝑦^𝑖1⋯𝑦^𝑖𝑚. (39) Further, differentiating (13) partially with respect to 𝑥^𝑘 and utilizing 𝜕_𝑘𝜌_{−(𝑚−1)}=(−(𝑚−1)/𝑚)𝜌_{−(2𝑚−1)}𝐴_𝑘, we have 𝜕_𝑘𝜕̇_𝑖1𝐹=−𝑚𝜌_{−(2𝑚−1)}𝐴_𝑘𝑎_𝑖+𝜌_{−(𝑚−1)}()𝜕𝑚−1^𝑘𝑎_𝑖. (40) Differentiating (10) partially with respect to 𝑥^𝑗, we get 𝜕_𝑗1𝛼=2𝜇₋₁𝐵_𝑗, (41) where 𝐵_𝑗=𝜕_𝑗𝑏_𝑖1𝑖2𝑦^𝑖1𝑦^𝑖2. (42) Also, we have 𝜕_𝑘𝜇₋₁=12𝜇₋₃𝐵_𝑘. (43) Next, differentiating (16) partially with respect to 𝑥^𝑘 and using (43), we obtain 𝜕_𝑘𝜕̇_𝑖1𝛼=2𝜇₋₃𝑏_𝑖𝐵_𝑘+𝜇₋₁𝜕_𝑘𝑏_𝑖. (44) Thus, we have the following.

Proposition 8. In an R-Randers mth-root space, the following holds good: 𝜕_𝑗𝜌𝐹=^{−(𝑚−1)}𝑚()𝐴𝑚−1^𝑗,𝜕_𝑗1𝛼=2𝜇₋₁𝐵_𝑗,𝜕_𝑘𝜕̇_𝑖1𝐹=−𝑚𝜌_{−(2𝑚−1)}𝐴_𝑘𝑎_𝑖+𝜌_{−(𝑚−1)}()𝜕𝑚−1^𝑘𝑎_𝑖,𝜕_𝑘𝜕̇_𝑖1𝛼=2𝜇₋₃𝑏_𝑖𝐵_𝑘+𝜇₋₁𝜕_𝑘𝑏_𝑖. (45)

If we differentiate (8) partially with respect to 𝑥^𝑘 and use (38) and (41), it follows that 𝜕_𝑘𝜌𝐿=^{−(𝑚−1)}𝑚()𝐴𝑚−1^𝑘+12𝜇₋₁𝐵_𝑘. (46) Next, differentiating (20) partially with respect to 𝑥^𝑘 and using (40) and (44), we get 𝜕_𝑘𝜕̇_𝑖1𝐿=−𝑚𝜌_{−(2𝑚−1)}𝐴_𝑘𝑎_𝑖+𝜌_{−(𝑚−1)}()𝜕𝑚−1^𝑘𝑎_𝑖+12𝜇₋₃𝑏_𝑖𝐵_𝑘+𝜇₋₁𝜕_𝑘𝑏_𝑖. (47) In view of (46) and (47), (6) gives 𝐺^𝑖=14𝑔^𝑖𝑗2𝑚𝜌²_{−()𝑚−1}(𝑚−1)²−𝐿𝜌_{−(2𝑚−1)}𝑎_𝑗𝐴_𝑘𝑦^𝑘+𝜇²₋₁+𝐿𝜇₋₃𝑏_𝑗𝐵_𝑘𝑦^𝑘+𝜌_{−(𝑚−1)}()𝜇𝑚−1⁻¹2𝑏_𝑗𝐴_𝑘+𝑎_𝐽𝐵_𝑘𝑦^𝑘𝜌+2𝐿^{−(𝑚−1)}()𝜕𝑚−1^𝑘𝑎_𝑗𝑦^𝑘+2𝐿𝜇₋₁𝜕_𝑘𝑏_𝑗𝑦^𝑘𝜌−2𝐿^{−(𝑚−1)}()𝐴𝑚−1^𝑗−𝐿𝜇₋₁𝐵_𝑗, (48) that is 𝐺^𝑖=14𝑔^𝑖𝑗2𝑚()𝜌𝑚−1^{−(2𝑚−2)}𝑎_𝑗𝐴₀+𝜇₋₂𝑏_𝑗𝐵₀+1𝜌𝑚−1^{−(𝑚−1)}𝜇₋₁2𝑏_𝑗𝐴₀+𝑎_𝑗𝐵₀+2()𝜌𝑚−1)(𝑚−2^{−(𝑚−2)}𝜕_𝑘𝑎_𝑗𝑦^𝑘−𝐴_𝑗+𝜂₋₂2𝜕_𝑘𝑏_𝑗𝑦^𝑘−𝐵_𝑗, (49) where 𝐴₀=𝐴_𝑘𝑦^𝑘,𝐵₀=𝐵_𝑘𝑦^𝑘,𝜂₋₂=𝐿𝜇₋₁, (50) and 𝑔^𝑖𝑗 is given by (26). Thus, we have the following.

Theorem 9. In an R-Randers mth-root space, the spray coefficients are given by (49).

In view of (5) and Theorem 9, we have the following.

Corollary 10. In an R-Randers mth-root space, the equation of geodesics is given by 𝑑²𝑥^𝑖𝑑𝑡²+2𝐺^𝑖=0, (51) where the spray coefficients 𝐺^𝑖 are given by (49).

5. Nonlinear Connection

In this section, we obtain the coefficients of nonlinear connection of the space under consideration.

Differentiating (49) partially with respect to 𝑥̇^𝑘, we have 𝜕̇_𝑘𝐺^𝑖=14𝜕̇_𝑘𝑔^𝑖𝑗2𝑚()𝜌𝑚−1^{−(2𝑚−2)}𝑎_𝑗𝐴₀+𝜇₋₂𝑏_𝑗𝐵₀+1𝜌𝑚−1^{−(𝑚−1)}𝜇₋₁2𝑏_𝑗𝐴₀+𝑎_𝑗𝐵₀+2()𝜌𝑚−1)(𝑚−2^{−(𝑚−2)}𝜕_𝑘𝑎_𝑗𝑦^𝑗−𝐴_𝑗+𝜂₋₂2𝜕_𝑘𝑏_𝑗𝑦^𝑘−𝐵_𝑗+14𝑔^𝑖𝑗2𝑚()𝜕̇𝑚−1^𝑘𝜌_{−(2𝑚−2)}𝑎_𝑗𝐴₀+𝜌_{−(2𝑚−2)}𝐴₀𝜕̇_𝑘𝑎_𝑗+𝑎_𝑗𝜕̇_𝑘𝐴₀+𝜕̇_𝑘𝜇₋₂𝑏_𝑗𝐵₀+𝜇₋₂𝑏_𝑗𝜕̇_𝑘𝐵₀+𝐵₀𝜕̇_𝑘𝑏_𝑗+1̇𝜕𝑚−1^𝑘𝜌_{−(𝑚−1)}𝜇₋₁+𝜌_{−(𝑚−1)}𝜕̇_𝑘𝜇₋₁2𝑏_𝑗𝐴₀+𝑎_𝑗𝐵₀+1𝜌𝑚−1^{−(𝑚−1)}𝜇₋₁×2𝜕̇_𝑘𝑏_𝑗𝐴₀+2𝑏_𝑗𝜕̇_𝑘𝐴₀+𝜕̇_𝑘𝑎_𝑗𝐵₀+𝑎_𝑗𝜕̇_𝑘𝐵₀+2()×𝜕̇𝑚−1)(𝑚−2_𝑘𝜌_{−(𝑚−2)}𝜕_𝑙𝑎_𝑗𝑦^𝑙−𝐴_𝑗+𝜌_{−(𝑚−2)}×𝜕̇_𝑘𝜕_𝑙𝑎_𝑗𝑦^𝑙+𝜕_𝑙𝑎_𝑗𝜕̇_𝑘𝑦^𝑙−𝜕̇_𝑘𝐴_𝑗+𝜕̇_𝑘𝜂₋₂2𝜕_𝑘𝑏_𝑗𝑦^𝑘−𝐵_𝑗+𝜂₋₂×2𝜕̇_𝑘𝜕_𝑘𝑏_𝑗𝑦^𝑘+2𝜕_𝑘𝑏_𝑗𝜕̇_𝑘𝑦^𝑘−𝜕̇_𝑘𝐵_𝑗. (52) Differentiation of 𝑔^𝑖𝑗𝑔_𝑗𝑚=𝛿^𝑖_𝑚 partially with respect to 𝑥̇^𝑘 yields 𝜕̇_𝑘𝑔^𝑖𝑙=−2𝐶^𝑖_𝑘𝑚𝑔^𝑚𝑙, (53) where 𝐶^𝑖_𝑘𝑚=𝐶_𝑗𝑘𝑚𝑔^𝑖𝑗.

Next, differentiating 𝜌_{−(2𝑚−2)}=(𝜌²_{−(𝑚−1)}/(𝑚−1))−𝐿(𝑚−1)𝜌_{−(2𝑚−1)} partially with respect to 𝑥̇^𝑘 and using (15), (20), and (29), we get 𝜕̇_𝑘𝜌_{−(2𝑚−2)}=(𝑚−1)−3𝜌_{−(3𝑚−2)}𝑎_𝑘−𝜌_{−(2𝑚−1)}𝜇₋₁𝑏_𝑘+(2𝑚−1)𝐿𝜌_{−(3𝑚−1)}𝑎_𝑘, (54) where 𝜌_{−(3𝑚−2)}=(𝑚−1)𝐹^{−(3𝑚−2)}.

Differentiating 𝐴₀=𝐴_𝑙𝑦^𝑙 and 𝐵₀=𝐵_𝑙𝑦^𝑙 partially with respect to 𝑥̇^𝑘, we, respectively, have 𝜕̇_𝑘𝐴₀=𝑚𝜕_𝑙𝑎_𝑘𝑦^𝑙+𝐴_𝑘,𝜕̇_𝑘𝐵₀=2𝜕_𝑙𝑏_𝑘𝑦^𝑙+𝐵_𝑘. (55) Also, we have 𝜕̇_𝑘𝜕_𝑙𝑎_𝑗=(𝑚−1)𝜕_𝑙𝑎_𝑗𝑘,𝜕̇_𝑘𝜕_𝑙𝑏_𝑗=𝜕_𝑙𝑏_𝑗𝑘,𝜕̇_𝑘𝐴_𝑗=𝑚𝜕_𝑗𝑎_𝑘,𝜕̇_𝑘𝐵_𝑗=2𝜕_𝑗𝑏_𝑘. (56) Using (7), (15), (18), (29), (30), (33), and (53)–(56) in (52), we have 𝑁^𝑖_𝑘=−2𝐶^𝑖_𝑘𝑚𝐺^𝑚+14𝑔^𝑖𝑗2𝑚−3𝜌_{−(3𝑚−2)}−(2𝑚−1)𝜌_{−(3𝑚−1)}𝑎_𝑘𝑎_𝑗𝐴₀−𝜌_{−(2𝑚−1)}𝜇₋₁𝑏_𝑘𝑎_𝑗𝐴₀+2𝑎_𝑘𝑏_𝑗𝐴₀+𝑎_𝑘𝑎_𝑗𝐵₀+2𝜌_{−(2𝑚−2)}𝑚()𝑚𝜕𝑚−1^𝑙𝑎_𝑘𝑦^𝑙𝑎_𝑗+𝐴_𝑘𝑎_𝑗+(𝑚−1)𝐴_𝑗𝑘𝐴₀+𝜕_𝑙𝑎_𝑗𝑦^𝑙𝑎_𝑘−𝐴_𝑗𝑎_𝑘+3𝜇₋₄−𝜇₋₁𝜇₋₃𝑏_𝑘𝑏_𝑗𝐵₀+1()𝜌𝑚−1^{−(𝑚−1)}𝜇₋₃𝑎_𝑘𝑏_𝑗𝐵₀+2𝑏_𝑗𝐴₀+𝑎_𝑗𝐵₀𝑏_𝑘+𝜇₋₂𝑏_𝑗𝑘𝐵₀+2𝜕_𝑙𝑏_𝑘𝑦^𝑙𝑏_𝑗+𝐵_𝑘𝑏_𝑗+𝑏_𝑘2𝜕_𝑙𝑏_𝑗𝑦^𝑙−𝐵_𝑗+𝜌_{−(𝑚−1)}𝜇₋₁(𝑚−1)2𝑏_𝑗𝑘𝐴₀+2𝑚𝑏_𝑗𝜕_𝑙𝑎_𝑘𝑦^𝑙+2𝑏_𝑗𝐴_𝑘+(𝑚−1)𝑎_𝑗𝑘𝐵₀+2𝑎_𝑗𝜕_𝑙𝑏_𝑘𝑦^𝑙+𝐵_𝑘𝑎_𝑗+2𝜕_𝑙𝑏_𝑗𝑦^𝑙𝑎_𝑘−𝐵_𝑗𝑎_𝑘+2𝜕_𝑙𝑎_𝑗𝑦^𝑙𝑏_𝑘−2𝐴_𝑗𝑏_𝑘+2𝜌_{−(𝑚−2)}())𝜕𝑚−1)(𝑚−2(𝑚−1_𝑙𝑎_𝑗𝑘𝑦^𝑙+𝜕_𝑘𝑎_𝑗−𝑚𝜕_𝑗𝑎_𝑘+𝜂₋₂2𝜕_𝑙𝑏_𝑗𝑘𝑦^𝑙+2𝜕_𝑘𝑏_𝑗−𝜕_𝑗𝑏_𝑘. (57)

Thus, we have the following.

Theorem 11. The local coefficients of the nonlinear connection of an R-Randers mth-root space are given by (57).

Acknowledgments

The authors are thankful to the reviewers for their valuable comments and suggestions. S. Saxena gratefully acknowledges the financial support provided by the UGC, Government of India.