Abstract

We initiate a study on the geometry of an almost 𝜑-Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and - and 𝑣-deflection tensors in an APL-space. Corresponding expressions in a 𝜑-Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.

1. Introduction

In the last three decades, various meaningful generalizations of Finsler spaces have been considered. These generalizations have been found much applicable to mechanics, theoretical physics, variational calculus, optimal control, complex analysis, biology, ecology, and so forth. The geometry of Lagrange spaces is one such generalization of the geometry of Finsler spaces which was introduced and studied by Miron [1, 2]. He [1, 2] introduced the most natural generalization of Lagrange spaces named as generalized Lagrange space. Since the introduction of Lagrange spaces and generalized Lagrange spaces, many geometers and physicists have been engaged in the exploration, development, and application of these concepts [313]. Antonelli and Hrimiuc [14, 15] introduced a special type of regular Lagrangian called 𝜑-Lagrangian. Applications of such Lagrangian have been discussed by Antonelli et al. in the monograph [16]. In the present paper, we generalize the notion of 𝜑-Lagrangian and introduce the concept of almost 𝜑-Lagrange spaces. We hope that the results obtained in the paper will be interesting for the researchers working on the application of Lagrange spaces in various fields of science.

Let 𝐹𝑛=(𝑀,𝐹(𝑥,𝑦)) be an 𝑛-dimensional Finsler space, and let 𝜑+ be a smooth function. The composition 𝐿=𝜑(𝐹2) defines a differentiable Lagrangian. This was regarded by Antonelli and Hrimiuc [14, 15] as 𝜑-Lagrangian associated to the Finsler space 𝐹𝑛. They [14] proved that if the function 𝜑 has the following properties:(a)𝜑((𝑡)0,b)𝜑(𝑡)+𝜑(𝑡)0,forevery𝐹𝑡Im2,(1.1) then 𝐿 is a regular Lagrangian and thus 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) is a Lagrange space, called a 𝜑-Lagrange space.

In this paper, we consider a more general Lagrangian as follows:𝐹𝐿(𝑥,𝑦)=𝜑2+𝐴𝑖(𝑥)𝑦𝑖+𝑈(𝑥),(1.2) where 𝜑 is the same as discussed earlier, 𝐴𝑖(𝑥) is a covector, and 𝑈(𝑥) is a smooth function.

In Section 2, we show that if the function 𝜑 has the properties (1.1), then 𝐿(𝑥,𝑦) is a regular Lagrangian and thus the pair 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) is a Lagrange space. We call this space as an almost 𝜑-Lagrange space (shortly APL-space).

An APL-space reduces to a 𝜑-Lagrange space if and only if 𝐴𝑖(𝑥)=0 and 𝑈(𝑥)=0.

If 𝜑(𝑡)=𝑡,forall𝑡Im(𝐹2), then the Lagrangian in (1.2) takes the form𝐿(𝑥,𝑦)=𝐹2+𝐴𝑖(𝑥)𝑦𝑖+𝑈(𝑥).(1.3) This defines a regular Lagrangian, and the pair 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) is called an almost Finsler Lagrange space (shortly AFL-space). Such Lagrange space was introduced by Miron and Anastasiei (vide Chapter IX of [17]).

We take𝑔𝑖𝑗=12̇𝜕𝑖̇𝜕𝑗𝐹2,𝑎𝑖𝑗=12̇𝜕𝑖̇𝜕𝑗̇𝜕𝐿,𝑖𝜕𝜕𝑦𝑖.(1.4) Henceforth, we will indicate all the geometrical objects related to 𝐹𝑛 by a small circle “” put over them.

In a Finsler space, the geodesics, parameterized by arc length (the extremals of the length integral), coincide with the extremals of action integral or with the autoparallel curves of the Cartan nonlinear connection [16]:𝑁𝑖𝑗=𝛾𝑖𝑗0𝐶𝑖𝑗𝑘𝛾𝑘00,(1.5) where𝛾𝑖𝑗𝑘=12𝑔𝑖𝜕𝑗𝑔𝑘+𝜕𝑘𝑔𝑗𝜕𝑔𝑗𝑘;𝜕𝑗𝜕𝜕𝑥𝑗,𝐶𝑖𝑗𝑘=12𝑔𝑖̇𝜕𝑔𝑗𝑘,𝛾𝑖𝑗0=𝛾𝑖𝑗𝑘𝑦𝑘,𝛾𝑖00=𝛾𝑖𝑗𝑘𝑦𝑗𝑦𝑘.(1.6) These geodesics are the integral curves of the spray [16] (i.e., (2) p-homogeneous):𝐺𝑖=14𝑔𝑖𝑗𝑦𝑘̇𝜕𝑗𝜕𝑘𝐹2𝜕𝑗𝐹2,(1.7) that is, solutions of the differential equations𝑑2𝑥𝑖𝑑𝑠2+2𝐺𝑖𝑥(𝑠),𝑑𝑥𝑑𝑠=0.(1.8) We have the following equalities:(a)𝐺𝑖=12𝛾𝑖00,(b)𝑁𝑖𝑗=̇𝜕𝑗𝐺𝑖.(1.9) In a general Lagrange space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)), the geodesics are the extremals of the action integral and coincide with the integral curves of the semispray [17, 18] (i.e., may not be a spray):𝐺𝑖=14𝑎𝑖𝑗𝑦𝑘̇𝜕𝑗𝜕𝑘𝐿𝜕𝑗𝐿.(1.10) As in a Finsler space, a remarkable nonlinear connection can be considered in a Lagrange space:𝑁𝑖𝑗=̇𝜕𝑗𝐺𝑖.(1.11) Such nonlinear connection is a canonical nonlinear connection [17, 18] as it depends only on the fundamental function 𝐿(𝑥,𝑦) of the Lagrange space.

In general, the autoparallel curves of (𝑁𝑖𝑗) are different from the geodesics of 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) (cf. [17]).

Given a nonlinear connection (𝑁𝑖𝑗) on a Lagrange space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)), there is a unique - and 𝑣-metrical 𝑑-connection (cf. [17, 19]) 𝐶Γ(𝑁)=(𝑁𝑖𝑗,𝐿𝑖𝑗𝑘,𝐶𝑖𝑗𝑘) with torsions 𝑇𝑖𝑗𝑘=0 and 𝑆𝑖𝑗𝑘=0, called the canonical metrical 𝑑-connection. This connection is linear and its coefficients are given by𝐿𝑖𝑗𝑘=12𝑎𝑖𝛿𝑗𝑎𝑘+𝛿𝑘𝑎𝑗𝛿𝑎𝑗𝑘,(1.12)𝐶𝑖𝑗𝑘=12𝑎𝑖̇𝜕𝑗𝑎𝑘+̇𝜕𝑘𝑎𝑗̇𝜕𝑎𝑗𝑘,(1.13) where 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟 is the Lagrange differentiation operator.

If 𝐶Γ(𝑁)=(𝑁𝑖𝑗,𝐿𝑖𝑗𝑘,𝐶𝑖𝑗𝑘) is the Cartan connection of the Finsler space 𝐹𝑛=(𝑀,𝐹(𝑥,𝑦)), then its coefficients are given by𝐿𝑖𝑗𝑘=12𝑔𝑖𝛿𝑗𝑔𝑘+𝛿𝑘𝑔𝑗𝛿𝑔𝑗𝑘,(1.14)𝐶𝑖𝑗𝑘=12𝑔𝑖̇𝜕𝑗𝑔𝑘+̇𝜕𝑘𝑔𝑗̇𝜕𝑔𝑗𝑘,(1.15) where 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟.

The - and 𝑣-deflection tensor fields 𝐷𝑖𝑗 and 𝑑𝑖𝑗, respectively, of a Lagrange space 𝐿𝑛 are defined by (cf. [19])𝐷𝑖𝑗=𝑦𝑖|𝑗=𝑦𝑠𝐿𝑖𝑠𝑗𝑁𝑖𝑗,(1.16)𝑑𝑖𝑗=𝑦𝑖|𝑗=𝛿𝑖𝑗+𝑦𝑠𝐶𝑖𝑠𝑗,(1.17) where | and |, respectively, denote the - and 𝑣-covariant derivatives with respect to 𝐶Γ.

If 𝐷𝑖𝑗 is the h-deflection tensor field and 𝑑𝑖𝑗 is the 𝑣-deflection tensor field of the Finsler space 𝐹𝑛, then𝐷𝑖𝑗=𝑦𝑖|𝑗=𝑦𝑠𝐿𝑖𝑠𝑗𝑁𝑖𝑗=0,(1.18)𝑑𝑖𝑗=𝑦𝑖|||𝑗=𝛿𝑖𝑗,(1.19) where | and |, respectively, denote the - and 𝑣-covariant derivatives with respect to 𝐶Γ.

For basic terminology and notations related to a Finsler space and a Lagrange space, we refer to the books [17, 20].

2. Almost 𝜑-Lagrange Spaces

As discussed earlier, we consider the Lagrangian given by (1.2) in which the function 𝜑 satisfies (1.1). We prove that it is a regular Lagrangian and the pair 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) is a Lagrange space which we term as an almost 𝜑-Lagrange space (APL-space in short).

Theorem 2.1. If the function 𝜑 satisfies the conditions (1.1), then 𝐿(𝑥,𝑦), given by (1.2), is a regular Lagrangian and 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) is a Lagrange space.

Proof. Differentiating (1.2) partially with respect to 𝑦𝑖, we get ̇𝜕𝑖𝐿=𝜑𝐹2̇𝜕𝑖𝐹2+𝐴𝑖(𝑥).(2.1) Again differentiating (2.1) partially with respect to 𝑦𝑗, we obtain ̇𝜕𝑗̇𝜕𝑖𝐿=𝜑𝐹2̇𝜕𝑖𝐹2̇𝜕𝑗𝐹2+𝜑𝐹2̇𝜕𝑖̇𝜕𝑗𝐹2,(2.2) which, in view of (1.4), provides 𝑎𝑖𝑗=2𝐹2𝜑𝐹2̇𝜕𝑖𝐹̇𝜕𝑗𝐹+𝜑𝐹2𝑔𝑖𝑗.(2.3) Now 𝐹̇𝜕𝑖1𝐹=2̇𝜕𝑖𝐹2=12̇𝜕𝑖𝑔𝑗𝑘𝑦𝑗𝑦𝑘=𝑔𝑖𝑘𝑦𝑘=𝑦𝑖.(2.4) In view of (2.4), (2.3) takes the form 𝑎𝑖𝑗=𝜑𝑔𝑖𝑗+2𝜑𝜑𝑦𝑖𝑦𝑗.(2.5) Under the hypothesis, the matrix (𝑎𝑖𝑗) is invertible and its inverse is (see Lemma 6.2.2.1, page 891 in [20]) 𝑎𝑖𝑗=1𝑔𝜑𝑖𝑗2𝜑𝜑+2𝐹2𝜑𝑦𝑖𝑦𝑗.(2.6) This proves the theorem.

Remarks 1. (i) If 𝐴𝑖(𝑥)=0 and 𝑈(𝑥)=0 in (1.2), then expression (2.5) remains unchanged. Hence, the symmetric metric tensor of a 𝜑-Lagrange space is the same as that of an APL-space.
(ii) If 𝜑(𝐹2)=𝐹2, then 𝜑=1 and 𝜑=0. Hence, the symmetric metric tensor of an AFL-space coincides with that of the associated Finsler space.

3. Semispray, Integral Curves of Euler-Lagrange Equations

In this section, we obtain the coefficients of the canonical semispray of the APL-space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) and deduce corresponding expressions for a 𝜑-Lagrange space and an AFL-space. Next, we obtain the differential equations whose solution curves are the integral curves of Euler-Lagrange equations in an APL-space. We deduce corresponding differential equations for a 𝜑-Lagrange space and an AFL-space.

If we differentiate (1.2) partially with respect to 𝑥𝑘, we have𝜕𝑘𝐿=𝜑𝐹2𝜕𝑘𝐹2+𝑦𝑖𝜕𝑘𝐴𝑖(𝑥)+𝜕𝑘𝑈(𝑥).(3.1) Differentiating (3.1) partially with respect to 𝑦𝑗, we obtaiṅ𝜕𝑗𝜕𝑘𝐿=2𝜑𝐹2𝐹̇𝜕𝑗𝐹𝜕𝑘𝐹2+𝜑𝐹2̇𝜕𝑗𝜕𝑘𝐹2+𝜕𝑘𝐴𝑗(𝑥),(3.2) which, in view of (2.4), takes the forṁ𝜕𝑗𝜕𝑘𝐿=2𝜑𝐹2𝑦𝑗𝜕𝑘𝐹2+𝜑𝐹2̇𝜕𝑗𝜕𝑘𝐹2+𝜕𝑘𝐴𝑗(𝑥).(3.3) Using (3.1) and (3.3) in (1.10), we have𝐺𝑖=14𝑎𝑖𝑗2𝜑𝐹2𝑦𝑗𝑦𝑘𝜕𝑘𝐹2+𝜑𝐹2𝑦𝑘̇𝜕𝑗𝜕𝑘𝐹2𝜕𝑗𝐹22𝑦𝑘𝐹𝑗𝑘𝜕𝑗𝑈,(3.4) where𝐹𝑗𝑘1(𝑥)=2𝜕𝑗𝐴𝑘𝜕𝑘𝐴𝑗(3.5) is electromagnetic tensor field of the potentials 𝐴𝑖(𝑥).

Applying (2.6) in (3.4) and using 𝑦𝑖𝑦𝑖=𝐹2,𝑔𝑖𝑗𝑦𝑗=𝑦𝑖, and 𝑦𝑗̇𝜕𝑗𝜕𝑘𝐹2=2𝜕𝑘𝐹2 (by Euler’s theorem on homogeneous functions), we obtain𝐺𝑖=12𝜑𝜑12𝜑𝐹2𝜑+2𝐹2𝜑𝑦𝑖𝑦𝑘𝜕𝑘𝐹2+14𝑔𝑖𝑗𝑦𝑘̇𝜕𝑗𝜕𝑘𝐹2𝜕𝑗𝐹22𝜑𝐹2𝜑+2𝐹2𝜑𝑦𝑖𝑦𝑘𝜕𝑘𝐹214𝑎𝑖𝑗2𝐹𝑗𝑘𝑦𝑘+𝜕𝑗𝑈.(3.6) Using (1.7) in (3.6) and simplifying, we get𝐺𝑖=𝐺𝑖14𝑎𝑖𝑗2𝐹𝑗𝑘𝑦𝑘+𝜕𝑗𝑈.(3.7) Thus, we have the following.

Theorem 3.1. The canonical semispray of an APL-space has the local coefficients given by 𝐺𝑖=𝐺𝑖14𝑎𝑖𝑗2𝐹𝑗𝑘𝑦𝑘+𝜕𝑗𝑈,(3.8) where 𝐺𝑖 are the local coefficients of the spray of 𝐹𝑛.

For a 𝜑-Lagrange space, 𝐴𝑖(𝑥)=0 and 𝑈(𝑥)=0. Hence, from (3.5), we have 𝐹𝑗𝑘=0. Therefore, (3.7) reduces to𝐺𝑖=𝐺𝑖.(3.9) Thus, we may state the following.

Corollary 3.2 (see [14]). The canonical semispray of a 𝜑-Lagrange space becomes a spray and coincides with that of the associated Finsler space.

For an AFL-space, 𝑎𝑖𝑗=𝑔𝑖𝑗 (see Remark (ii)). Hence, (3.7) takes the form𝐺𝑖=𝐺𝑖14𝑔𝑖𝑗2𝐹𝑗𝑘𝑦𝑘+𝜕𝑗𝑈.(3.10) Thus, we have the following.

Corollary 3.3 (see [17, 20]). The canonical semispray of an AFL-space has the local coefficients given by (3.10).

In a Lagrange space, the integral curves of the Euler-Lagrange equations:𝐸𝑖(𝐿)=𝜕𝑖𝑑𝐿̇𝜕𝑑𝑡𝑖𝐿=0(3.11) are the solution curves of the equations [20]𝑑2𝑥𝑖𝑑𝑡2+2𝐺𝑖(𝑥,𝑦)=0.(3.12) Using (3.7) in (3.12), we obtain𝑑2𝑥𝑖𝑑𝑡2+2𝐺𝑖=122𝐹𝑖𝑘𝑦𝑘+𝑎𝑖𝑗𝜕𝑗𝑈,(3.13) where 𝐹𝑖𝑘=𝑎𝑖𝑗𝐹𝑗𝑘.

Using (1.9) (a) in (3.13), we have𝑑2𝑥𝑖𝑑𝑡2+𝛾𝑖00=122𝐹𝑖𝑘𝑦𝑘+𝑎𝑖𝑗𝜕𝑗𝑈.(3.14) Thus, we have the following.

Theorem 3.4. In an APL-space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)), the integral curves of the Euler-Lagrange equations 𝐸𝑖(𝐿)=0 are the solution curves of (3.14).

For a 𝜑-Lagrange space, equations (3.14) take the following simple form:𝑑2𝑥𝑖𝑑𝑡2+𝛾𝑖00=0.(3.15) This enables us to state the following.

Corollary 3.5 (see [14]). In a 𝜑-Lagrange space, the integral curves of the Euler-Lagrange equations are the solution curves of (3.15).

For an AFL-space, 𝑎𝑖𝑗=𝑔𝑖𝑗. Therefore, equations (3.14) become𝑑2𝑥𝑖𝑑𝑡2+𝛾𝑖00=𝐹𝑖𝑘𝑦𝑘+12𝑔𝑖𝑗𝜕𝑗𝑈,(3.16) where 𝐹𝑖𝑘=𝑔𝑖𝑗𝐹𝑗𝑘.

Thus, we have the following.

Corollary 3.6 (see [17, 20]). In an AFL-space, the integral curves of the Euler-Lagrange equations 𝐸𝑖(𝐿)=0 are the solution curves of (3.16).

4. Nonlinear Connection, Autoparallel Curves

In this section, we find the coefficients of the nonlinear connection of an APL-space and obtain the differential equations of the autoparallel curves of the nonlinear connection. Corresponding results have been deduced for a 𝜑-Lagrange space and an AFL-space.

Partial differentiation of (2.5) with respect to 𝑦𝑘 yieldṡ𝜕𝑘𝑎𝑖𝑗=2𝐶𝑖𝑗𝑘=2𝜑𝐶𝑖𝑗𝑘+2𝜑𝑔𝑖𝑗𝑦𝑘+𝑔𝑗𝑘𝑦𝑖+𝑔𝑘𝑖𝑦𝑗+4𝜑𝑦𝑖𝑦𝑗𝑦𝑘.(4.1) Using (3.7) in (1.11) and taking (1.9) (b), (2.6), (4.1), 𝐶𝑝𝑞𝑗𝑦𝑗=0, 𝑦𝑖𝑦𝑖=𝐹2, and 𝑔𝑖𝑗𝑦𝑗=𝑦𝑖 into account, we obtain𝑁𝑖𝑗=𝑁𝑖𝑗12𝐹𝑖𝑗+12𝜑𝐶𝑖𝑞𝑗𝑔𝑞𝑟+12𝜑𝜑2𝑔𝑖𝑟𝑦𝑗+𝜑2𝜑𝜑+2𝐹2𝜑𝛿𝑟𝑗𝑦𝑖+𝛿𝑖𝑗𝑦𝑟+𝜑2𝜑2𝜑3𝐹24𝜑𝜑22𝜑2𝜑+2𝐹2𝜑2𝑦𝑖𝑦𝑗𝑦𝑟2𝐹𝑟𝑘𝑦𝑘+𝜕𝑟𝑈.(4.2) If we take𝑆𝑗𝑖𝑟=12𝜑𝐶𝑖𝑞𝑗𝑔𝑞𝑟+12𝜑𝜑2𝑔𝑖𝑟𝑦𝑗+𝜑𝛿𝑟𝑗𝑦𝑖+𝛿𝑖𝑗𝑦𝑟2𝜑𝜑+2𝐹2𝜑+𝜑2𝜑2𝜑3𝐹24𝜑𝜑22𝜑2𝜑+2𝐹2𝜑2𝑦𝑖𝑦𝑗𝑦𝑟,(4.3) the last expression becomes𝑁𝑖𝑗=𝑁𝑖𝑗12𝐹𝑖𝑗+𝑆𝑗𝑖𝑟2𝐹𝑟𝑘𝑦𝑘+𝜕𝑟𝑈,(4.4) that is,𝑁𝑖𝑗=𝑁𝑖𝑗𝑉𝑖𝑗,(4.5) where𝑉𝑖𝑗=12𝐹𝑖𝑗𝑆𝑗𝑖𝑟2𝐹𝑟𝑘𝑦𝑘+𝜕𝑟𝑈.(4.6) Thus, we have the following.

Theorem 4.1. The canonical nonlinear connection of an APL-space 𝐿𝑛 has the local coefficients given by (4.5).

For a 𝜑-Lagrange space, we have 𝐹𝑟𝑘=0,𝐹𝑖𝑗=0 and 𝑈=0 and hence 𝑉𝑖𝑗=0. Therefore, (4.5) reduces to𝑁𝑖𝑗=𝑁𝑖𝑗.(4.7) Thus, we have the following.

Corollary 4.2 (see [14]). The canonical nonlinear connection of a 𝜑-Lagrange space coincides with the nonlinear connection of the associated Finsler space.

For an AFL-space, (4.3) reduces to𝑆𝑗𝑖𝑟=12𝐶𝑖𝑞𝑗𝑔𝑞𝑟(4.8) and hence (4.6) gives𝑉𝑖𝑗=12𝐹𝑖𝑗𝐶𝑖𝑞𝑗𝐹𝑞𝑘𝑦𝑘12𝐶𝑖𝑞𝑗𝑔𝑞𝑟𝜕𝑟𝑈=𝐵𝑖𝑗.(4.9) Therefore, (4.5) takes the form𝑁𝑖𝑗=𝑁𝑖𝑗𝐵𝑖𝑗.(4.10) Thus, we have the following.

Corollary 4.3 (see [17, 20]). The canonical nonlinear connection of an AFL-space 𝐿𝑛 has the local coefficients given by (4.10).

Transvecting (4.5) by 𝑦𝑖 and using 𝑁𝑖𝑗𝑦𝑗=𝛾𝑖00, we obtain𝑁𝑖𝑗𝑦𝑗=𝛾𝑖00𝑉𝑖0,(4.11) where 𝑉𝑖0=𝑉𝑖𝑗𝑦𝑗.

The autoparallel curves of the canonical nonlinear connection 𝑁=(𝑁𝑖𝑗) of a Lagrange space are given by the following system of differential equations (vide [20]):𝑑2𝑥𝑖𝑑𝑡2+𝑁𝑖𝑗(𝑥,𝑦)𝑦𝑗=0.(4.12) Equations (4.12), in view of (4.11), take the form𝑑2𝑥𝑖𝑑𝑡2+𝛾𝑖00=𝑉𝑖0.(4.13) Thus, we have the following.

Theorem 4.4. The autoparallel curves of the canonical nonlinear connection 𝑁=(𝑁𝑖𝑗) of an APL-space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) are given by the system of differential equations (4.13).

For a 𝜑-Lagrange space, 𝑉𝑖𝑗=0 and hence 𝑉𝑖0=0. Therefore, (4.13) reduces to𝑑2𝑥𝑖𝑑𝑡2+𝛾𝑖00=0.(4.14) Thus, we have the following.

Corollary 4.5 (see [14]). The autoparallel curves of the canonical nonlinear connection of a 𝜑-Lagrange space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) are given by the system of differential equations (4.14).

For an AFL-space,𝑉𝑖𝑗=𝐵𝑖𝑗1=2𝐹𝑖𝑗𝐶𝑖𝑞𝑗𝐹𝑞𝑘𝑦𝑘12𝐶𝑖𝑞𝑗𝑔𝑞𝑟𝜕𝑟𝑈(4.15) and hence, by virtue of 𝐶𝑖𝑞𝑗𝑦𝑗=0, we have 𝑉𝑖0=(1/2)𝐹𝑖𝑗𝑦𝑗. Therefore, equations (4.12) take the form𝑑2𝑥𝑖𝑑𝑡2+𝛾𝑖00=12𝐹𝑖𝑗𝑦𝑗.(4.16) Thus, we deduce the following.

Corollary 4.6 (see [17, 20]). The autoparallel curves of the nonlinear connection 𝑁=(𝑁𝑖𝑗) of an AFL-space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)) are given by the system of differential equations (4.16).

If we compare (3.14), (3.15), and (3.16), respectively, with (4.13), (4.14), and (4.16), we observe that, in an APL-space as well as in an AFL-space, solution curves of Euler-Lagrange equations do not coincide with the autoparallel curves of the canonical nonlinear connection whereas in a 𝜑-Lagrange space they do. Therefore, in a 𝜑-Lagrange space, geodesics are autoparallel curves whereas in an APL-space and in an AFL-space they are not so.

5. Canonical Metrical 𝑑-Connection

Let 𝐶Γ(𝑁)=(𝑁𝑖𝑗,𝐿𝑖𝑗𝑘,𝐶𝑖𝑗𝑘) be the canonical metrical 𝑑-connection of the APL-space 𝐿𝑛=(𝑀,𝐿(𝑥,𝑦)), and let 𝐶Γ(𝑁)=(𝑁𝑖𝑗,𝐿𝑖𝑗𝑘,𝐶𝑖𝑗𝑘) be the Cartan connection of the associated Finsler space 𝐹𝑛=(𝑀,𝐹(𝑥,𝑦)). In this section, we obtain the expressions for the coefficients of 𝐶Γ(𝑁) and we investigate some properties of 𝐶Γ(𝑁). We deduce corresponding results for a 𝜑-Lagrange space and an AFL-space.

Using (4.1) in (1.13) and taking (1.15) into account, we find𝐶𝑖𝑗𝑘=𝐶𝑖𝑗𝑘+𝜑𝛿𝜑𝑖𝑗𝑦𝑘+𝛿𝑖𝑘𝑦𝑗+𝜑𝜑+2𝐹2𝜑𝑔𝑗𝑘𝑦𝑖+2𝜑𝜑2𝜑2𝜑𝜑+2𝐹2𝜑𝑦𝑖𝑦𝑗𝑦𝑘.(5.1)

For any 𝐶-class function 𝜓+, taking 𝑓(𝑥,𝑦)=𝜓(𝐹2(𝑥,𝑦)), we have𝛿𝑘𝑓=𝑓𝐹2|𝑘(5.2) which, in view of 𝐹2|𝑘=0 (see proposition 9.4, page 1037 of [20]), gives𝛿𝑘𝑓=0.(5.3) Since 0=𝑦𝑖|𝑘=𝛿𝑘𝑦𝑖𝐿𝑟𝑖𝑘𝑦𝑟 (see proposition 9.4, page 1037 of [20]), we have𝛿𝑘𝑦𝑖=𝐿𝑟𝑖𝑘𝑦𝑟.(5.4) If we operate 𝛿𝑘 on (2.5) and utilize (5.3) and (5.4), it follows that𝛿𝑘𝑎𝑖𝑗=𝜑𝛿𝑘𝑔𝑖𝑗+2𝜑𝑦𝑟𝐿𝑟𝑖𝑘𝑦𝑗+𝐿𝑟𝑗𝑘𝑦𝑖.(5.5) In view of 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟, (4.5), and 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟, we get𝛿𝑘𝑎𝑖𝑗=𝛿𝑘𝑎𝑖𝑗+𝑉𝑟𝑘̇𝜕𝑟𝑎𝑖𝑗,(5.6) which, on account of (4.1) and (5.5), becomes𝛿𝑘𝑎𝑖𝑗=𝜑𝛿𝑘𝑔𝑖𝑗+2𝜑𝑦𝑟𝐿𝑟𝑖𝑘𝑦𝑗+𝐿𝑟𝑗𝑘𝑦𝑖+2𝑉𝑟𝑘𝐶𝑖𝑗𝑟.(5.7) Using (5.7) in (1.12) and taking (1.14) and 𝑎𝑖𝑙𝐶𝑗𝑘𝑙=𝐶𝑖𝑗𝑘 into account, we obtain𝐿𝑖𝑗𝑘=𝐿𝑖𝑗𝑘+𝑉𝑟𝑘𝐶𝑖𝑗𝑟+𝑉𝑟𝑗𝐶𝑖𝑘𝑟+𝑉𝑟𝑝𝑎𝑖𝑝𝐶𝑟𝑘𝑗.(5.8) Equations (5.1) and (5.8) enable us to state the following.

Theorem 5.1. The coefficients of the canonical metrical 𝑑-connection 𝐶Γ(𝑁) of an APL-space 𝐿𝑛 are given by (5.1) and (5.8).

For a 𝜑-Lagrange space, 𝑉𝑖𝑗=0. Hence, (5.1) remains unchanged whereas (5.8) reduces to𝐿𝑖𝑗𝑘=𝐿𝑖𝑗𝑘.(5.9) Thus, we have the following.

Corollary 5.2 (see [14]). The coefficients of the canonical metrical 𝑑-connection 𝐶Γ(𝑁) of a 𝜑-Lagrange space 𝐿𝑛 are given by (5.1) and (5.9).

For an AFL-space, 𝜑(𝐹2)=𝐹2,𝜑(𝐹2)=1,𝜑(𝐹2)=0, and 𝑎𝑖𝑗=𝑔𝑖𝑗. Therefore, we have 𝐶𝑖𝑗𝑘=𝐶𝑖𝑗𝑘 and 𝑉𝑟𝑗=𝐵𝑟𝑗.

In view of these facts, (5.1) reduces to𝐶𝑖𝑗𝑘=𝐶𝑖𝑗𝑘,(5.10) whereas (5.8) gives the following:𝐿𝑖𝑗𝑘=𝐿𝑖𝑗𝑘+𝐵𝑟𝑘𝐶𝑖𝑗𝑟+𝐵𝑟𝑗𝐶𝑖𝑘𝑟+𝐵𝑟𝑝𝑔𝑖𝑝𝐶𝑟𝑘𝑗,(5.11) where 𝐵𝑟𝑘 is given by (4.9). Thus, we have the following.

Corollary 5.3 (see [17, 20]). The coefficients of the canonical metrical 𝑑-connection 𝐶Γ(𝑁) of an AFL-space 𝐿𝑛 are given by (5.10) and (5.11).

Now, we investigate some properties of the canonical metrical 𝑑-connection 𝐶Γ(𝑁) of an APL-space and deduce the corresponding properties for a 𝜑-Lagrange space and an AFL-space.

Theorem 5.4. The canonical metrical 𝑑-connection 𝐶Γ(𝑁) of an APL-space has the following properties: (1)𝐷𝑖𝑘=𝑦𝑖|𝑘=𝑉𝑖𝑘+𝑉𝑝𝑘𝐶𝑖𝑝𝑟𝑦𝑟+𝑉𝑝𝑟𝐶𝑖𝑘𝑝𝑦𝑟+𝑉𝑝𝑠𝑎𝑖𝑠𝐶𝑝𝑘𝑟𝑦𝑟,(5.12)𝑦𝑖|𝑘=𝑉𝑠𝑘𝑎𝑠𝑖+𝐶𝑠𝑖𝑗𝑦𝑗𝑉𝑠𝑖𝐶𝑘𝑠𝑝𝑦𝑝𝑉𝑠𝑝𝐶𝑠𝑘𝑖𝑦𝑝,(5.13) where 𝑦𝑖=𝑎𝑖𝑗𝑦𝑗, (2)𝑑𝑖𝑘=𝑦𝑖|𝑘=𝜑+𝜑𝐹2𝛿𝜑𝑖𝑘+𝐵𝑦𝑘𝑦𝑖,𝑦𝑖|𝑘=𝑎𝑖𝑘+𝐶𝑖𝑘𝑗𝑦𝑗,(5.14) where 𝐵=2{𝜑𝜑+𝐹2(𝜑𝜑𝜑2)}/𝜑(𝜑+2𝐹2𝜑), (3)𝐿|𝑘=𝑋𝑘+2𝜑𝜑+2𝐹2𝜑𝑉𝑟𝑘𝑦𝑟,𝐿|𝑘=2𝜑𝜑+2𝐹2𝜑𝑦𝑘+𝐴𝑘,(5.15) where 𝑋𝑘=𝑦𝑟𝜕𝑘𝐴𝑟𝑁𝑝𝑘𝐴𝑝+𝜕𝑘𝑈.

Proof. (1) Using (5.8) and (4.5) in (1.16), we have 𝐷𝑖𝑘=𝑦𝑟𝐿𝑖𝑟𝑘+𝑉𝑝𝑘𝐶𝑖𝑟𝑝+𝑉𝑝𝑟𝐶𝑖𝑘𝑝+𝑉𝑝𝑠𝑎𝑖𝑠𝐶𝑝𝑘𝑟𝑁𝑖𝑘+𝑉𝑖𝑘,(5.16) which, in view of (1.18), reduces to 𝐷𝑖𝑘=𝑉𝑖𝑘+𝑦𝑟𝑉𝑝𝑘𝐶𝑖𝑟𝑝+𝑉𝑝𝑟𝐶𝑖𝑘𝑝+𝑉𝑝𝑠𝑎𝑖𝑠𝐶𝑝𝑘𝑟.(5.17) Next, if we use (2.5) in 𝑦𝑖=𝑎𝑖𝑗𝑦𝑗, then it follows that 𝑦𝑖=𝜑+2𝐹2𝜑𝑦𝑖.(5.18) Now, applying successively 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟, (4.5), and 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟 in 𝑦𝑖|𝑘=𝛿𝑘𝑦𝑖𝑦𝑟𝐿𝑟𝑖𝑘 and keeping (5.8) and (5.18) in view, we have 𝑦𝑖|𝑘=𝛿𝑘𝜑+2𝐹2𝜑𝑦𝑖𝜑+2𝐹2𝜑𝑦𝑟𝐿𝑟𝑖𝑘+𝑉𝑟𝑘̇𝜕𝑟𝑦𝑖𝑦𝑟𝑉𝑠𝑘𝐶𝑟𝑖𝑠+𝑉𝑠𝑖𝐶𝑟𝑘𝑠+𝑉𝑠𝑝𝑎𝑟𝑝𝐶𝑠𝑘𝑖.(5.19) Differentiating 𝑦𝑖=𝑎𝑖𝑗𝑦𝑗 partially with respect to 𝑦𝑟, we have ̇𝜕𝑟𝑦𝑖=𝑎𝑖𝑟+2𝐶𝑖𝑟𝑗𝑦𝑗.(5.20) Also, 𝑦𝑙𝐶𝑙𝑗𝑘=𝑎𝑙𝑦𝐶𝑙𝑗𝑘=𝑦𝐶𝑗𝑘.(5.21) In view of (5.3), we have 𝛿𝑘𝜑+2𝐹2𝜑=0.(5.22) Using (5.20), (5.21), and (5.22) in (5.19), we obtain 𝑦𝑖|𝑘=𝜑+2𝐹2𝜑𝛿𝑘𝑦𝑖𝑦𝑟𝐿𝑟𝑖𝑘+𝑉𝑠𝑘𝑎𝑠𝑖+𝐶𝑠𝑖𝑗𝑦𝑗𝑉𝑠𝑖𝐶𝑘𝑠𝑝+𝑉𝑠𝑝𝐶𝑘𝑠𝑖𝑦𝑝,(5.23) which, in view of (5.4), gives the desired result.
(2) Using (5.1) in (1.17), we get 𝑑𝑖𝑘=𝜑+𝜑𝐹2𝛿𝜑𝑖𝑘+𝐵𝑦𝑘𝑦𝑖,(5.24) where 𝐵=2{𝜑𝜑+𝐹2(𝜑𝜑𝜑2)}/𝜑(𝜑+2𝐹2𝜑).
In view of (5.20) and (5.21), it follows, from 𝑦𝑖|𝑘=̇𝜕𝑘𝑦𝑖𝑦𝑟𝐶𝑟𝑖𝑘, that 𝑦𝑖|𝑘=𝑎𝑖𝑘+2𝐶𝑖𝑘𝑗𝑦𝑗𝐶𝑖𝑗𝑘𝑦𝑗,(5.25) that is, 𝑦𝑖|𝑘=𝑎𝑖𝑘+𝐶𝑖𝑘𝑗𝑦𝑗 as 𝐶𝑖𝑗𝑘 is totally symmetric.
(3) Utilizing successively 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟, (4.5), and 𝛿𝑖=𝜕𝑖𝑁𝑟𝑖̇𝜕𝑟 in 𝐿|𝑘=𝛿𝑘𝐿, we get 𝐿|𝑘=𝛿𝑘𝐿+𝑉𝑟𝑘̇𝜕𝑟𝐿.(5.26) Using (1.2) and (2.1) in (5.26), we have 𝐿|𝑘=𝛿𝑘𝜑+𝐴𝑟𝑦𝑟+𝑈+𝑉𝑟𝑘2𝜑𝑦𝑟+𝐴𝑟,(5.27) which, in view of (5.3), gives 𝐿|𝑘=𝛿𝑘𝐴𝑟𝑦𝑟+𝑈+𝑉𝑟𝑘2𝜑𝑦𝑟+𝐴𝑟.(5.28) Using 𝛿𝑘=𝜕𝑘𝑁𝑟𝑘̇𝜕𝑟 and (5.18) in (5.28) and keeping (4.5) in view, we find 𝐿|𝑘=𝑦𝑟𝜕𝑘𝐴𝑟𝑁𝑝𝑘𝐴𝑝+𝜕𝑘𝑈+2𝜑𝜑+2𝐹2𝜑𝑉𝑟𝑘𝑦𝑟.(5.29) If we take 𝑋𝑘=𝑦𝑟𝜕𝑘𝐴𝑟𝑁𝑝𝑘𝐴𝑝+𝜕𝑘𝑈, then the last expression takes the form 𝐿|𝑘=𝑋𝑘+2𝜑𝜑+2𝐹2𝜑𝑉𝑟𝑘𝑦𝑟.(5.30) Next, using (2.1) in 𝐿|𝑘=̇𝜕𝑘𝐿, we get 𝐿|𝑘=2𝜑𝑦𝑘+𝐴𝑘,(5.31) which, in view of (5.18), gives the required result.

Corollary 5.5 (see [14]). The canonical metrical 𝑑-connection 𝐶Γ(𝑁) of a 𝜑-Lagrange space has the following properties: (1)𝐷𝑖𝑘=𝑦𝑖|𝑘=0,𝑦𝑖|𝑘=0,(5.32)(2)𝑑𝑖𝑘=𝑦𝑖|𝑘=𝜑+𝜑𝐹2𝛿𝜑𝑖𝑘+𝐵𝑦𝑘𝑦𝑖,𝑦𝑖|𝑘=𝑎𝑖𝑘+𝐶𝑖𝑘𝑗𝑦𝑗,(5.33) where 𝐵=2{𝜑𝜑+𝐹2(𝜑𝜑𝜑2)}/𝜑(𝜑+2𝐹2𝜑), (3)𝐿|𝑘=0,𝐿|𝑘=2𝜑𝜑+2𝐹2𝜑𝑦𝑘.(5.34)

Proof. Applying 𝐴𝑖(𝑥)=0, 𝑈(𝑥)=0, and 𝑉𝑖𝑗=0 in Theorem 5.4, we have the corollary.

Corollary 5.6. The canonical metrical 𝑑-connection 𝐶Γ(𝑁) of an AFL-space has the following properties: (1)𝐷𝑖𝑘=𝐵𝑖𝑘+𝐵𝑝𝑟𝐶𝑖𝑘𝑝𝑦𝑟,𝑦𝑖|𝑘=𝑔𝑠𝑖𝐵𝑠𝑘𝐵𝑙𝑝𝑦𝑝𝐶𝑠𝑙𝑘,(5.35) where 𝑦𝑖=𝑔𝑖𝑗𝑦𝑗, (2)𝑑𝑖𝑘=𝛿𝑖𝑘,𝑦𝑖|𝑘=𝑔𝑖𝑘,(5.36)(3)𝐿|𝑘=𝑦𝑟𝜕𝑘𝐴𝑟𝑁𝑝𝑘𝐴𝑝+𝜕𝑘𝑈+2𝐵𝑟𝑘𝑦𝑟,𝐿|𝑘=2𝑦𝑘+𝐴𝑘.(5.37)

Proof. Using 𝜑(𝐹2)=𝐹2,𝜑(𝐹2)=1,𝜑(𝐹2)=0=𝜑(𝐹2),𝑎𝑖𝑗=𝑔𝑖𝑗,𝐶𝑖𝑗𝑘=𝐶𝑖𝑗𝑘,𝐶𝑖𝑗𝑘𝑦𝑗=0,𝐶𝑖𝑗𝑘𝑦𝑘=0, and 𝑉𝑟𝑗=𝐵𝑟𝑗 in Theorem 5.4, we have the corollary.

Acknowledgment

S. K. Shukla gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), India.