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ISRN Geometry
VolumeΒ 2011Β (2011), Article IDΒ 505161, 16 pages
http://dx.doi.org/10.5402/2011/505161
Research Article

On Almost 𝝋-Lagrange Spaces

Department of Mathematics, University of Allahabad, Allahabad 211 002, India

Received 12 October 2011; Accepted 13 November 2011

Academic Editors: A.Β Belhaj and M.Β Margenstern

Copyright Β© 2011 P. N. Pandey and Suresh K. Shukla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We initiate a study on the geometry of an almost πœ‘-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 β„Ž- and 𝑣-deflection tensors in an APL-space. Corresponding expressions in a πœ‘-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 [3–13]. Antonelli and Hrimiuc [14, 15] introduced a special type of regular Lagrangian called πœ‘-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 πœ‘-Lagrangian and introduce the concept of almost πœ‘-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(π‘₯,𝑦)) be an 𝑛-dimensional Finsler space, and let πœ‘βˆΆβ„+→ℝ be a smooth function. The composition 𝐿∢=πœ‘(𝐹2) defines a differentiable Lagrangian. This was regarded by Antonelli and Hrimiuc [14, 15] as πœ‘-Lagrangian associated to the Finsler space 𝐹𝑛. They [14] proved that if the function πœ‘ has the following properties:(a)πœ‘ξ…ž((𝑑)β‰ 0,b)πœ‘ξ…ž(𝑑)+πœ‘ξ…žξ…ž(𝑑)β‰ 0,foreveryξ€·πΉπ‘‘βˆˆIm2ξ€Έ,(1.1) then 𝐿 is a regular Lagrangian and thus 𝐿𝑛=(𝑀,𝐿(π‘₯,𝑦)) is a Lagrange space, called a πœ‘-Lagrange space.

In this paper, we consider a more general Lagrangian as follows:𝐹𝐿(π‘₯,𝑦)=πœ‘2ξ€Έ+𝐴𝑖(π‘₯)𝑦𝑖+π‘ˆ(π‘₯),(1.2) where πœ‘ is the same as discussed earlier, 𝐴𝑖(π‘₯) is a covector, and π‘ˆ(π‘₯) is a smooth function.

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

An APL-space reduces to a πœ‘-Lagrange space if and only if 𝐴𝑖(π‘₯)=0 and π‘ˆ(π‘₯)=0.

If πœ‘(𝑑)=𝑑,forallπ‘‘βˆˆIm(𝐹2), then the Lagrangian in (1.2) takes the form𝐿(π‘₯,𝑦)=𝐹2+𝐴𝑖(π‘₯)𝑦𝑖+π‘ˆ(π‘₯).(1.3) This defines a regular Lagrangian, and the pair 𝐿𝑛=(𝑀,𝐿(π‘₯,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=12Μ‡πœ•π‘–Μ‡πœ•π‘—πΉ2,π‘Žπ‘–π‘—=12Μ‡πœ•π‘–Μ‡πœ•π‘—Μ‡πœ•πΏ,π‘–β‰‘πœ•πœ•π‘¦π‘–.(1.4) Henceforth, we will indicate all the geometrical objects related to 𝐹𝑛 by a small circle β€œβˆ˜β€ 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]:βˆ˜π‘π‘–π‘—=βˆ˜π›Ύπ‘–π‘—0βˆ’βˆ˜πΆπ‘–βˆ˜π‘—π‘˜π›Ύπ‘˜00,(1.5) whereβˆ˜π›Ύπ‘–π‘—π‘˜=12π‘”π‘–β„Žξ€·πœ•π‘—π‘”β„Žπ‘˜+πœ•π‘˜π‘”π‘—β„Žβˆ’πœ•β„Žπ‘”π‘—π‘˜ξ€Έ;πœ•π‘—β‰‘πœ•πœ•π‘₯𝑗,βˆ˜πΆπ‘–π‘—π‘˜=12π‘”π‘–β„ŽΜ‡πœ•β„Žπ‘”π‘—π‘˜,βˆ˜π›Ύπ‘–π‘—0=βˆ˜π›Ύπ‘–π‘—π‘˜π‘¦π‘˜,βˆ˜π›Ύπ‘–00=βˆ˜π›Ύπ‘–π‘—π‘˜π‘¦π‘—π‘¦π‘˜.(1.6) These geodesics are the integral curves of the spray [16] (i.e., (2) p-homogeneous):βˆ˜πΊπ‘–=14π‘”π‘–π‘—ξ€·π‘¦π‘˜Μ‡πœ•π‘—πœ•π‘˜πΉ2βˆ’πœ•π‘—πΉ2ξ€Έ,(1.7) that is, solutions of the differential equations𝑑2π‘₯𝑖𝑑𝑠2+2βˆ˜πΊπ‘–ξ‚€π‘₯(𝑠),𝑑π‘₯𝑑𝑠=0.(1.8) We have the following equalities:(a)βˆ˜πΊπ‘–=12βˆ˜π›Ύπ‘–00,(b)βˆ˜π‘π‘–π‘—=Μ‡πœ•π‘—βˆ˜πΊπ‘–.(1.9) In a general Lagrange space 𝐿𝑛=(𝑀,𝐿(π‘₯,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=14π‘Žπ‘–π‘—ξ€·π‘¦π‘˜Μ‡πœ•π‘—πœ•π‘˜πΏβˆ’πœ•π‘—πΏξ€Έ.(1.10) As in a Finsler space, a remarkable nonlinear connection can be considered in a Lagrange space:𝑁𝑖𝑗=Μ‡πœ•π‘—πΊπ‘–.(1.11) Such nonlinear connection is a canonical nonlinear connection [17, 18] as it depends only on the fundamental function 𝐿(π‘₯,𝑦) of the Lagrange space.

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

Given a nonlinear connection (𝑁𝑖𝑗) on a Lagrange space 𝐿𝑛=(𝑀,𝐿(π‘₯,𝑦)), there is a unique β„Ž- and 𝑣-metrical 𝑑-connection (cf. [17, 19]) 𝐢Γ(𝑁)=(𝑁𝑖𝑗,πΏπ‘–π‘—π‘˜,πΆπ‘–π‘—π‘˜) with torsions π‘‡π‘–π‘—π‘˜=0 and π‘†π‘–π‘—π‘˜=0, called the canonical metrical 𝑑-connection. This connection is linear and its coefficients are given byπΏπ‘–π‘—π‘˜=12π‘Žπ‘–β„Žξ€·π›Ώπ‘—π‘Žβ„Žπ‘˜+π›Ώπ‘˜π‘Žπ‘—β„Žβˆ’π›Ώβ„Žπ‘Žπ‘—π‘˜ξ€Έ,(1.12)πΆπ‘–π‘—π‘˜=12π‘Žπ‘–β„Žξ€·Μ‡πœ•π‘—π‘Žβ„Žπ‘˜+Μ‡πœ•π‘˜π‘Žπ‘—β„Žβˆ’Μ‡πœ•β„Žπ‘Žπ‘—π‘˜ξ€Έ,(1.13) where 𝛿𝑖=πœ•π‘–βˆ’π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ is the Lagrange differentiation operator.

If βˆ˜πΆΞ“(βˆ˜π‘)=(βˆ˜π‘π‘–π‘—,βˆ˜πΏπ‘–π‘—π‘˜,βˆ˜πΆπ‘–π‘—π‘˜) is the Cartan connection of the Finsler space 𝐹𝑛=(𝑀,𝐹(π‘₯,𝑦)), then its coefficients are given byβˆ˜πΏπ‘–π‘—π‘˜=12π‘”π‘–β„Žξ‚΅βˆ˜π›Ώπ‘—π‘”β„Žπ‘˜+βˆ˜π›Ώπ‘˜π‘”π‘—β„Žβˆ’βˆ˜π›Ώβ„Žπ‘”π‘—π‘˜ξ‚Ά,(1.14)βˆ˜πΆπ‘–π‘—π‘˜=12π‘”π‘–β„Žξ€·Μ‡πœ•π‘—π‘”β„Žπ‘˜+Μ‡πœ•π‘˜π‘”π‘—β„Žβˆ’Μ‡πœ•β„Žπ‘”π‘—π‘˜ξ€Έ,(1.15) where βˆ˜π›Ώπ‘–=πœ•π‘–βˆ’βˆ˜π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ.

The β„Ž- and 𝑣-deflection tensor fields 𝐷𝑖𝑗 and 𝑑𝑖𝑗, respectively, of a Lagrange space 𝐿𝑛 are defined by (cf. [19])π·π‘–π‘—βˆΆ=𝑦𝑖|𝑗=π‘¦π‘ πΏπ‘–π‘ π‘—βˆ’π‘π‘–π‘—,(1.16)π‘‘π‘–π‘—βˆΆ=𝑦𝑖|𝑗=𝛿𝑖𝑗+𝑦𝑠𝐢𝑖𝑠𝑗,(1.17) where | and |, respectively, denote the β„Ž- and 𝑣-covariant derivatives with respect to 𝐢Γ.

If βˆ˜π·π‘–π‘— is the h-deflection tensor field and βˆ˜π‘‘π‘–π‘— is the 𝑣-deflection tensor field of the Finsler space 𝐹𝑛, thenβˆ˜π·π‘–π‘—βˆΆ=π‘¦π‘–βˆ˜|𝑗=π‘¦π‘ βˆ˜πΏπ‘–π‘ π‘—βˆ’βˆ˜π‘π‘–π‘—=0,(1.18)βˆ˜π‘‘π‘–π‘—βˆΆ=π‘¦π‘–βˆ˜|||𝑗=𝛿𝑖𝑗,(1.19) where ∘| and ∘|, respectively, denote the β„Ž- and 𝑣-covariant derivatives with respect to βˆ˜πΆΞ“.

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

2. Almost πœ‘-Lagrange Spaces

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

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

Proof. Differentiating (1.2) partially with respect to 𝑦𝑖, we get Μ‡πœ•π‘–πΏ=πœ‘ξ…žξ€·πΉ2ξ€ΈΜ‡πœ•π‘–πΉ2+𝐴𝑖(π‘₯).(2.1) Again differentiating (2.1) partially with respect to 𝑦𝑗, we obtain Μ‡πœ•π‘—Μ‡πœ•π‘–πΏ=πœ‘ξ…žξ…žξ€·πΉ2ξ€ΈΜ‡πœ•π‘–πΉ2Μ‡πœ•π‘—πΉ2+πœ‘ξ…žξ€·πΉ2ξ€ΈΜ‡πœ•π‘–Μ‡πœ•π‘—πΉ2,(2.2) which, in view of (1.4), provides π‘Žπ‘–π‘—=2𝐹2πœ‘ξ…žξ…žξ€·πΉ2ξ€ΈΜ‡πœ•π‘–πΉΜ‡πœ•π‘—πΉ+πœ‘ξ…žξ€·πΉ2𝑔𝑖𝑗.(2.3) Now πΉΜ‡πœ•π‘–1𝐹=2Μ‡πœ•π‘–πΉ2=12Μ‡πœ•π‘–ξ€·π‘”π‘—π‘˜π‘¦π‘—π‘¦π‘˜ξ€Έ=π‘”π‘–π‘˜π‘¦π‘˜βˆΆ=βˆ˜π‘¦π‘–.(2.4) In view of (2.4), (2.3) takes the form π‘Žπ‘–π‘—=πœ‘ξ…žβ‹…ξ‚΅π‘”π‘–π‘—+2πœ‘ξ…žξ…žπœ‘ξ…žβˆ˜π‘¦π‘–βˆ˜π‘¦π‘—ξ‚Ά.(2.5) Under the hypothesis, the matrix (π‘Žπ‘–π‘—) is invertible and its inverse is (see Lemma 6.2.2.1, page 891 in [20]) π‘Žπ‘–π‘—=1ξ‚΅π‘”πœ‘β€²π‘–π‘—βˆ’2πœ‘ξ…žξ…žπœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žπ‘¦π‘–π‘¦π‘—ξ‚Ά.(2.6) This proves the theorem.

Remarks 1. (i) If 𝐴𝑖(π‘₯)=0 and π‘ˆ(π‘₯)=0 in (1.2), then expression (2.5) remains unchanged. Hence, the symmetric metric tensor of a πœ‘-Lagrange space is the same as that of an APL-space.
(ii) If πœ‘(𝐹2)=𝐹2, then πœ‘ξ…ž=1 and πœ‘ξ…žξ…ž=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(π‘₯,𝑦)) and deduce corresponding expressions for a πœ‘-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 πœ‘-Lagrange space and an AFL-space.

If we differentiate (1.2) partially with respect to π‘₯π‘˜, we haveπœ•π‘˜πΏ=πœ‘ξ…žξ€·πΉ2ξ€Έπœ•π‘˜πΉ2+π‘¦π‘–πœ•π‘˜π΄π‘–(π‘₯)+πœ•π‘˜π‘ˆ(π‘₯).(3.1) Differentiating (3.1) partially with respect to 𝑦𝑗, we obtainΜ‡πœ•π‘—πœ•π‘˜πΏ=2πœ‘ξ…žξ…žξ€·πΉ2ξ€ΈπΉΜ‡πœ•π‘—πΉπœ•π‘˜πΉ2+πœ‘ξ…žξ€·πΉ2ξ€ΈΜ‡πœ•π‘—πœ•π‘˜πΉ2+πœ•π‘˜π΄π‘—(π‘₯),(3.2) which, in view of (2.4), takes the formΜ‡πœ•π‘—πœ•π‘˜πΏ=2πœ‘ξ…žξ…žξ€·πΉ2ξ€Έβˆ˜π‘¦π‘—πœ•π‘˜πΉ2+πœ‘ξ…žξ€·πΉ2ξ€ΈΜ‡πœ•π‘—πœ•π‘˜πΉ2+πœ•π‘˜π΄π‘—(π‘₯).(3.3) Using (3.1) and (3.3) in (1.10), we have𝐺𝑖=14π‘Žπ‘–π‘—ξ‚»2πœ‘ξ…žξ…žξ€·πΉ2ξ€Έβˆ˜π‘¦π‘—π‘¦π‘˜πœ•π‘˜πΉ2+πœ‘ξ…žξ€·πΉ2π‘¦ξ€Έξ€·π‘˜Μ‡πœ•π‘—πœ•π‘˜πΉ2βˆ’πœ•π‘—πΉ2ξ€Έβˆ’2π‘¦π‘˜πΉπ‘—π‘˜βˆ’πœ•π‘—π‘ˆξ‚Ό,(3.4) whereπΉπ‘—π‘˜1(π‘₯)=2ξ€·πœ•π‘—π΄π‘˜βˆ’πœ•π‘˜π΄π‘—ξ€Έ(3.5) is electromagnetic tensor field of the potentials 𝐴𝑖(π‘₯).

Applying (2.6) in (3.4) and using βˆ˜π‘¦π‘–π‘¦π‘–=𝐹2,π‘”βˆ˜π‘–π‘—π‘¦π‘—=𝑦𝑖, and π‘¦π‘—Μ‡πœ•π‘—πœ•π‘˜πΉ2=2πœ•π‘˜πΉ2 (by Euler’s theorem on homogeneous functions), we obtain𝐺𝑖=12πœ‘ξ…žξ…žπœ‘ξ…žξ‚΅1βˆ’2πœ‘ξ…žξ…žπΉ2πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ‚Άπ‘¦π‘–π‘¦π‘˜πœ•π‘˜πΉ2+14ξ‚»π‘”π‘–π‘—ξ€·π‘¦π‘˜Μ‡πœ•π‘—πœ•π‘˜πΉ2βˆ’πœ•π‘—πΉ2ξ€Έβˆ’2πœ‘ξ…žξ…žπΉ2πœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘¦π‘–π‘¦π‘˜πœ•π‘˜πΉ2ξ‚Όβˆ’14π‘Žπ‘–π‘—ξ€·2πΉπ‘—π‘˜π‘¦π‘˜+πœ•π‘—π‘ˆξ€Έ.(3.6) Using (1.7) in (3.6) and simplifying, we get𝐺𝑖=βˆ˜πΊπ‘–βˆ’14π‘Žπ‘–π‘—ξ€·2πΉπ‘—π‘˜π‘¦π‘˜+πœ•π‘—π‘ˆξ€Έ.(3.7) Thus, we have the following.

Theorem 3.1. The canonical semispray of an APL-space has the local coefficients given by 𝐺𝑖=βˆ˜πΊπ‘–βˆ’14π‘Žπ‘–π‘—ξ€·2πΉπ‘—π‘˜π‘¦π‘˜+πœ•π‘—π‘ˆξ€Έ,(3.8) where βˆ˜πΊπ‘– are the local coefficients of the spray of 𝐹𝑛.

For a πœ‘-Lagrange space, 𝐴𝑖(π‘₯)=0 and π‘ˆ(π‘₯)=0. Hence, from (3.5), we have πΉπ‘—π‘˜=0. Therefore, (3.7) reduces to𝐺𝑖=βˆ˜πΊπ‘–.(3.9) Thus, we may state the following.

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

For an AFL-space, π‘Žπ‘–π‘—=𝑔𝑖𝑗 (see Remark (ii)). Hence, (3.7) takes the form𝐺𝑖=βˆ˜πΊπ‘–βˆ’14𝑔𝑖𝑗2πΉπ‘—π‘˜π‘¦π‘˜+πœ•π‘—π‘ˆξ€Έ.(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)∢=πœ•π‘–π‘‘πΏβˆ’ξ€·Μ‡πœ•π‘‘π‘‘π‘–πΏξ€Έ=0(3.11) are the solution curves of the equations [20]𝑑2π‘₯𝑖𝑑𝑑2+2𝐺𝑖(π‘₯,𝑦)=0.(3.12) Using (3.7) in (3.12), we obtain𝑑2π‘₯𝑖𝑑𝑑2+2βˆ˜πΊπ‘–=12ξ€·2πΉπ‘–π‘˜π‘¦π‘˜+π‘Žπ‘–π‘—πœ•π‘—π‘ˆξ€Έ,(3.13) where πΉπ‘–π‘˜=π‘Žπ‘–π‘—πΉπ‘—π‘˜.

Using (1.9) (a) in (3.13), we have𝑑2π‘₯𝑖𝑑𝑑2+βˆ˜π›Ύπ‘–00=12ξ€·2πΉπ‘–π‘˜π‘¦π‘˜+π‘Žπ‘–π‘—πœ•π‘—π‘ˆξ€Έ.(3.14) Thus, we have the following.

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

For a πœ‘-Lagrange space, equations (3.14) take the following simple form:𝑑2π‘₯𝑖𝑑𝑑2+βˆ˜π›Ύπ‘–00=0.(3.15) This enables us to state the following.

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

For an AFL-space, π‘Žπ‘–π‘—=𝑔𝑖𝑗. Therefore, equations (3.14) become𝑑2π‘₯𝑖𝑑𝑑2+βˆ˜π›Ύπ‘–00=πΉπ‘–π‘˜π‘¦π‘˜+12π‘”π‘–π‘—πœ•π‘—π‘ˆ,(3.16) where πΉπ‘–π‘˜=π‘”π‘–π‘—πΉπ‘—π‘˜.

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 πœ‘-Lagrange space and an AFL-space.

Partial differentiation of (2.5) with respect to π‘¦π‘˜ yieldsΜ‡πœ•π‘˜π‘Žπ‘–π‘—=∢2πΆπ‘–π‘—π‘˜=2πœ‘β€²βˆ˜πΆπ‘–π‘—π‘˜+2πœ‘ξ…žξ…žξ‚΅π‘”βˆ˜π‘–π‘—π‘¦π‘˜+π‘”βˆ˜π‘—π‘˜π‘¦π‘–+π‘”βˆ˜π‘˜π‘–π‘¦π‘—ξ‚Ά+4πœ‘βˆ˜ξ…žξ…žξ…žπ‘¦π‘–βˆ˜π‘¦π‘—βˆ˜π‘¦π‘˜.(4.1) Using (3.7) in (1.11) and taking (1.9) (b), (2.6), (4.1), βˆ˜πΆπ‘π‘žπ‘—π‘¦π‘—=0, π‘¦π‘–βˆ˜π‘¦π‘–=𝐹2, and π‘”βˆ˜π‘–π‘—π‘¦π‘—=𝑦𝑖 into account, we obtain𝑁𝑖𝑗=βˆ˜π‘π‘–π‘—βˆ’12𝐹𝑖𝑗+12πœ‘ξ…žβˆ˜πΆπ‘–π‘žπ‘—π‘”π‘žπ‘Ÿ+12πœ‘ξ…žξ…žπœ‘ξ…ž2π‘”βˆ˜π‘–π‘Ÿπ‘¦π‘—+πœ‘ξ…žξ…ž2πœ‘ξ…žξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έξ€·π›Ώπ‘Ÿπ‘—π‘¦π‘–+π›Ώπ‘–π‘—π‘¦π‘Ÿξ€Έ+πœ‘ξ…ž2πœ‘ξ…žξ…žξ…žβˆ’2πœ‘3ξ…žξ…žπΉ2βˆ’4πœ‘ξ…žπœ‘ξ…žξ…ž22πœ‘ξ…ž2ξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έ2π‘¦π‘–βˆ˜π‘¦π‘—π‘¦π‘Ÿξƒ­ξ€·2πΉπ‘Ÿπ‘˜π‘¦π‘˜+πœ•π‘Ÿπ‘ˆξ€Έ.(4.2) If we takeπ‘†π‘—π‘–π‘Ÿ=12πœ‘ξ…žβˆ˜πΆπ‘–π‘žπ‘—π‘”π‘žπ‘Ÿ+12πœ‘ξ…žξ…žπœ‘ξ…ž2π‘”βˆ˜π‘–π‘Ÿπ‘¦π‘—+πœ‘ξ…žξ…žξ€·π›Ώπ‘Ÿπ‘—π‘¦π‘–+π›Ώπ‘–π‘—π‘¦π‘Ÿξ€Έ2πœ‘ξ…žξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έ+πœ‘ξ…ž2πœ‘ξ…žξ…žξ…žβˆ’2πœ‘ξ…žξ…ž3𝐹2βˆ’4πœ‘β€²πœ‘ξ…žξ…ž22πœ‘ξ…ž2ξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έ2π‘¦π‘–βˆ˜π‘¦π‘—π‘¦π‘Ÿ,(4.3) the last expression becomes𝑁𝑖𝑗=βˆ˜π‘π‘–π‘—βˆ’12𝐹𝑖𝑗+π‘†π‘—π‘–π‘Ÿξ€·2πΉπ‘Ÿπ‘˜π‘¦π‘˜+πœ•π‘Ÿπ‘ˆξ€Έ,(4.4) that is,𝑁𝑖𝑗=βˆ˜π‘π‘–π‘—βˆ’π‘‰π‘–π‘—,(4.5) where𝑉𝑖𝑗=12πΉπ‘–π‘—βˆ’π‘†π‘—π‘–π‘Ÿξ€·2πΉπ‘Ÿπ‘˜π‘¦π‘˜+πœ•π‘Ÿπ‘ˆξ€Έ.(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 πœ‘-Lagrange space, we have πΉπ‘Ÿπ‘˜=0,𝐹𝑖𝑗=0 and π‘ˆ=0 and hence 𝑉𝑖𝑗=0. Therefore, (4.5) reduces to𝑁𝑖𝑗=βˆ˜π‘π‘–π‘—.(4.7) Thus, we have the following.

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

For an AFL-space, (4.3) reduces toπ‘†π‘—π‘–π‘Ÿ=12βˆ˜πΆπ‘–π‘žπ‘—π‘”π‘žπ‘Ÿ(4.8) and hence (4.6) gives𝑉𝑖𝑗=12πΉπ‘–π‘—βˆ’βˆ˜πΆπ‘–π‘žπ‘—πΉπ‘žπ‘˜π‘¦π‘˜βˆ’12βˆ˜πΆπ‘–π‘žπ‘—π‘”π‘žπ‘Ÿπœ•π‘Ÿπ‘ˆβˆΆ=𝐡𝑖𝑗.(4.9) Therefore, (4.5) takes the form𝑁𝑖𝑗=βˆ˜π‘π‘–π‘—βˆ’π΅π‘–π‘—.(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 βˆ˜π‘π‘–π‘—π‘¦π‘—=βˆ˜π›Ύπ‘–00, we obtain𝑁𝑖𝑗𝑦𝑗=βˆ˜π›Ύπ‘–00βˆ’π‘‰π‘–0,(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]):𝑑2π‘₯𝑖𝑑𝑑2+𝑁𝑖𝑗(π‘₯,𝑦)𝑦𝑗=0.(4.12) Equations (4.12), in view of (4.11), take the form𝑑2π‘₯𝑖𝑑𝑑2+βˆ˜π›Ύπ‘–00=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(π‘₯,𝑦)) are given by the system of differential equations (4.13).

For a πœ‘-Lagrange space, 𝑉𝑖𝑗=0 and hence 𝑉𝑖0=0. Therefore, (4.13) reduces to𝑑2π‘₯𝑖𝑑𝑑2+βˆ˜π›Ύπ‘–00=0.(4.14) Thus, we have the following.

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

For an AFL-space,𝑉𝑖𝑗=𝐡𝑖𝑗1=∢2πΉπ‘–π‘—βˆ’βˆ˜πΆπ‘–π‘žπ‘—πΉπ‘žπ‘˜π‘¦π‘˜βˆ’12βˆ˜πΆπ‘–π‘žπ‘—π‘”π‘žπ‘Ÿπœ•π‘Ÿπ‘ˆ(4.15) and hence, by virtue of βˆ˜πΆπ‘–π‘žπ‘—π‘¦π‘—=0, we have 𝑉𝑖0=(1/2)𝐹𝑖𝑗𝑦𝑗. Therefore, equations (4.12) take the form𝑑2π‘₯𝑖𝑑𝑑2+βˆ˜π›Ύπ‘–00=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(π‘₯,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 πœ‘-Lagrange space they do. Therefore, in a πœ‘-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 𝐢Γ(𝑁)=(𝑁𝑖𝑗,πΏπ‘–π‘—π‘˜,πΆπ‘–π‘—π‘˜) be the canonical metrical 𝑑-connection of the APL-space 𝐿𝑛=(𝑀,𝐿(π‘₯,𝑦)), and let βˆ˜πΆΞ“(βˆ˜π‘)=(βˆ˜π‘π‘–π‘—,βˆ˜πΏπ‘–π‘—π‘˜,βˆ˜πΆπ‘–π‘—π‘˜) be the Cartan connection of the associated Finsler space 𝐹𝑛=(𝑀,𝐹(π‘₯,𝑦)). In this section, we obtain the expressions for the coefficients of 𝐢Γ(𝑁) and we investigate some properties of 𝐢Γ(𝑁). We deduce corresponding results for a πœ‘-Lagrange space and an AFL-space.

Using (4.1) in (1.13) and taking (1.15) into account, we findπΆπ‘–π‘—π‘˜=βˆ˜πΆπ‘–π‘—π‘˜+πœ‘ξ…žξ…žξ‚΅π›Ώπœ‘β€²π‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚Ά+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–+2ξ€·πœ‘ξ…žξ…žξ…žπœ‘β€²βˆ’2πœ‘ξ…žξ…ž2ξ€Έξ€·πœ‘β€²πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘–βˆ˜π‘¦π‘—βˆ˜π‘¦π‘˜.(5.1)

For any 𝐢∞-class function πœ“βˆΆβ„+→ℝ, taking 𝑓(π‘₯,𝑦)=πœ“(𝐹2(π‘₯,𝑦)), we haveβˆ˜π›Ώπ‘˜π‘“=𝑓′𝐹2∘|π‘˜(5.2) which, in view of 𝐹2∘|π‘˜=0 (see proposition 9.4, page 1037 of [20]), givesβˆ˜π›Ώπ‘˜π‘“=0.(5.3) Since 0=π‘¦π‘–βˆ˜|π‘˜βˆΆ=βˆ˜π›Ώπ‘˜βˆ˜π‘¦π‘–βˆ’βˆ˜πΏπ‘Ÿβˆ˜π‘–π‘˜π‘¦π‘Ÿ (see proposition 9.4, page 1037 of [20]), we haveβˆ˜π›Ώπ‘˜βˆ˜π‘¦π‘–=βˆ˜πΏπ‘Ÿβˆ˜π‘–π‘˜π‘¦π‘Ÿ.(5.4) If we operate βˆ˜π›Ώπ‘˜ on (2.5) and utilize (5.3) and (5.4), it follows thatβˆ˜π›Ώπ‘˜π‘Žπ‘–π‘—=πœ‘β€²βˆ˜π›Ώπ‘˜π‘”π‘–π‘—+2πœ‘βˆ˜ξ…žξ…žπ‘¦π‘Ÿξ‚΅βˆ˜πΏπ‘Ÿβˆ˜π‘–π‘˜π‘¦π‘—+βˆ˜πΏπ‘Ÿβˆ˜π‘—π‘˜π‘¦π‘–ξ‚Ά.(5.5) In view of 𝛿𝑖=πœ•π‘–βˆ’π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ, (4.5), and βˆ˜π›Ώπ‘–=πœ•π‘–βˆ’βˆ˜π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ, we getπ›Ώπ‘˜π‘Žπ‘–π‘—=βˆ˜π›Ώπ‘˜π‘Žπ‘–π‘—+π‘‰π‘Ÿπ‘˜Μ‡πœ•π‘Ÿπ‘Žπ‘–π‘—,(5.6) which, on account of (4.1) and (5.5), becomesπ›Ώπ‘˜π‘Žπ‘–π‘—=πœ‘β€²βˆ˜π›Ώπ‘˜π‘”π‘–π‘—+2πœ‘βˆ˜ξ…žξ…žπ‘¦π‘Ÿξ‚΅βˆ˜πΏπ‘Ÿβˆ˜π‘–π‘˜π‘¦π‘—+βˆ˜πΏπ‘Ÿβˆ˜π‘—π‘˜π‘¦π‘–ξ‚Ά+2π‘‰π‘Ÿπ‘˜πΆπ‘–π‘—π‘Ÿ.(5.7) Using (5.7) in (1.12) and taking (1.14) and π‘Žπ‘–π‘™πΆπ‘—π‘˜π‘™=πΆπ‘–π‘—π‘˜ into account, we obtainπΏπ‘–π‘—π‘˜=βˆ˜πΏπ‘–π‘—π‘˜+π‘‰π‘Ÿπ‘˜πΆπ‘–π‘—π‘Ÿ+π‘‰π‘Ÿπ‘—πΆπ‘–π‘˜π‘Ÿ+π‘‰π‘Ÿπ‘π‘Žπ‘–π‘πΆπ‘Ÿπ‘˜π‘—.(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 𝐢Γ(𝑁) of an APL-space 𝐿𝑛 are given by (5.1) and (5.8).

For a πœ‘-Lagrange space, 𝑉𝑖𝑗=0. Hence, (5.1) remains unchanged whereas (5.8) reduces toπΏπ‘–π‘—π‘˜=βˆ˜πΏπ‘–π‘—π‘˜.(5.9) Thus, we have the following.

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

For an AFL-space, πœ‘(𝐹2)=𝐹2,πœ‘β€²(𝐹2)=1,πœ‘ξ…žξ…ž(𝐹2)=0, and π‘Žπ‘–π‘—=𝑔𝑖𝑗. Therefore, we have πΆπ‘–π‘—π‘˜=βˆ˜πΆπ‘–π‘—π‘˜ and π‘‰π‘Ÿπ‘—=π΅π‘Ÿπ‘—.

In view of these facts, (5.1) reduces toπΆπ‘–π‘—π‘˜=βˆ˜πΆπ‘–π‘—π‘˜,(5.10) whereas (5.8) gives the following:πΏπ‘–π‘—π‘˜=βˆ˜πΏπ‘–π‘—π‘˜+π΅π‘Ÿπ‘˜βˆ˜πΆπ‘–π‘—π‘Ÿ+π΅π‘Ÿπ‘—βˆ˜πΆπ‘–π‘˜π‘Ÿ+π΅π‘Ÿπ‘π‘”βˆ˜π‘–π‘πΆπ‘Ÿπ‘˜π‘—,(5.11) where π΅π‘Ÿπ‘˜ is given by (4.9). Thus, we have the following.

Corollary 5.3 (see [17, 20]). The coefficients of the canonical metrical 𝑑-connection 𝐢Γ(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 𝐢Γ(𝑁) of an APL-space and deduce the corresponding properties for a πœ‘-Lagrange space and an AFL-space.

Theorem 5.4. The canonical metrical 𝑑-connection 𝐢Γ(𝑁) of an APL-space has the following properties: (1)π·π‘–π‘˜βˆΆ=𝑦𝑖|π‘˜=π‘‰π‘–π‘˜+π‘‰π‘π‘˜πΆπ‘–π‘π‘Ÿπ‘¦π‘Ÿ+π‘‰π‘π‘ŸπΆπ‘–π‘˜π‘π‘¦π‘Ÿ+π‘‰π‘π‘ π‘Žπ‘–π‘ πΆπ‘π‘˜π‘Ÿπ‘¦π‘Ÿ,(5.12)𝑦𝑖|π‘˜=π‘‰π‘ π‘˜ξ€·π‘Žπ‘ π‘–+πΆπ‘ π‘–π‘—π‘¦π‘—ξ€Έβˆ’π‘‰π‘ π‘–πΆπ‘˜π‘ π‘π‘¦π‘βˆ’π‘‰π‘ π‘πΆπ‘ π‘˜π‘–π‘¦π‘,(5.13) where 𝑦𝑖=βˆΆπ‘Žπ‘–π‘—π‘¦π‘—, (2)π‘‘π‘–π‘˜βˆΆ=𝑦𝑖|π‘˜=πœ‘β€²+πœ‘ξ…žξ…žπΉ2π›Ώπœ‘β€²π‘–π‘˜+π΅βˆ˜π‘¦π‘˜π‘¦π‘–,𝑦𝑖|π‘˜=π‘Žπ‘–π‘˜+πΆπ‘–π‘˜π‘—π‘¦π‘—,(5.14) where 𝐡=2{πœ‘ξ…žπœ‘ξ…žξ…ž+𝐹2(πœ‘ξ…žξ…žξ…žπœ‘ξ…žβˆ’πœ‘ξ…žξ…ž2)}/πœ‘β€²(πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…ž), (3)𝐿|π‘˜=π‘‹π‘˜+2πœ‘β€²ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘‰π‘Ÿπ‘˜π‘¦π‘Ÿ,𝐿|π‘˜=2πœ‘β€²ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘˜+π΄π‘˜,(5.15) where π‘‹π‘˜=π‘¦π‘Ÿπœ•π‘˜π΄π‘Ÿβˆ’π‘π‘π‘˜π΄π‘+πœ•π‘˜π‘ˆ.

Proof. (1) Using (5.8) and (4.5) in (1.16), we have π·π‘–π‘˜=π‘¦π‘Ÿξ‚΅βˆ˜πΏπ‘–π‘Ÿπ‘˜+π‘‰π‘π‘˜πΆπ‘–π‘Ÿπ‘+π‘‰π‘π‘ŸπΆπ‘–π‘˜π‘+π‘‰π‘π‘ π‘Žπ‘–π‘ πΆπ‘π‘˜π‘Ÿξ‚Άβˆ’βˆ˜π‘π‘–π‘˜+π‘‰π‘–π‘˜,(5.16) which, in view of (1.18), reduces to π·π‘–π‘˜=π‘‰π‘–π‘˜+π‘¦π‘Ÿξ‚€π‘‰π‘π‘˜πΆπ‘–π‘Ÿπ‘+π‘‰π‘π‘ŸπΆπ‘–π‘˜π‘+π‘‰π‘π‘ π‘Žπ‘–π‘ πΆπ‘π‘˜π‘Ÿξ‚.(5.17) Next, if we use (2.5) in 𝑦𝑖=π‘Žπ‘–π‘—π‘¦π‘—, then it follows that 𝑦𝑖=ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έβˆ˜π‘¦π‘–.(5.18) Now, applying successively 𝛿𝑖=πœ•π‘–βˆ’π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ, (4.5), and βˆ˜π›Ώπ‘–=πœ•π‘–βˆ’βˆ˜π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ in 𝑦𝑖|π‘˜=π›Ώπ‘˜π‘¦π‘–βˆ’π‘¦π‘ŸπΏπ‘Ÿπ‘–π‘˜ and keeping (5.8) and (5.18) in view, we have 𝑦𝑖|π‘˜=βˆ˜π›Ώπ‘˜ξ‚»ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έβˆ˜π‘¦π‘–ξ‚Όβˆ’ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έβˆ˜π‘¦π‘Ÿβˆ˜πΏπ‘Ÿπ‘–π‘˜+π‘‰π‘Ÿπ‘˜Μ‡πœ•π‘Ÿπ‘¦π‘–βˆ’π‘¦π‘Ÿξ€·π‘‰π‘ π‘˜πΆπ‘Ÿπ‘–π‘ +π‘‰π‘ π‘–πΆπ‘Ÿπ‘˜π‘ +π‘‰π‘ π‘π‘Žπ‘Ÿπ‘πΆπ‘ π‘˜π‘–ξ€Έ.(5.19) Differentiating 𝑦𝑖=π‘Žπ‘–π‘—π‘¦π‘— partially with respect to π‘¦π‘Ÿ, we have Μ‡πœ•π‘Ÿπ‘¦π‘–=π‘Žπ‘–π‘Ÿ+2πΆπ‘–π‘Ÿπ‘—π‘¦π‘—.(5.20) Also, π‘¦π‘™πΆπ‘™π‘—π‘˜=π‘Žπ‘™β„Žπ‘¦β„ŽπΆπ‘™π‘—π‘˜=π‘¦β„ŽπΆβ„Žπ‘—π‘˜.(5.21) In view of (5.3), we have βˆ˜π›Ώπ‘˜ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έ=0.(5.22) Using (5.20), (5.21), and (5.22) in (5.19), we obtain 𝑦𝑖|π‘˜=ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έξ‚΅βˆ˜π›Ώπ‘˜βˆ˜π‘¦π‘–βˆ’βˆ˜π‘¦π‘Ÿβˆ˜πΏπ‘Ÿπ‘–π‘˜ξ‚Ά+π‘‰π‘ π‘˜ξ€·π‘Žπ‘ π‘–+πΆπ‘ π‘–π‘—π‘¦π‘—ξ€Έβˆ’ξ€·π‘‰π‘ π‘–πΆπ‘˜π‘ π‘+π‘‰π‘ π‘πΆπ‘˜π‘ π‘–ξ€Έπ‘¦π‘,(5.23) which, in view of (5.4), gives the desired result.
(2) Using (5.1) in (1.17), we get π‘‘π‘–π‘˜=πœ‘β€²+πœ‘ξ…žξ…žπΉ2π›Ώπœ‘β€²π‘–π‘˜+π΅βˆ˜π‘¦π‘˜π‘¦π‘–,(5.24) where 𝐡=2{πœ‘ξ…žπœ‘ξ…žξ…ž+𝐹2(πœ‘ξ…žξ…žξ…žπœ‘ξ…žβˆ’πœ‘ξ…žξ…ž2)}/πœ‘β€²(πœ‘β€²+2𝐹2πœ‘ξ…žξ…ž).
In view of (5.20) and (5.21), it follows, from 𝑦𝑖|π‘˜=Μ‡πœ•π‘˜π‘¦π‘–βˆ’π‘¦π‘ŸπΆπ‘Ÿπ‘–π‘˜, that 𝑦𝑖|π‘˜=π‘Žπ‘–π‘˜+2πΆπ‘–π‘˜π‘—π‘¦π‘—βˆ’πΆπ‘–π‘—π‘˜π‘¦π‘—,(5.25) that is, 𝑦𝑖|π‘˜=π‘Žπ‘–π‘˜+πΆπ‘–π‘˜π‘—π‘¦π‘— as πΆπ‘–π‘—π‘˜ is totally symmetric.
(3) Utilizing successively 𝛿𝑖=πœ•π‘–βˆ’π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ, (4.5), and βˆ˜π›Ώπ‘–=πœ•π‘–βˆ’βˆ˜π‘π‘Ÿπ‘–Μ‡πœ•π‘Ÿ in 𝐿|π‘˜=π›Ώπ‘˜πΏ, we get 𝐿|π‘˜=βˆ˜π›Ώπ‘˜πΏ+π‘‰π‘Ÿπ‘˜Μ‡πœ•π‘ŸπΏ.(5.26) Using (1.2) and (2.1) in (5.26), we have 𝐿|π‘˜=βˆ˜π›Ώπ‘˜ξ€·πœ‘+π΄π‘Ÿπ‘¦π‘Ÿξ€Έ+π‘ˆ+π‘‰π‘Ÿπ‘˜ξ‚΅2πœ‘β€²βˆ˜π‘¦π‘Ÿ+π΄π‘Ÿξ‚Ά,(5.27) which, in view of (5.3), gives 𝐿|π‘˜=βˆ˜π›Ώπ‘˜ξ€·π΄π‘Ÿπ‘¦π‘Ÿξ€Έ+π‘ˆ+π‘‰π‘Ÿπ‘˜ξ‚΅2πœ‘β€²βˆ˜π‘¦π‘Ÿ+π΄π‘Ÿξ‚Ά.(5.28) Using βˆ˜π›Ώπ‘˜=πœ•π‘˜βˆ’βˆ˜π‘π‘Ÿπ‘˜Μ‡πœ•π‘Ÿ and (5.18) in (5.28) and keeping (4.5) in view, we find 𝐿|π‘˜=π‘¦π‘Ÿπœ•π‘˜π΄π‘Ÿβˆ’π‘π‘π‘˜π΄π‘+πœ•π‘˜π‘ˆ+2πœ‘β€²πœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘‰π‘Ÿπ‘˜π‘¦π‘Ÿ.(5.29) If we take π‘‹π‘˜=π‘¦π‘Ÿπœ•π‘˜π΄π‘Ÿβˆ’π‘π‘π‘˜π΄π‘+πœ•π‘˜π‘ˆ, then the last expression takes the form 𝐿|π‘˜=π‘‹π‘˜+2πœ‘β€²πœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘‰π‘Ÿπ‘˜π‘¦π‘Ÿ.(5.30) Next, using (2.1) in 𝐿|π‘˜=Μ‡πœ•π‘˜πΏ, we get 𝐿|π‘˜=2πœ‘β€²βˆ˜π‘¦π‘˜+π΄π‘˜,(5.31) which, in view of (5.18), gives the required result.

Corollary 5.5 (see [14]). The canonical metrical 𝑑-connection 𝐢Γ(𝑁) of a πœ‘-Lagrange space has the following properties: (1)π·π‘–π‘˜βˆΆ=𝑦𝑖|π‘˜=0,𝑦𝑖|π‘˜=0,(5.32)(2)π‘‘π‘–π‘˜βˆΆ=𝑦𝑖|π‘˜=πœ‘β€²+πœ‘ξ…žξ…žπΉ2π›Ώπœ‘β€²π‘–π‘˜+π΅βˆ˜π‘¦π‘˜π‘¦π‘–,𝑦𝑖|π‘˜=π‘Žπ‘–π‘˜+πΆπ‘–π‘˜π‘—π‘¦π‘—,(5.33) where 𝐡=2{πœ‘ξ…žπœ‘ξ…žξ…ž+𝐹2(πœ‘ξ…žξ…žξ…žπœ‘ξ…žβˆ’πœ‘ξ…žξ…ž2)}/πœ‘β€²(πœ‘β€²+2𝐹2πœ‘ξ…žξ…ž), (3)𝐿|π‘˜=0,𝐿|π‘˜=2πœ‘β€²ξ€·πœ‘β€²+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘˜.(5.34)

Proof. Applying 𝐴𝑖(π‘₯)=0, π‘ˆ(π‘₯)=0, and 𝑉𝑖𝑗=0 in Theorem 5.4, we have the corollary.

Corollary 5.6. The canonical metrical 𝑑-connection 𝐢Γ(𝑁) of an AFL-space has the following properties: (1)π·π‘–π‘˜=π΅π‘–π‘˜+π΅π‘π‘Ÿβˆ˜πΆπ‘–π‘˜π‘π‘¦π‘Ÿ,𝑦𝑖|π‘˜=π‘”π‘ π‘–ξ‚΅π΅π‘ π‘˜βˆ’π΅π‘™π‘π‘¦π‘βˆ˜πΆπ‘ π‘™π‘˜ξ‚Ά,(5.35) where 𝑦𝑖=𝑔𝑖𝑗𝑦𝑗, (2)π‘‘π‘–π‘˜=π›Ώπ‘–π‘˜,𝑦𝑖|π‘˜=π‘”π‘–π‘˜,(5.36)(3)𝐿|π‘˜=π‘¦π‘Ÿπœ•π‘˜π΄π‘Ÿβˆ’π‘π‘π‘˜π΄π‘+πœ•π‘˜π‘ˆ+2π΅π‘Ÿπ‘˜π‘¦π‘Ÿ,𝐿|π‘˜=2π‘¦π‘˜+π΄π‘˜.(5.37)

Proof. Using πœ‘(𝐹2)=𝐹2,πœ‘β€²(𝐹2)=1,πœ‘ξ…žξ…ž(𝐹2)=0=πœ‘ξ…žξ…žξ…ž(𝐹2),π‘Žπ‘–π‘—=𝑔𝑖𝑗,πΆπ‘–π‘—π‘˜=βˆ˜πΆπ‘–π‘—π‘˜,βˆ˜πΆπ‘–π‘—π‘˜π‘¦π‘—=0,βˆ˜πΆπ‘–π‘—π‘˜π‘¦π‘˜=0, and π‘‰π‘Ÿπ‘—=π΅π‘Ÿπ‘— 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.

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