International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 505161 | https://doi.org/10.5402/2011/505161

P. N. Pandey, Suresh K. Shukla, "On Almost š‹-Lagrange Spaces", International Scholarly Research Notices, vol. 2011, Article ID 505161, 16 pages, 2011. https://doi.org/10.5402/2011/505161

On Almost š‹-Lagrange Spaces

Academic Editor: M. Margenstern
Received12 Oct 2011
Accepted13 Nov 2011
Published27 Dec 2011

Abstract

We initiate a study on the geometry of an almost š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 ā„Ž- and š‘£-deflection tensors in an APL-space. Corresponding expressions in a š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 [3ā€“13]. Antonelli and Hrimiuc [14, 15] introduced a special type of regular Lagrangian called š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 šœ‘-Lagrangian and introduce the concept of almost š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 š¹š‘›=(š‘€,š¹(š‘„,š‘¦)) be an š‘›-dimensional Finsler space, and let šœ‘āˆ¶ā„+ā†’ā„ be a smooth function. The composition šæāˆ¶=šœ‘(š¹2) defines a differentiable Lagrangian. This was regarded by Antonelli and Hrimiuc [14, 15] as šœ‘-Lagrangian associated to the Finsler space š¹š‘›. They [14] proved that if the function šœ‘ has the following properties:(a)šœ‘ī…ž((š‘”)ā‰ 0,b)šœ‘ī…ž(š‘”)+šœ‘ī…žī…ž(š‘”)ā‰ 0,foreveryī€·š¹š‘”āˆˆIm2ī€ø,(1.1) then šæ is a regular Lagrangian and thus šæš‘›=(š‘€,šæ(š‘„,š‘¦)) is a Lagrange space, called a šœ‘-Lagrange space.

In this paper, we consider a more general Lagrangian as follows:ī€·š¹šæ(š‘„,š‘¦)=šœ‘2ī€ø+š“š‘–(š‘„)š‘¦š‘–+š‘ˆ(š‘„),(1.2) where šœ‘ is the same as discussed earlier, š“š‘–(š‘„) is a covector, and š‘ˆ(š‘„) is a smooth function.

In Section 2, we show that if the function šœ‘ has the properties (1.1), then šæ(š‘„,š‘¦) is a regular Lagrangian and thus the pair šæš‘›=(š‘€,šæ(š‘„,š‘¦)) is a Lagrange space. We call this space as an almost šœ‘-Lagrange space (shortly APL-space).

An APL-space reduces to a šœ‘-Lagrange space if and only if š“š‘–(š‘„)=0 and š‘ˆ(š‘„)=0.

If šœ‘(š‘”)=š‘”,forallš‘”āˆˆIm(š¹2), then the Lagrangian in (1.2) takes the formšæ(š‘„,š‘¦)=š¹2+š“š‘–(š‘„)š‘¦š‘–+š‘ˆ(š‘„).(1.3) This defines a regular Lagrangian, and the pair šæš‘›=(š‘€,šæ(š‘„,š‘¦)) 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 takeš‘”š‘–š‘—=12Ģ‡šœ•š‘–Ģ‡šœ•š‘—š¹2,š‘Žš‘–š‘—=12Ģ‡šœ•š‘–Ģ‡šœ•š‘—Ģ‡šœ•šæ,š‘–ā‰”šœ•šœ•š‘¦š‘–.(1.4) Henceforth, we will indicate all the geometrical objects related to š¹š‘› by a small circle ā€œāˆ˜ā€ 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]:āˆ˜š‘š‘–š‘—=āˆ˜š›¾š‘–š‘—0āˆ’āˆ˜š¶š‘–āˆ˜š‘—š‘˜š›¾š‘˜00,(1.5) whereāˆ˜š›¾š‘–š‘—š‘˜=12š‘”š‘–ā„Žī€·šœ•š‘—š‘”ā„Žš‘˜+šœ•š‘˜š‘”š‘—ā„Žāˆ’šœ•ā„Žš‘”š‘—š‘˜ī€ø;šœ•š‘—ā‰”šœ•šœ•š‘„š‘—,āˆ˜š¶š‘–š‘—š‘˜=12š‘”š‘–ā„ŽĢ‡šœ•ā„Žš‘”š‘—š‘˜,āˆ˜š›¾š‘–š‘—0=āˆ˜š›¾š‘–š‘—š‘˜š‘¦š‘˜,āˆ˜š›¾š‘–00=āˆ˜š›¾š‘–š‘—š‘˜š‘¦š‘—š‘¦š‘˜.(1.6) These geodesics are the integral curves of the spray [16] (i.e., (2) p-homogeneous):āˆ˜šŗš‘–=14š‘”š‘–š‘—ī€·š‘¦š‘˜Ģ‡šœ•š‘—šœ•š‘˜š¹2āˆ’šœ•š‘—š¹2ī€ø,(1.7) that is, solutions of the differential equationsš‘‘2š‘„š‘–š‘‘š‘ 2+2āˆ˜šŗš‘–ī‚€š‘„(š‘ ),š‘‘š‘„ī‚š‘‘š‘ =0.(1.8) We have the following equalities:(a)āˆ˜šŗš‘–=12āˆ˜š›¾š‘–00,(b)āˆ˜š‘š‘–š‘—=Ģ‡šœ•š‘—āˆ˜šŗš‘–.(1.9) In a general Lagrange space šæš‘›=(š‘€,šæ(š‘„,š‘¦)), 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):šŗš‘–=14š‘Žš‘–š‘—ī€·š‘¦š‘˜Ģ‡šœ•š‘—šœ•š‘˜šæāˆ’šœ•š‘—šæī€ø.(1.10) As in a Finsler space, a remarkable nonlinear connection can be considered in a Lagrange space:š‘š‘–š‘—=Ģ‡šœ•š‘—šŗš‘–.(1.11) Such nonlinear connection is a canonical nonlinear connection [17, 18] as it depends only on the fundamental function šæ(š‘„,š‘¦) of the Lagrange space.

In general, the autoparallel curves of (š‘š‘–š‘—) are different from the geodesics of šæš‘›=(š‘€,šæ(š‘„,š‘¦)) (cf. [17]).

Given a nonlinear connection (š‘š‘–š‘—) on a Lagrange space šæš‘›=(š‘€,šæ(š‘„,š‘¦)), there is a unique ā„Ž- and š‘£-metrical š‘‘-connection (cf. [17, 19]) š¶Ī“(š‘)=(š‘š‘–š‘—,šæš‘–š‘—š‘˜,š¶š‘–š‘—š‘˜) with torsions š‘‡š‘–š‘—š‘˜=0 and š‘†š‘–š‘—š‘˜=0, called the canonical metrical š‘‘-connection. This connection is linear and its coefficients are given byšæš‘–š‘—š‘˜=12š‘Žš‘–ā„Žī€·š›æš‘—š‘Žā„Žš‘˜+š›æš‘˜š‘Žš‘—ā„Žāˆ’š›æā„Žš‘Žš‘—š‘˜ī€ø,(1.12)š¶š‘–š‘—š‘˜=12š‘Žš‘–ā„Žī€·Ģ‡šœ•š‘—š‘Žā„Žš‘˜+Ģ‡šœ•š‘˜š‘Žš‘—ā„Žāˆ’Ģ‡šœ•ā„Žš‘Žš‘—š‘˜ī€ø,(1.13) where š›æš‘–=šœ•š‘–āˆ’š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ is the Lagrange differentiation operator.

If āˆ˜š¶Ī“(āˆ˜š‘)=(āˆ˜š‘š‘–š‘—,āˆ˜šæš‘–š‘—š‘˜,āˆ˜š¶š‘–š‘—š‘˜) is the Cartan connection of the Finsler space š¹š‘›=(š‘€,š¹(š‘„,š‘¦)), then its coefficients are given byāˆ˜šæš‘–š‘—š‘˜=12š‘”š‘–ā„Žī‚µāˆ˜š›æš‘—š‘”ā„Žš‘˜+āˆ˜š›æš‘˜š‘”š‘—ā„Žāˆ’āˆ˜š›æā„Žš‘”š‘—š‘˜ī‚¶,(1.14)āˆ˜š¶š‘–š‘—š‘˜=12š‘”š‘–ā„Žī€·Ģ‡šœ•š‘—š‘”ā„Žš‘˜+Ģ‡šœ•š‘˜š‘”š‘—ā„Žāˆ’Ģ‡šœ•ā„Žš‘”š‘—š‘˜ī€ø,(1.15) where āˆ˜š›æš‘–=šœ•š‘–āˆ’āˆ˜š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ.

The ā„Ž- and š‘£-deflection tensor fields š·š‘–š‘— and š‘‘š‘–š‘—, respectively, of a Lagrange space šæš‘› are defined by (cf. [19])š·š‘–š‘—āˆ¶=š‘¦š‘–|š‘—=š‘¦š‘ šæš‘–š‘ š‘—āˆ’š‘š‘–š‘—,(1.16)š‘‘š‘–š‘—āˆ¶=š‘¦š‘–|š‘—=š›æš‘–š‘—+š‘¦š‘ š¶š‘–š‘ š‘—,(1.17) where | and |, respectively, denote the ā„Ž- and š‘£-covariant derivatives with respect to š¶Ī“.

If āˆ˜š·š‘–š‘— is the h-deflection tensor field and āˆ˜š‘‘š‘–š‘— is the š‘£-deflection tensor field of the Finsler space š¹š‘›, thenāˆ˜š·š‘–š‘—āˆ¶=š‘¦š‘–āˆ˜|š‘—=š‘¦š‘ āˆ˜šæš‘–š‘ š‘—āˆ’āˆ˜š‘š‘–š‘—=0,(1.18)āˆ˜š‘‘š‘–š‘—āˆ¶=š‘¦š‘–āˆ˜|||š‘—=š›æš‘–š‘—,(1.19) where āˆ˜| and āˆ˜|, respectively, denote the ā„Ž- and š‘£-covariant derivatives with respect to āˆ˜š¶Ī“.

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

2. Almost šœ‘-Lagrange Spaces

As discussed earlier, we consider the Lagrangian given by (1.2) in which the function šœ‘ satisfies (1.1). We prove that it is a regular Lagrangian and the pair šæš‘›=(š‘€,šæ(š‘„,š‘¦)) is a Lagrange space which we term as an almost šœ‘-Lagrange space (APL-space in short).

Theorem 2.1. If the function šœ‘ satisfies the conditions (1.1), then šæ(š‘„,š‘¦), given by (1.2), is a regular Lagrangian and šæš‘›=(š‘€,šæ(š‘„,š‘¦)) is a Lagrange space.

Proof. Differentiating (1.2) partially with respect to š‘¦š‘–, we get Ģ‡šœ•š‘–šæ=šœ‘ī…žī€·š¹2ī€øĢ‡šœ•š‘–š¹2+š“š‘–(š‘„).(2.1) Again differentiating (2.1) partially with respect to š‘¦š‘—, we obtain Ģ‡šœ•š‘—Ģ‡šœ•š‘–šæ=šœ‘ī…žī…žī€·š¹2ī€øĢ‡šœ•š‘–š¹2Ģ‡šœ•š‘—š¹2+šœ‘ī…žī€·š¹2ī€øĢ‡šœ•š‘–Ģ‡šœ•š‘—š¹2,(2.2) which, in view of (1.4), provides š‘Žš‘–š‘—=2š¹2šœ‘ī…žī…žī€·š¹2ī€øĢ‡šœ•š‘–š¹Ģ‡šœ•š‘—š¹+šœ‘ī…žī€·š¹2ī€øš‘”š‘–š‘—.(2.3) Now š¹Ģ‡šœ•š‘–1š¹=2Ģ‡šœ•š‘–š¹2=12Ģ‡šœ•š‘–ī€·š‘”š‘—š‘˜š‘¦š‘—š‘¦š‘˜ī€ø=š‘”š‘–š‘˜š‘¦š‘˜āˆ¶=āˆ˜š‘¦š‘–.(2.4) In view of (2.4), (2.3) takes the form š‘Žš‘–š‘—=šœ‘ī…žā‹…ī‚µš‘”š‘–š‘—+2šœ‘ī…žī…žšœ‘ī…žāˆ˜š‘¦š‘–āˆ˜š‘¦š‘—ī‚¶.(2.5) Under the hypothesis, the matrix (š‘Žš‘–š‘—) is invertible and its inverse is (see Lemma 6.2.2.1, page 891 in [20]) š‘Žš‘–š‘—=1ī‚µš‘”šœ‘ā€²š‘–š‘—āˆ’2šœ‘ī…žī…žšœ‘ī…ž+2š¹2šœ‘ī…žī…žš‘¦š‘–š‘¦š‘—ī‚¶.(2.6) This proves the theorem.

Remarks 1. (i) If š“š‘–(š‘„)=0 and š‘ˆ(š‘„)=0 in (1.2), then expression (2.5) remains unchanged. Hence, the symmetric metric tensor of a šœ‘-Lagrange space is the same as that of an APL-space.
(ii) If šœ‘(š¹2)=š¹2, then šœ‘ī…ž=1 and šœ‘ī…žī…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 šæš‘›=(š‘€,šæ(š‘„,š‘¦)) and deduce corresponding expressions for a š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 šœ‘-Lagrange space and an AFL-space.

If we differentiate (1.2) partially with respect to š‘„š‘˜, we havešœ•š‘˜šæ=šœ‘ī…žī€·š¹2ī€øšœ•š‘˜š¹2+š‘¦š‘–šœ•š‘˜š“š‘–(š‘„)+šœ•š‘˜š‘ˆ(š‘„).(3.1) Differentiating (3.1) partially with respect to š‘¦š‘—, we obtainĢ‡šœ•š‘—šœ•š‘˜šæ=2šœ‘ī…žī…žī€·š¹2ī€øš¹Ģ‡šœ•š‘—š¹šœ•š‘˜š¹2+šœ‘ī…žī€·š¹2ī€øĢ‡šœ•š‘—šœ•š‘˜š¹2+šœ•š‘˜š“š‘—(š‘„),(3.2) which, in view of (2.4), takes the formĢ‡šœ•š‘—šœ•š‘˜šæ=2šœ‘ī…žī…žī€·š¹2ī€øāˆ˜š‘¦š‘—šœ•š‘˜š¹2+šœ‘ī…žī€·š¹2ī€øĢ‡šœ•š‘—šœ•š‘˜š¹2+šœ•š‘˜š“š‘—(š‘„).(3.3) Using (3.1) and (3.3) in (1.10), we havešŗš‘–=14š‘Žš‘–š‘—ī‚»2šœ‘ī…žī…žī€·š¹2ī€øāˆ˜š‘¦š‘—š‘¦š‘˜šœ•š‘˜š¹2+šœ‘ī…žī€·š¹2š‘¦ī€øī€·š‘˜Ģ‡šœ•š‘—šœ•š‘˜š¹2āˆ’šœ•š‘—š¹2ī€øāˆ’2š‘¦š‘˜š¹š‘—š‘˜āˆ’šœ•š‘—š‘ˆī‚¼,(3.4) whereš¹š‘—š‘˜1(š‘„)=2ī€·šœ•š‘—š“š‘˜āˆ’šœ•š‘˜š“š‘—ī€ø(3.5) is electromagnetic tensor field of the potentials š“š‘–(š‘„).

Applying (2.6) in (3.4) and using āˆ˜š‘¦š‘–š‘¦š‘–=š¹2,š‘”āˆ˜š‘–š‘—š‘¦š‘—=š‘¦š‘–, and š‘¦š‘—Ģ‡šœ•š‘—šœ•š‘˜š¹2=2šœ•š‘˜š¹2 (by Eulerā€™s theorem on homogeneous functions), we obtainšŗš‘–=12šœ‘ī…žī…žšœ‘ī…žī‚µ1āˆ’2šœ‘ī…žī…žš¹2šœ‘ī…ž+2š¹2šœ‘ī…žī…žī‚¶š‘¦š‘–š‘¦š‘˜šœ•š‘˜š¹2+14ī‚»š‘”š‘–š‘—ī€·š‘¦š‘˜Ģ‡šœ•š‘—šœ•š‘˜š¹2āˆ’šœ•š‘—š¹2ī€øāˆ’2šœ‘ī…žī…žš¹2šœ‘ā€²+2š¹2šœ‘ī…žī…žš‘¦š‘–š‘¦š‘˜šœ•š‘˜š¹2ī‚¼āˆ’14š‘Žš‘–š‘—ī€·2š¹š‘—š‘˜š‘¦š‘˜+šœ•š‘—š‘ˆī€ø.(3.6) Using (1.7) in (3.6) and simplifying, we getšŗš‘–=āˆ˜šŗš‘–āˆ’14š‘Žš‘–š‘—ī€·2š¹š‘—š‘˜š‘¦š‘˜+šœ•š‘—š‘ˆī€ø.(3.7) Thus, we have the following.

Theorem 3.1. The canonical semispray of an APL-space has the local coefficients given by šŗš‘–=āˆ˜šŗš‘–āˆ’14š‘Žš‘–š‘—ī€·2š¹š‘—š‘˜š‘¦š‘˜+šœ•š‘—š‘ˆī€ø,(3.8) where āˆ˜šŗš‘– are the local coefficients of the spray of š¹š‘›.

For a šœ‘-Lagrange space, š“š‘–(š‘„)=0 and š‘ˆ(š‘„)=0. Hence, from (3.5), we have š¹š‘—š‘˜=0. Therefore, (3.7) reduces tošŗš‘–=āˆ˜šŗš‘–.(3.9) Thus, we may state the following.

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

For an AFL-space, š‘Žš‘–š‘—=š‘”š‘–š‘— (see Remark (ii)). Hence, (3.7) takes the formšŗš‘–=āˆ˜šŗš‘–āˆ’14š‘”š‘–š‘—ī€·2š¹š‘—š‘˜š‘¦š‘˜+šœ•š‘—š‘ˆī€ø.(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:šøš‘–(šæ)āˆ¶=šœ•š‘–š‘‘šæāˆ’ī€·Ģ‡šœ•š‘‘š‘”š‘–šæī€ø=0(3.11) are the solution curves of the equations [20]š‘‘2š‘„š‘–š‘‘š‘”2+2šŗš‘–(š‘„,š‘¦)=0.(3.12) Using (3.7) in (3.12), we obtainš‘‘2š‘„š‘–š‘‘š‘”2+2āˆ˜šŗš‘–=12ī€·2š¹š‘–š‘˜š‘¦š‘˜+š‘Žš‘–š‘—šœ•š‘—š‘ˆī€ø,(3.13) where š¹š‘–š‘˜=š‘Žš‘–š‘—š¹š‘—š‘˜.

Using (1.9) (a) in (3.13), we haveš‘‘2š‘„š‘–š‘‘š‘”2+āˆ˜š›¾š‘–00=12ī€·2š¹š‘–š‘˜š‘¦š‘˜+š‘Žš‘–š‘—šœ•š‘—š‘ˆī€ø.(3.14) Thus, we have the following.

Theorem 3.4. In an APL-space šæš‘›=(š‘€,šæ(š‘„,š‘¦)), the integral curves of the Euler-Lagrange equations šøš‘–(šæ)=0 are the solution curves of (3.14).

For a šœ‘-Lagrange space, equations (3.14) take the following simple form:š‘‘2š‘„š‘–š‘‘š‘”2+āˆ˜š›¾š‘–00=0.(3.15) This enables us to state the following.

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

For an AFL-space, š‘Žš‘–š‘—=š‘”š‘–š‘—. Therefore, equations (3.14) becomeš‘‘2š‘„š‘–š‘‘š‘”2+āˆ˜š›¾š‘–00=š¹š‘–š‘˜š‘¦š‘˜+12š‘”š‘–š‘—šœ•š‘—š‘ˆ,(3.16) where š¹š‘–š‘˜=š‘”š‘–š‘—š¹š‘—š‘˜.

Thus, we have the following.

Corollary 3.6 (see [17, 20]). In an AFL-space, the integral curves of the Euler-Lagrange equations šøš‘–(šæ)=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 šœ‘-Lagrange space and an AFL-space.

Partial differentiation of (2.5) with respect to š‘¦š‘˜ yieldsĢ‡šœ•š‘˜š‘Žš‘–š‘—=āˆ¶2š¶š‘–š‘—š‘˜=2šœ‘ā€²āˆ˜š¶š‘–š‘—š‘˜+2šœ‘ī…žī…žī‚µš‘”āˆ˜š‘–š‘—š‘¦š‘˜+š‘”āˆ˜š‘—š‘˜š‘¦š‘–+š‘”āˆ˜š‘˜š‘–š‘¦š‘—ī‚¶+4šœ‘āˆ˜ī…žī…žī…žš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜.(4.1) Using (3.7) in (1.11) and taking (1.9) (b), (2.6), (4.1), āˆ˜š¶š‘š‘žš‘—š‘¦š‘—=0, š‘¦š‘–āˆ˜š‘¦š‘–=š¹2, and š‘”āˆ˜š‘–š‘—š‘¦š‘—=š‘¦š‘– into account, we obtainš‘š‘–š‘—=āˆ˜š‘š‘–š‘—āˆ’12š¹š‘–š‘—+īƒ¬12šœ‘ī…žāˆ˜š¶š‘–š‘žš‘—š‘”š‘žš‘Ÿ+12šœ‘ī…žī…žšœ‘ī…ž2š‘”āˆ˜š‘–š‘Ÿš‘¦š‘—+šœ‘ī…žī…ž2šœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€øī€·š›æš‘Ÿš‘—š‘¦š‘–+š›æš‘–š‘—š‘¦š‘Ÿī€ø+šœ‘ī…ž2šœ‘ī…žī…žī…žāˆ’2šœ‘3ī…žī…žš¹2āˆ’4šœ‘ī…žšœ‘ī…žī…ž22šœ‘ī…ž2ī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€ø2š‘¦š‘–āˆ˜š‘¦š‘—š‘¦š‘Ÿīƒ­ī€·2š¹š‘Ÿš‘˜š‘¦š‘˜+šœ•š‘Ÿš‘ˆī€ø.(4.2) If we takeš‘†š‘—š‘–š‘Ÿ=12šœ‘ī…žāˆ˜š¶š‘–š‘žš‘—š‘”š‘žš‘Ÿ+12šœ‘ī…žī…žšœ‘ī…ž2š‘”āˆ˜š‘–š‘Ÿš‘¦š‘—+šœ‘ī…žī…žī€·š›æš‘Ÿš‘—š‘¦š‘–+š›æš‘–š‘—š‘¦š‘Ÿī€ø2šœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€ø+šœ‘ī…ž2šœ‘ī…žī…žī…žāˆ’2šœ‘ī…žī…ž3š¹2āˆ’4šœ‘ā€²šœ‘ī…žī…ž22šœ‘ī…ž2ī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€ø2š‘¦š‘–āˆ˜š‘¦š‘—š‘¦š‘Ÿ,(4.3) the last expression becomesš‘š‘–š‘—=āˆ˜š‘š‘–š‘—āˆ’12š¹š‘–š‘—+š‘†š‘—š‘–š‘Ÿī€·2š¹š‘Ÿš‘˜š‘¦š‘˜+šœ•š‘Ÿš‘ˆī€ø,(4.4) that is,š‘š‘–š‘—=āˆ˜š‘š‘–š‘—āˆ’š‘‰š‘–š‘—,(4.5) whereš‘‰š‘–š‘—=12š¹š‘–š‘—āˆ’š‘†š‘—š‘–š‘Ÿī€·2š¹š‘Ÿš‘˜š‘¦š‘˜+šœ•š‘Ÿš‘ˆī€ø.(4.6) Thus, we have the following.

Theorem 4.1. The canonical nonlinear connection of an APL-space šæš‘› has the local coefficients given by (4.5).

For a šœ‘-Lagrange space, we have š¹š‘Ÿš‘˜=0,š¹š‘–š‘—=0 and š‘ˆ=0 and hence š‘‰š‘–š‘—=0. Therefore, (4.5) reduces toš‘š‘–š‘—=āˆ˜š‘š‘–š‘—.(4.7) Thus, we have the following.

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

For an AFL-space, (4.3) reduces toš‘†š‘—š‘–š‘Ÿ=12āˆ˜š¶š‘–š‘žš‘—š‘”š‘žš‘Ÿ(4.8) and hence (4.6) givesš‘‰š‘–š‘—=12š¹š‘–š‘—āˆ’āˆ˜š¶š‘–š‘žš‘—š¹š‘žš‘˜š‘¦š‘˜āˆ’12āˆ˜š¶š‘–š‘žš‘—š‘”š‘žš‘Ÿšœ•š‘Ÿš‘ˆāˆ¶=šµš‘–š‘—.(4.9) Therefore, (4.5) takes the formš‘š‘–š‘—=āˆ˜š‘š‘–š‘—āˆ’šµš‘–š‘—.(4.10) Thus, we have the following.

Corollary 4.3 (see [17, 20]). The canonical nonlinear connection of an AFL-space šæš‘› has the local coefficients given by (4.10).

Transvecting (4.5) by š‘¦š‘– and using āˆ˜š‘š‘–š‘—š‘¦š‘—=āˆ˜š›¾š‘–00, we obtainš‘š‘–š‘—š‘¦š‘—=āˆ˜š›¾š‘–00āˆ’š‘‰š‘–0,(4.11) where š‘‰š‘–0=š‘‰š‘–š‘—š‘¦š‘—.

The autoparallel curves of the canonical nonlinear connection š‘=(š‘š‘–š‘—) of a Lagrange space are given by the following system of differential equations (vide [20]):š‘‘2š‘„š‘–š‘‘š‘”2+š‘š‘–š‘—(š‘„,š‘¦)š‘¦š‘—=0.(4.12) Equations (4.12), in view of (4.11), take the formš‘‘2š‘„š‘–š‘‘š‘”2+āˆ˜š›¾š‘–00=š‘‰š‘–0.(4.13) Thus, we have the following.

Theorem 4.4. The autoparallel curves of the canonical nonlinear connection š‘=(š‘š‘–š‘—) of an APL-space šæš‘›=(š‘€,šæ(š‘„,š‘¦)) are given by the system of differential equations (4.13).

For a šœ‘-Lagrange space, š‘‰š‘–š‘—=0 and hence š‘‰š‘–0=0. Therefore, (4.13) reduces toš‘‘2š‘„š‘–š‘‘š‘”2+āˆ˜š›¾š‘–00=0.(4.14) Thus, we have the following.

Corollary 4.5 (see [14]). The autoparallel curves of the canonical nonlinear connection of a šœ‘-Lagrange space šæš‘›=(š‘€,šæ(š‘„,š‘¦)) are given by the system of differential equations (4.14).

For an AFL-space,š‘‰š‘–š‘—=šµš‘–š‘—1=āˆ¶2š¹š‘–š‘—āˆ’āˆ˜š¶š‘–š‘žš‘—š¹š‘žš‘˜š‘¦š‘˜āˆ’12āˆ˜š¶š‘–š‘žš‘—š‘”š‘žš‘Ÿšœ•š‘Ÿš‘ˆ(4.15) and hence, by virtue of āˆ˜š¶š‘–š‘žš‘—š‘¦š‘—=0, we have š‘‰š‘–0=(1/2)š¹š‘–š‘—š‘¦š‘—. Therefore, equations (4.12) take the formš‘‘2š‘„š‘–š‘‘š‘”2+āˆ˜š›¾š‘–00=12š¹š‘–š‘—š‘¦š‘—.(4.16) Thus, we deduce the following.

Corollary 4.6 (see [17, 20]). The autoparallel curves of the nonlinear connection š‘=(š‘š‘–š‘—) of an AFL-space šæš‘›=(š‘€,šæ(š‘„,š‘¦)) 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 šœ‘-Lagrange space they do. Therefore, in a šœ‘-Lagrange space, geodesics are autoparallel curves whereas in an APL-space and in an AFL-space they are not so.

5. Canonical Metrical š‘‘-Connection

Let š¶Ī“(š‘)=(š‘š‘–š‘—,šæš‘–š‘—š‘˜,š¶š‘–š‘—š‘˜) be the canonical metrical š‘‘-connection of the APL-space šæš‘›=(š‘€,šæ(š‘„,š‘¦)), and let āˆ˜š¶Ī“(āˆ˜š‘)=(āˆ˜š‘š‘–š‘—,āˆ˜šæš‘–š‘—š‘˜,āˆ˜š¶š‘–š‘—š‘˜) be the Cartan connection of the associated Finsler space š¹š‘›=(š‘€,š¹(š‘„,š‘¦)). In this section, we obtain the expressions for the coefficients of š¶Ī“(š‘) and we investigate some properties of š¶Ī“(š‘). We deduce corresponding results for a šœ‘-Lagrange space and an AFL-space.

Using (4.1) in (1.13) and taking (1.15) into account, we findš¶š‘–š‘—š‘˜=āˆ˜š¶š‘–š‘—š‘˜+šœ‘ī…žī…žī‚µš›æšœ‘ā€²š‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚¶+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜.(5.1)

For any š¶āˆž-class function šœ“āˆ¶ā„+ā†’ā„, taking š‘“(š‘„,š‘¦)=šœ“(š¹2(š‘„,š‘¦)), we haveāˆ˜š›æš‘˜š‘“=š‘“ā€²š¹2āˆ˜|š‘˜(5.2) which, in view of š¹2āˆ˜|š‘˜=0 (see proposition 9.4, page 1037 of [20]), givesāˆ˜š›æš‘˜š‘“=0.(5.3) Since 0=š‘¦š‘–āˆ˜|š‘˜āˆ¶=āˆ˜š›æš‘˜āˆ˜š‘¦š‘–āˆ’āˆ˜šæš‘Ÿāˆ˜š‘–š‘˜š‘¦š‘Ÿ (see proposition 9.4, page 1037 of [20]), we haveāˆ˜š›æš‘˜āˆ˜š‘¦š‘–=āˆ˜šæš‘Ÿāˆ˜š‘–š‘˜š‘¦š‘Ÿ.(5.4) If we operate āˆ˜š›æš‘˜ on (2.5) and utilize (5.3) and (5.4), it follows thatāˆ˜š›æš‘˜š‘Žš‘–š‘—=šœ‘ā€²āˆ˜š›æš‘˜š‘”š‘–š‘—+2šœ‘āˆ˜ī…žī…žš‘¦š‘Ÿī‚µāˆ˜šæš‘Ÿāˆ˜š‘–š‘˜š‘¦š‘—+āˆ˜šæš‘Ÿāˆ˜š‘—š‘˜š‘¦š‘–ī‚¶.(5.5) In view of š›æš‘–=šœ•š‘–āˆ’š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ, (4.5), and āˆ˜š›æš‘–=šœ•š‘–āˆ’āˆ˜š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ, we getš›æš‘˜š‘Žš‘–š‘—=āˆ˜š›æš‘˜š‘Žš‘–š‘—+š‘‰š‘Ÿš‘˜Ģ‡šœ•š‘Ÿš‘Žš‘–š‘—,(5.6) which, on account of (4.1) and (5.5), becomesš›æš‘˜š‘Žš‘–š‘—=šœ‘ā€²āˆ˜š›æš‘˜š‘”š‘–š‘—+2šœ‘āˆ˜ī…žī…žš‘¦š‘Ÿī‚µāˆ˜šæš‘Ÿāˆ˜š‘–š‘˜š‘¦š‘—+āˆ˜šæš‘Ÿāˆ˜š‘—š‘˜š‘¦š‘–ī‚¶+2š‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ.(5.7) Using (5.7) in (1.12) and taking (1.14) and š‘Žš‘–š‘™š¶š‘—š‘˜š‘™=š¶š‘–š‘—š‘˜ into account, we obtainšæš‘–š‘—š‘˜=āˆ˜šæš‘–š‘—š‘˜+š‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—.(5.8) Equations (5.1) and (5.8) enable us to state the following.

Theorem 5.1. The coefficients of the canonical metrical š‘‘-connection š¶Ī“(š‘) of an APL-space šæš‘› are given by (5.1) and (5.8).

For a šœ‘-Lagrange space, š‘‰š‘–š‘—=0. Hence, (5.1) remains unchanged whereas (5.8) reduces tošæš‘–š‘—š‘˜=āˆ˜šæš‘–š‘—š‘˜.(5.9) Thus, we have the following.

Corollary 5.2 (see [14]). The coefficients of the canonical metrical š‘‘-connection š¶Ī“(š‘) of a šœ‘-Lagrange space šæš‘› are given by (5.1) and (5.9).

For an AFL-space, šœ‘(š¹2)=š¹2,šœ‘ā€²(š¹2)=1,šœ‘ī…žī…ž(š¹2)=0, and š‘Žš‘–š‘—=š‘”š‘–š‘—. Therefore, we have š¶š‘–š‘—š‘˜=āˆ˜š¶š‘–š‘—š‘˜ and š‘‰š‘Ÿš‘—=šµš‘Ÿš‘—.

In view of these facts, (5.1) reduces toš¶š‘–š‘—š‘˜=āˆ˜š¶š‘–š‘—š‘˜,(5.10) whereas (5.8) gives the following:šæš‘–š‘—š‘˜=āˆ˜šæš‘–š‘—š‘˜+šµš‘Ÿš‘˜āˆ˜š¶š‘–š‘—š‘Ÿ+šµš‘Ÿš‘—āˆ˜š¶š‘–š‘˜š‘Ÿ+šµš‘Ÿš‘š‘”āˆ˜š‘–š‘š¶š‘Ÿš‘˜š‘—,(5.11) where šµš‘Ÿš‘˜ is given by (4.9). Thus, we have the following.

Corollary 5.3 (see [17, 20]). The coefficients of the canonical metrical š‘‘-connection š¶Ī“(š‘) of an AFL-space šæš‘› are given by (5.10) and (5.11).

Now, we investigate some properties of the canonical metrical š‘‘-connection š¶Ī“(š‘) of an APL-space and deduce the corresponding properties for a šœ‘-Lagrange space and an AFL-space.

Theorem 5.4. The canonical metrical š‘‘-connection š¶Ī“(š‘) of an APL-space has the following properties: (1)š·š‘–š‘˜āˆ¶=š‘¦š‘–|š‘˜=š‘‰š‘–š‘˜+š‘‰š‘š‘˜š¶š‘–š‘š‘Ÿš‘¦š‘Ÿ+š‘‰š‘š‘Ÿš¶š‘–š‘˜š‘š‘¦š‘Ÿ+š‘‰š‘š‘ š‘Žš‘–š‘ š¶š‘š‘˜š‘Ÿš‘¦š‘Ÿ,(5.12)š‘¦š‘–|š‘˜=š‘‰š‘ š‘˜ī€·š‘Žš‘ š‘–+š¶š‘ š‘–š‘—š‘¦š‘—ī€øāˆ’š‘‰š‘ š‘–š¶š‘˜š‘ š‘š‘¦š‘āˆ’š‘‰š‘ š‘š¶š‘ š‘˜š‘–š‘¦š‘,(5.13) where š‘¦š‘–=āˆ¶š‘Žš‘–š‘—š‘¦š‘—, (2)š‘‘š‘–š‘˜āˆ¶=š‘¦š‘–|š‘˜=šœ‘ā€²+šœ‘ī…žī…žš¹2š›æšœ‘ā€²š‘–š‘˜+šµāˆ˜š‘¦š‘˜š‘¦š‘–,š‘¦š‘–|š‘˜=š‘Žš‘–š‘˜+š¶š‘–š‘˜š‘—š‘¦š‘—,(5.14) where šµ=2{šœ‘ī…žšœ‘ī…žī…ž+š¹2(šœ‘ī…žī…žī…žšœ‘ī…žāˆ’šœ‘ī…žī…ž2)}/šœ‘ā€²(šœ‘ī…ž+2š¹2šœ‘ī…žī…ž), (3)šæ|š‘˜=š‘‹š‘˜+2šœ‘ā€²ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘‰š‘Ÿš‘˜š‘¦š‘Ÿ,šæ|š‘˜=2šœ‘ā€²ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘˜+š“š‘˜,(5.15) where š‘‹š‘˜=š‘¦š‘Ÿšœ•š‘˜š“š‘Ÿāˆ’š‘š‘š‘˜š“š‘+šœ•š‘˜š‘ˆ.

Proof. (1) Using (5.8) and (4.5) in (1.16), we have š·š‘–š‘˜=š‘¦š‘Ÿī‚µāˆ˜šæš‘–š‘Ÿš‘˜+š‘‰š‘š‘˜š¶š‘–š‘Ÿš‘+š‘‰š‘š‘Ÿš¶š‘–š‘˜š‘+š‘‰š‘š‘ š‘Žš‘–š‘ š¶š‘š‘˜š‘Ÿī‚¶āˆ’āˆ˜š‘š‘–š‘˜+š‘‰š‘–š‘˜,(5.16) which, in view of (1.18), reduces to š·š‘–š‘˜=š‘‰š‘–š‘˜+š‘¦š‘Ÿī‚€š‘‰š‘š‘˜š¶š‘–š‘Ÿš‘+š‘‰š‘š‘Ÿš¶š‘–š‘˜š‘+š‘‰š‘š‘ š‘Žš‘–š‘ š¶š‘š‘˜š‘Ÿī‚.(5.17) Next, if we use (2.5) in š‘¦š‘–=š‘Žš‘–š‘—š‘¦š‘—, then it follows that š‘¦š‘–=ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øāˆ˜š‘¦š‘–.(5.18) Now, applying successively š›æš‘–=šœ•š‘–āˆ’š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ, (4.5), and āˆ˜š›æš‘–=šœ•š‘–āˆ’āˆ˜š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ in š‘¦š‘–|š‘˜=š›æš‘˜š‘¦š‘–āˆ’š‘¦š‘Ÿšæš‘Ÿš‘–š‘˜ and keeping (5.8) and (5.18) in view, we have š‘¦š‘–|š‘˜=āˆ˜š›æš‘˜ī‚»ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øāˆ˜š‘¦š‘–ī‚¼āˆ’ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øāˆ˜š‘¦š‘Ÿāˆ˜šæš‘Ÿš‘–š‘˜+š‘‰š‘Ÿš‘˜Ģ‡šœ•š‘Ÿš‘¦š‘–āˆ’š‘¦š‘Ÿī€·š‘‰š‘ š‘˜š¶š‘Ÿš‘–š‘ +š‘‰š‘ š‘–š¶š‘Ÿš‘˜š‘ +š‘‰š‘ š‘š‘Žš‘Ÿš‘š¶š‘ š‘˜š‘–ī€ø.(5.19) Differentiating š‘¦š‘–=š‘Žš‘–š‘—š‘¦š‘— partially with respect to š‘¦š‘Ÿ, we have Ģ‡šœ•š‘Ÿš‘¦š‘–=š‘Žš‘–š‘Ÿ+2š¶š‘–š‘Ÿš‘—š‘¦š‘—.(5.20) Also, š‘¦š‘™š¶š‘™š‘—š‘˜=š‘Žš‘™ā„Žš‘¦ā„Žš¶š‘™š‘—š‘˜=š‘¦ā„Žš¶ā„Žš‘—š‘˜.(5.21) In view of (5.3), we have āˆ˜š›æš‘˜ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€ø=0.(5.22) Using (5.20), (5.21), and (5.22) in (5.19), we obtain š‘¦š‘–|š‘˜=ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øī‚µāˆ˜š›æš‘˜āˆ˜š‘¦š‘–āˆ’āˆ˜š‘¦š‘Ÿāˆ˜šæš‘Ÿš‘–š‘˜ī‚¶+š‘‰š‘ š‘˜ī€·š‘Žš‘ š‘–+š¶š‘ š‘–š‘—š‘¦š‘—ī€øāˆ’ī€·š‘‰š‘ š‘–š¶š‘˜š‘ š‘+š‘‰š‘ š‘š¶š‘˜š‘ š‘–ī€øš‘¦š‘,(5.23) which, in view of (5.4), gives the desired result.
(2) Using (5.1) in (1.17), we get š‘‘š‘–š‘˜=šœ‘ā€²+šœ‘ī…žī…žš¹2š›æšœ‘ā€²š‘–š‘˜+šµāˆ˜š‘¦š‘˜š‘¦š‘–,(5.24) where šµ=2{šœ‘ī…žšœ‘ī…žī…ž+š¹2(šœ‘ī…žī…žī…žšœ‘ī…žāˆ’šœ‘ī…žī…ž2)}/šœ‘ā€²(šœ‘ā€²+2š¹2šœ‘ī…žī…ž).
In view of (5.20) and (5.21), it follows, from š‘¦š‘–|š‘˜=Ģ‡šœ•š‘˜š‘¦š‘–āˆ’š‘¦š‘Ÿš¶š‘Ÿš‘–š‘˜, that š‘¦š‘–|š‘˜=š‘Žš‘–š‘˜+2š¶š‘–š‘˜š‘—š‘¦š‘—āˆ’š¶š‘–š‘—š‘˜š‘¦š‘—,(5.25) that is, š‘¦š‘–|š‘˜=š‘Žš‘–š‘˜+š¶š‘–š‘˜š‘—š‘¦š‘— as š¶š‘–š‘—š‘˜ is totally symmetric.
(3) Utilizing successively š›æš‘–=šœ•š‘–āˆ’š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ, (4.5), and āˆ˜š›æš‘–=šœ•š‘–āˆ’āˆ˜š‘š‘Ÿš‘–Ģ‡šœ•š‘Ÿ in šæ|š‘˜=š›æš‘˜šæ, we get šæ|š‘˜=āˆ˜š›æš‘˜šæ+š‘‰š‘Ÿš‘˜Ģ‡šœ•š‘Ÿšæ.(5.26) Using (1.2) and (2.1) in (5.26), we have šæ|š‘˜=āˆ˜š›æš‘˜ī€·šœ‘+š“š‘Ÿš‘¦š‘Ÿī€ø+š‘ˆ+š‘‰š‘Ÿš‘˜ī‚µ2šœ‘ā€²āˆ˜š‘¦š‘Ÿ+š“š‘Ÿī‚¶,(5.27) which, in view of (5.3), gives šæ|š‘˜=āˆ˜š›æš‘˜ī€·š“š‘Ÿš‘¦š‘Ÿī€ø+š‘ˆ+š‘‰š‘Ÿš‘˜ī‚µ2šœ‘ā€²āˆ˜š‘¦š‘Ÿ+š“š‘Ÿī‚¶.(5.28) Using āˆ˜š›æš‘˜=šœ•š‘˜āˆ’āˆ˜š‘š‘Ÿš‘˜Ģ‡šœ•š‘Ÿ and (5.18) in (5.28) and keeping (4.5) in view, we find šæ|š‘˜=š‘¦š‘Ÿšœ•š‘˜š“š‘Ÿāˆ’š‘š‘š‘˜š“š‘+šœ•š‘˜š‘ˆ+2šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žš‘‰š‘Ÿš‘˜š‘¦š‘Ÿ.(5.29) If we take š‘‹š‘˜=š‘¦š‘Ÿšœ•š‘˜š“š‘Ÿāˆ’š‘š‘š‘˜š“š‘+šœ•š‘˜š‘ˆ, then the last expression takes the form šæ|š‘˜=š‘‹š‘˜+2šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žš‘‰š‘Ÿš‘˜š‘¦š‘Ÿ.(5.30) Next, using (2.1) in šæ|š‘˜=Ģ‡šœ•š‘˜šæ, we get šæ|š‘˜=2šœ‘ā€²āˆ˜š‘¦š‘˜+š“š‘˜,(5.31) which, in view of (5.18), gives the required result.

Corollary 5.5 (see [14]). The canonical metrical š‘‘-connection š¶Ī“(š‘) of a šœ‘-Lagrange space has the following properties: (1)š·š‘–š‘˜āˆ¶=š‘¦š‘–|š‘˜=0,š‘¦š‘–|š‘˜=0,(5.32)(2)š‘‘š‘–š‘˜āˆ¶=š‘¦š‘–|š‘˜=šœ‘ā€²+šœ‘ī…žī…žš¹2š›æšœ‘ā€²š‘–š‘˜+šµāˆ˜š‘¦š‘˜š‘¦š‘–,š‘¦š‘–|š‘˜=š‘Žš‘–š‘˜+š¶š‘–š‘˜š‘—š‘¦š‘—,(5.33) where šµ=2{šœ‘ī…žšœ‘ī…žī…ž+š¹2(šœ‘ī…žī…žī…žšœ‘ī…žāˆ’šœ‘ī…žī…ž2)}/šœ‘ā€²(šœ‘ā€²+2š¹2šœ‘ī…žī…ž), (3)šæ|š‘˜=0,šæ|š‘˜=2šœ‘ā€²ī€·šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘˜.(5.34)

Proof. Applying š“š‘–(š‘„)=0, š‘ˆ(š‘„)=0, and š‘‰š‘–š‘—=0 in Theorem 5.4, we have the corollary.

Corollary 5.6. The canonical metrical š‘‘-connection š¶Ī“(š‘) of an AFL-space has the following properties: (1)š·š‘–š‘˜=šµš‘–š‘˜+šµš‘š‘Ÿāˆ˜š¶š‘–š‘˜š‘š‘¦š‘Ÿ,š‘¦š‘–|š‘˜=š‘”š‘ š‘–ī‚µšµš‘ š‘˜āˆ’šµš‘™š‘š‘¦š‘āˆ˜š¶š‘ š‘™š‘˜ī‚¶,(5.35) where š‘¦š‘–=š‘”š‘–š‘—š‘¦š‘—, (2)š‘‘š‘–š‘˜=š›æš‘–š‘˜,š‘¦š‘–|š‘˜=š‘”š‘–š‘˜,(5.36)(3)šæ|š‘˜=š‘¦š‘Ÿšœ•š‘˜š“š‘Ÿāˆ’š‘š‘š‘˜š“š‘+šœ•š‘˜š‘ˆ+2šµš‘Ÿš‘˜š‘¦š‘Ÿ,šæ|š‘˜=2š‘¦š‘˜+š“š‘˜.(5.37)

Proof. Using šœ‘(š¹2)=š¹2,šœ‘ā€²(š¹2)=1,šœ‘ī…žī…ž(š¹2)=0=šœ‘ī…žī…žī…ž(š¹2),š‘Žš‘–š‘—=š‘”š‘–š‘—,š¶š‘–š‘—š‘˜=āˆ˜š¶š‘–š‘—š‘˜,āˆ˜š¶š‘–š‘—š‘˜š‘¦š‘—=0,āˆ˜š¶š‘–š‘—š‘˜š‘¦š‘˜=0, and š‘‰š‘Ÿš‘—=šµš‘Ÿš‘— 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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Copyright Ā© 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.


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