International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 981059 | 14 pages | https://doi.org/10.1155/2012/981059

On Subspaces of an Almost 𝝋 -Lagrange Space

Academic Editor: Zhongmin Shen
Received29 Mar 2012
Accepted04 Jun 2012
Published08 Aug 2012

Abstract

We discuss the subspaces of an almost πœ‘-Lagrange space (APL space in short). We obtain the induced nonlinear connection, coefficients of coupling, coefficients of induced tangent and induced normal connections, the Gauss-Weingarten formulae, and the Gauss-Codazzi equations for a subspace of an APL-space. Some consequences of the Gauss-Weingarten formulae have also been discussed.

1. Introduction

The credit for introducing the geometry of Lagrange spaces and their subspaces goes to the famous Romanian geometer Miron [1]. He developed the theory of subspaces of a Lagrange space together with Bejancu [2]. Miron and Anastasiei [3] and Sakaguchi [4] studied the subspaces of generalized Lagrange spaces (GL spaces in short). Antonelli and Hrimiuc [5, 6] introduced the concept of πœ‘-Lagrangians and studied πœ‘-Lagrange manifolds. Generalizing the notion of a πœ‘-Lagrange manifold, the present authors recently studied the geometry of an almost πœ‘-Lagrange space (APL space briefly) and obtained the fundamental entities related to such space [7]. This paper is devoted to the subspaces of an APL space.

Let 𝐹𝑛=(𝑀,𝐹(π‘₯,𝑦)) be an 𝑛-dimensional Finsler space and πœ‘βˆΆβ„+→ℝ a smooth function. 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, for every π‘‘βˆˆIm(𝐹2),then the Lagrangian given by 𝐹𝐿(π‘₯,𝑦)=πœ‘2ξ€Έ+𝐴𝑖(π‘₯)𝑦𝑖+π‘ˆ(π‘₯),(1.1) where 𝐴𝑖(π‘₯) is a covector and π‘ˆ(π‘₯) is a smooth function, is a regular Lagrangian [7]. The space 𝐿𝑛=(𝑀,𝐿(π‘₯,𝑦)) is a Lagrange space. The present authors [7] called such space as an almost πœ‘-Lagrange space (shortly APL space) associated to the Finsler space 𝐹𝑛. An APL space reduces to a πœ‘-Lagrange space if and only if 𝐴𝑖(π‘₯)=0 and π‘ˆ(π‘₯)=0. We take 𝑔𝑖𝑗=12Μ‡πœ•π‘–Μ‡πœ•π‘—πΉ2,π‘Žπ‘–π‘—=12Μ‡πœ•π‘–Μ‡πœ•π‘—πΏ,whereΜ‡πœ•π‘–β‰‘πœ•πœ•π‘¦π‘–.(1.2) We indicate all the geometrical objects related to 𝐹𝑛 by putting a small circle β€œβˆ˜β€ over them. Equations (1.2), in view of (1.1), provide the following expressions for π‘Žπ‘–π‘— and its inverse (cf. [7]): 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,π‘Žπ‘–π‘—=1ξ‚΅π‘”πœ‘β€²π‘–π‘—βˆ’2πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘¦π‘–π‘¦π‘—ξ‚Ά,(1.3) where 𝑔𝑖𝑗𝑦𝑗=βˆ˜π‘¦π‘–.

Let 𝑀 be a smooth manifold of dimension π‘š, 1<π‘š<𝑛, immersed in 𝑀 by immersion ξ‚π‘–βˆΆπ‘€β†’π‘€. The immersion 𝑖 induces an immersion π‘‡π‘–ξ‚βˆΆπ‘‡π‘€β†’π‘‡π‘€ making the following diagram commutative: π‘‡ξ‚π‘€π‘‡π‘–ξ‚π‘€βŸΆπ‘–βŸΆπ‘‡π‘€ξ„πœ‹β†“β†“πœ‹π‘€.(1.4)

Let (𝑒𝛼,𝑣𝛼) (throughout the paper, the Greek indices 𝛼,𝛽,𝛾,… run from 1 to π‘š) be local coordinates on 𝑇𝑀. The restriction of the Lagrangian 𝐿 on 𝑇𝑀 is 𝐿(𝑒,𝑣)=𝐿(π‘₯(𝑒),𝑦(𝑒,𝑣)). Let π‘Žπ›Όπ›½=(1/2)(πœ•2𝐿/πœ•π‘’π›Όπœ•π‘’π›½). Then, we have (cf. [8]) π‘Žπ›Όπ›½=π΅π‘–π›Όπ΅π‘—π›½π‘Žπ‘–π‘— where 𝐡𝑖𝛼(𝑒)=πœ•π‘₯𝑖/πœ•π‘’π›Ό are the projection factors. The pair ξ‚πΏπ‘šξ‚ξ„=(𝑀,𝐿(𝑒,𝑣)) is also a Lagrange space, called the subspace of 𝐿𝑛. For the natural bases (πœ•/πœ•π‘₯𝑖,πœ•/πœ•π‘¦π‘–) on TM and (πœ•/πœ•π‘’π›Ό,πœ•/πœ•π‘£π›Ό) on 𝑇𝑀, we have [8] πœ•πœ•π‘’π›Ό=π΅π‘–π›Όπœ•πœ•π‘₯𝑖+𝐡𝑖0π›Όπœ•πœ•π‘¦π‘–,πœ•πœ•π‘£π›Ό=π΅π‘–π›Όπœ•πœ•π‘¦π‘–,(1.5) where 𝐡𝑖0𝛼=𝐡𝑖𝛽𝛼𝑣𝛽,𝐡𝑖𝛽𝛼=πœ•2π‘₯𝑖/πœ•π‘’π›Όπœ•π‘’π›½.

For the bases (𝑑π‘₯𝑖,𝑑𝑦𝑖) and (𝑑𝑒𝛼,𝑑𝑣𝛼), we have 𝑑π‘₯𝑖=𝐡𝑖𝛼𝑑𝑒𝛼,𝑑𝑦𝑖=𝐡𝑖𝛼𝑑𝑣𝛼+𝐡𝑖0𝛼𝑑𝑒𝛼.(1.6) Since (𝐡𝑖𝛼) are π‘š linearly independent vector fields tangent to 𝑀, a vector field πœ‰π‘–(π‘₯,𝑦) is normal to 𝑀 along 𝑇𝑀 if on 𝑇𝑀, we have π‘Žπ‘–π‘—(π‘₯(𝑒),𝑦(𝑒,𝑣))π΅π‘–π›Όπœ‰π‘—=0,βˆ€π›Ό=1,2,…,π‘š.(1.7) There are, at least locally, (π‘›βˆ’π‘š) unit vector fields π΅π‘–π‘Ž(𝑒,𝑣)(π‘Ž=π‘š+1,π‘š+2,…,𝑛) normal to 𝑀 and mutually orthonormal, that is, π‘Žπ‘–π‘—π΅π‘–π›Όπ΅π‘—π‘=0,π‘Žπ‘–π‘—π΅π‘–π‘Žπ΅π‘—π‘=π›Ώπ‘Žπ‘,(π‘Ž,𝑏=π‘š+1,π‘š+2,…,𝑛).(1.8) Thus, at every point 𝑀(𝑒,𝑣)βˆˆπ‘‡, we have a moving frame β„œ=((𝑒,𝑣),𝐡𝑖𝛼(𝑒,𝑣),π΅π‘–π‘Ž(𝑒,𝑣)). Using (1.3) in the first expression of (1.8) and keeping βˆ˜π‘¦π‘–π΅π‘–π‘Ž=0 (this fact is clear from π‘”π‘–π‘—π‘¦π‘–π΅π‘—π‘Ž=0) in view, we observe that π΅π‘–π‘Ž's are normal to 𝑀 with respect to 𝐿𝑛 if and only if they are so with respect to 𝐹𝑛. The dual frame of β„œ is β„œβˆ—=((𝑒,𝑣),𝐡𝛼𝑖(𝑒,𝑣),π΅π‘Žπ‘–(𝑒,𝑣)) with the following duality conditions: 𝐡𝑖𝛼𝐡𝛽𝑖=𝛿𝛽𝛼,π΅π‘–π‘Žπ΅π›½π‘–=0,𝐡𝑖𝛼𝐡𝑏𝑖=0,π΅π‘–π‘Žπ΅π‘π‘–=π›Ώπ‘π‘Ž,π΅π‘–π‘Žπ΅π‘Žπ‘—+𝐡𝑖𝛼𝐡𝛼𝑗=𝛿𝑖𝑗.(1.9) We will make use of the following results due to the present authors [7], during further discussion.

Theorem 1.1 1.1 (cf. [7]). The canonical nonlinear connection of an APL space 𝐿𝑛 has the local coefficients given by 𝑁𝑖𝑗=βˆ˜π‘π‘–π‘—βˆ’π‘‰π‘–π‘—,(1.10) where 𝑉𝑖𝑗=(1/2)πΉπ‘–π‘—βˆ’π‘†π‘—π‘–π‘Ÿ(2πΉπ‘Ÿπ‘˜π‘¦π‘˜+πœ•π‘Ÿπ‘ˆ), π‘†π‘—π‘–π‘Ÿ=12πœ‘β€²βˆ˜πΆπ‘–π‘žπ‘—π‘”π‘žπ‘Ÿ+12πœ‘ξ…žξ…žπœ‘β€²2π‘”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‘β€²2πœ‘ξ…žξ…žξ…žβˆ’2πœ‘ξ…žξ…ž3𝐹2βˆ’4πœ‘β€²πœ‘ξ…žξ…ž22πœ‘β€²2ξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έ2π‘¦π‘–βˆ˜π‘¦π‘—π‘¦π‘Ÿ,πΉπ‘Ÿπ‘˜(1π‘₯)=2ξ€·πœ•π‘Ÿπ΄π‘˜βˆ’πœ•π‘˜π΄π‘Ÿξ€Έ,𝐹𝑖𝑗=π‘Žπ‘–π‘˜πΉπ‘˜π‘—.(1.11)

Theorem 1.2 (cf. [7]). The coefficients of the canonical metrical d-connection 𝐢Γ(𝑁) of an APL space 𝐿𝑛 are given by πΆπ‘–π‘—π‘˜=βˆ˜πΆπ‘–π‘—π‘˜+πœ‘ξ…žξ…žπœ‘ξ…žξ‚€π›Ώπ‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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€Έπœ‘ξ…žξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘–βˆ˜π‘¦π‘—βˆ˜π‘¦π‘˜,(1.12)π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‘—.(1.13)

For basic notations related to a Finsler space, a Lagrange space, and their subspaces, we refer to the books [8, 9].

2. Induced Nonlinear Connection

Let 𝑁𝑁=(𝛼𝛽(𝑒,𝑣)) be a nonlinear connection for ξ‚πΏπ‘šξ‚ξ„=(𝑀,𝐿(𝑒,𝑣)). The adapted basis of 𝑇(𝑒,𝑣)𝑇𝑀 induced by 𝑁 is (𝛿/𝛿𝑒𝛼=𝛿𝛼,πœ•/πœ•π‘£π›Ό=Μ‡πœ•π›Ό), where 𝛿𝛼=πœ•π›Όβˆ’ξ‚π‘π›½π›ΌΜ‡πœ•π›½.(2.1) The dual basis (cobasis) of the adapted basis (𝛿𝛼,Μ‡πœ•π›Ό) is (𝑑𝑒𝛼,𝛿𝑣𝛼=𝑑𝑣𝛼+𝑁𝛼𝛽𝑑𝑒𝛽).

Definition 2.1 (cf. [8]). A nonlinear connection 𝑁𝑁=(𝛼𝛽(𝑒,𝑣)) of ξ‚πΏπ‘š is said to be induced by the canonical nonlinear connection 𝑁 if the following equation holds good: 𝛿𝑣𝛼=𝐡𝛼𝑖𝛿𝑦𝑖.(2.2) The local coefficients of the induced nonlinear connection 𝑁𝑁=(𝛼𝛽(𝑒,𝑣)) for the subspace ξ‚πΏπ‘šξ‚ξ„=(𝑀,𝐿(𝑒,𝑣)) of a Lagrange space 𝐿𝑛=(𝑀,𝐿(π‘₯,𝑦)) are given by (cf. [8]) 𝑁𝛼𝛽=𝐡𝛼𝑖𝑁𝑖𝑗𝐡𝑗𝛽+𝐡𝑖0𝛽,(2.3)𝑁𝑖𝑗 being the local coefficients of canonical nonlinear connection 𝑁 of the Lagrange space 𝐿𝑛=(𝑀,𝐿(π‘₯,𝑦)). Now using (1.10) in (2.3), we get 𝑁𝛼𝛽=π΅π›Όπ‘–ξ‚΅βˆ˜π‘π‘–π‘—π΅π‘—π›½+𝐡𝑖0π›½ξ‚Άβˆ’π΅π›Όπ‘–π‘‰π‘–π‘—π΅π‘—π›½.(2.4) If we take βˆ˜ξ„π‘π›Όπ›½=𝐡𝛼𝑖(βˆ˜π‘π‘–π‘—π΅π‘—π›½+𝐡𝑖0𝛽), it follows from (2.4) that 𝑁𝛼𝛽=βˆ˜ξ„π‘π›Όπ›½βˆ’π΅π›Όπ‘–π‘‰π‘–π‘—π΅π‘—π›½.(2.5) Thus, we have the following.

Theorem 2.2. The local coefficients of the induced nonlinear connection 𝑁 of the subspace ξ‚πΏπ‘š of an APL space 𝐿𝑛 are given by (2.5).

In view of (2.5), (2.1) takes the following form, for the subspace ξ‚πΏπ‘š of an APL space 𝐿𝑛: 𝛿𝛽=βˆ˜π›Ώπ›½+π΅π›Όπ‘π‘‰π‘π‘—π΅π‘—π›½Μ‡πœ•π›Ό,(2.6) where βˆ˜π›Ώπ›½=πœ•π›½βˆ’βˆ˜ξ„π‘π›Όπ›½Μ‡πœ•π›Ό.

We can put (𝑑π‘₯𝑖,𝛿𝑦𝑖) as (cf. [8]) 𝑑π‘₯𝑖=𝐡𝑖𝛼𝑑𝑒𝛼,𝛿𝑦𝑖=𝐡𝑖𝛼𝛿𝑦𝛼+π΅π‘–π‘Žπ»π‘Žπ›Όπ‘‘π‘’π›Ό,(2.7) where π»π‘Žπ›Ό=π΅π‘Žπ‘–ξ‚€π‘π‘–π‘—π΅π‘—π›Ό+𝐡𝑖0𝛼.(2.8) Using (1.10) in (2.8) and simplifying, we get π»π‘Žπ›Ό=π΅π‘Žπ‘–ξ‚΅βˆ˜π‘π‘–π‘—π΅π‘—π›Ό+𝐡𝑖0π›Όξ‚Άβˆ’π΅π‘Žπ‘–π‘‰π‘–π‘—π΅π‘—π›Ό.(2.9) Taking βˆ˜π»π‘Žπ›Ό=π΅π‘Žπ‘–(βˆ˜π‘π‘–π‘—π΅π‘—π›Ό+𝐡𝑖0𝛼), in (2.9), it follows that π»π‘Žπ›Ό=βˆ˜π»π‘Žπ›Όβˆ’π΅π‘Žπ‘–π‘‰π‘–π‘—π΅π‘—π›Ό.(2.10) Now, 𝑑π‘₯𝑖=𝐡𝑖𝛼𝑑𝑒𝛼,𝛿𝑦𝑖=𝐡𝑖𝛼𝛿𝑦𝛼 if and only if π»π‘Žπ›Ό=0, that is, if and only if βˆ˜π»π‘Žπ›Ό=π΅π‘Žπ‘–π‘‰π‘–π‘—π΅π‘—π›Ό. Thus, we have the following.

Theorem 2.3. The adapted cobasis (𝑑π‘₯𝑖,𝛿𝑦𝑖) of the basis (πœ•/πœ•π‘₯𝑖,πœ•/πœ•π‘¦π‘–) induced by the nonlinear connection 𝑁 of an APL space 𝐿𝑛 is of the form 𝑑π‘₯𝑖=𝐡𝑖𝛼𝑑𝑒𝛼,𝛿𝑦𝑖=𝐡𝑖𝛼𝛿𝑦𝛼 if and only if βˆ˜π»π‘Žπ›Ό=π΅π‘Žπ‘–π‘‰π‘–π‘—π΅π‘—π›Ό.

Definition 2.4 (cf. [8]). Let 𝐷=𝐷Γ(𝑁) be the canonical metrical 𝑑-connection of 𝐿𝑛. An operator 𝐷 is said to be a coupling of 𝐷 with 𝑁 if 𝐷𝑋𝑖=𝑋𝑖|𝛼𝑑𝑒𝛼+𝑋𝑖|𝛼𝛿𝑣𝛼,(2.11) where 𝑋𝑖|𝛼=𝛿𝛼𝑋𝑖+𝑋𝑗𝐿𝑖𝑗𝛼,𝑋𝑖|𝛼=Μ‡πœ•π›Όπ‘‹π‘–+𝑋𝑗𝐢𝑖𝑗𝛼.
The coefficients (𝐿𝑖𝑗𝛼,𝐢𝑖𝑗𝛼) of coupling 𝐷 of 𝐷 with 𝑁 are given by 𝐿𝑖𝑗𝛼=πΏπ‘–π‘—π‘˜π΅π‘˜π›Ό+πΆπ‘–π‘—π‘˜π΅π‘˜π‘Žπ»π‘Žπ›Ό,(2.12)𝐢𝑖𝑗𝛼=πΆπ‘–π‘—π‘˜π΅π‘˜π›Ό.(2.13) Using (1.12) and (1.13) in (2.12), we have 𝐿𝑖𝑗𝛽=ξ‚΅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‘—ξ‚Άπ΅π‘˜π›½+ξ‚Έβˆ˜πΆπ‘–π‘—π‘˜+πœ‘ξ…žξ…žπœ‘ξ…žξ‚€π›Ώπ‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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‘ξ…žβˆ’2πœ‘ξ…žξ…ž2ξ€Έπœ‘ξ…žξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘–βˆ˜π‘¦π‘—βˆ˜π‘¦π‘˜ξƒ­π΅π‘˜π‘Žπ»π‘Žπ›½.(2.14) In view of (2.10) and βˆ˜π‘¦π‘–π΅π‘–π‘Ž=0, (2.14) becomes 𝐿𝑖𝑗𝛽=ξ‚΅βˆ˜πΏπ‘–π‘—π‘˜π΅π‘˜π›½+βˆ˜πΆπ‘–π‘—π‘˜π΅π‘˜π‘Žβˆ˜π»π‘Žπ›½ξ‚Ά+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘˜π›½+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘Žπ»π‘Žπ›½,(2.15) that is, 𝐿𝑖𝑗𝛽=βˆ˜ξ„πΏπ‘–π‘—π›½+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘˜π›½+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘Žπ»π‘Žπ›½,(2.16) where βˆ˜ξ„πΏπ‘–π‘—π›½=βˆ˜πΏπ‘–π‘—π‘˜π΅π‘˜π›½+βˆ˜πΆπ‘–π‘—π‘˜π΅π‘˜π‘Žβˆ˜π»π‘Žπ›½.
Using (1.12) in (2.13), we find that 𝐢𝑖𝑗𝛽=βˆ˜πΆπ‘–π‘—π‘˜π΅π‘˜π›½+ξƒ©πœ‘ξ…žξ…žξ‚€π›Ώπœ‘β€²π‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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‘˜ξƒͺπ΅π‘˜π›½,(2.17) that is, 𝐢𝑖𝑗𝛽=βˆ˜ξ„πΆπ‘–π‘—π›½+ξƒ©πœ‘ξ…žξ…žξ‚€π›Ώπœ‘β€²π‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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‘˜ξƒͺπ΅π‘˜π›½,(2.18) where βˆ˜ξ„πΆπ‘–π‘—π›½=βˆ˜πΆπ‘–π‘—π‘˜π΅π‘˜π›½. Thus, we have the following.

Theorem 2.5. The coefficients of coupling for the subspace ξ‚πΏπ‘š of an APL space 𝐿𝑛 are given by (2.16) and (2.18).

Definition 2.6 (cf. [8]). An operator 𝐷𝑇 given by 𝐷𝑇𝑋𝛼=𝑋𝛼|𝛽𝑑𝑒𝛽+𝑋𝛼|𝛽𝛿𝑣𝛽,(2.19) where 𝑋𝛼|𝛽=𝛿𝛽𝑋𝛼+𝑋𝛾𝐿𝛼𝛾𝛽,𝑋𝛼|𝛽=Μ‡πœ•π›½π‘‹π›Ό+𝑋𝛾𝐢𝛼𝛾𝛽, is called the induced tangent connection by 𝐷. This defines an 𝑁-linear connection for ξ‚πΏπ‘š.
The coefficients (𝐿𝛼𝛾𝛽,𝐢𝛼𝛾𝛽) of 𝐷𝑇 are given by 𝐿𝛼𝛽𝛾=𝐡𝛼𝑖𝐡𝑖𝛽𝛾+𝐡𝑗𝛽𝐿𝑖𝑗𝛾,(2.20)𝐢𝛼𝛽𝛾=𝐡𝛼𝑖𝐡𝑗𝛽𝐢𝑖𝑗𝛾.(2.21) Using (2.16) in (2.20), we get 𝐿𝛼𝛽𝛾=𝐡𝛼𝑖𝐡𝑖𝛽𝛾+π΅π‘—π›½π΅π›Όπ‘–ξƒ¬βˆ˜ξ„πΏπ‘–π‘—π›Ύ+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘˜π›Ύ+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘Žπ»π‘Žπ›Ύξ‚Ή,(2.22) that is, 𝐿𝛼𝛽𝛾=𝐡𝛼𝑖𝐡𝑖𝛽𝛾+βˆ˜ξ„πΏπ‘–π‘—π›Ύπ΅π‘—π›½ξƒͺ+π΅π›Όπ‘–π΅π‘—π›½π‘‰ξ‚Έξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘˜π›Ύ+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘Žπ»π‘Žπ›Ύξ‚Ή.(2.23) If we take βˆ˜πΏπ›Όπ›½π›Ύ=𝐡𝛼𝑖(𝐡𝑖𝛽𝛾+βˆ˜ξ„πΏπ‘–π‘—π›Ύπ΅π‘—π›½), the last expression gives 𝐿𝛼𝛽𝛾=βˆ˜πΏπ›Όπ›½π›Ύ+π΅π›Όπ‘–π΅π‘—π›½π‘‰ξ‚Έξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘˜π›Ύ+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘Žπ»π‘Žπ›Ύξ‚Ή.(2.24) Next, using (2.18) in (2.21), we obtain 𝐢𝛼𝛽𝛾=π΅π›Όπ‘–π΅π‘—π›½βˆ˜ξ„πΆπ‘–π‘—π›Ύ+ξƒ©πœ‘ξ…žξ…žξ‚€π›Ώπœ‘β€²π‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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‘˜ξƒͺπ΅π‘˜π›Ύπ΅π›Όπ‘–π΅π‘—π›½.(2.25) If we take βˆ˜πΆπ›Όπ›½π›Ύ=π΅π›Όπ‘–π΅π‘—π›½βˆ˜ξ„πΆπ‘–π‘—π›Ύ, (2.25) becomes 𝐢𝛼𝛽𝛾=βˆ˜πΆπ›Όπ›½π›Ύ+ξƒ©πœ‘ξ…žξ…žξ‚€π›Ώπœ‘β€²π‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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‘˜ξƒͺπ΅π‘˜π›Ύπ΅π›Όπ‘–π΅π‘—π›½.(2.26) Thus, we have the following.

Theorem 2.7. The coefficients of the induced tangent connection 𝐷𝑇 for the subspace ξ‚πΏπ‘š of an APL space are given by (2.24) and (2.26).

Remarks 2. The torsion 𝑇𝛼𝛽𝛾=πΏπ›Όπ›½π›Ύβˆ’πΏπ›Όπ›Ύπ›½ does not vanish, in general, while 𝑆𝛼𝛽𝛾=πΆπ›Όπ›½π›Ύβˆ’πΆπ›Όπ›Ύπ›½=0. These facts may be observed from (2.24) and (2.26).

Definition 2.8 (cf. [8]). An operator π·βŸ‚ given by π·βŸ‚π‘‹π‘Ž=π‘‹π‘Ž|𝛼𝑑𝑒𝛼+π‘‹π‘Ž|𝛼𝛿𝑣𝛼,(2.27) where π‘‹π‘Ž|𝛼=π›Ώπ›Όπ‘‹π‘Ž+π‘‹π‘πΏπ‘Žπ‘π›Ό,π‘‹π‘Ž|𝛼=Μ‡πœ•π›Όπ‘‹π‘Ž+π‘‹π‘πΆπ‘Žπ‘π›Ό, is called the induced normal connection by 𝐷.
The coefficients (πΏπ‘Žπ‘π›Ύ,πΆπ‘Žπ‘π›Ύ) of π·βŸ‚ are given by πΏπ‘Žπ‘π›Ύ=π΅π‘Žπ‘–ξ‚€π›Ώπ›Ύπ΅π‘–π‘+𝐡𝑗𝑏𝐿𝑖𝑗𝛾,(2.28)πΆπ‘Žπ‘π›Ύ=π΅π‘Žπ‘–ξ‚€Μ‡πœ•π›Ύπ΅π‘–π‘+𝐡𝑗𝑏𝐢𝑖𝑗𝛾.(2.29) Using (2.6) and (2.16) in (2.28), we find πΏπ‘Žπ‘π›Ύ=π΅π‘Žπ‘–βˆ˜π›Ώπ›Ύπ΅π‘–π‘+π΅π‘Žπ‘–π΅π›Όπ‘π‘‰π‘π‘—π΅π‘—π›ΎΜ‡πœ•π›Όπ΅π‘–π‘+π΅π‘—π‘π΅π‘Žπ‘–ξƒ¬βˆ˜ξ„πΏπ‘–π‘—π›Ύ+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘˜π›Ύ+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘π»π‘π›Ύξ‚Ή.(2.30) Taking βˆ˜πΏπ‘Žπ‘π›Ύ=π΅π‘Žπ‘–(βˆ˜π›Ώπ›Ύπ΅π‘–π‘+π΅π‘—π‘βˆ˜ξ„πΏπ‘–π‘—π›Ύ) and using βˆ˜π‘¦π‘—π΅π‘—π‘=0, (2.30) reduces to πΏπ‘Žπ‘π›Ύ=βˆ˜πΏπ‘Žπ‘π›Ύ+π΅π‘Žπ‘–π΅π›Όπ‘π‘‰π‘π‘—π΅π‘—π›ΎΜ‡πœ•π›Όπ΅π‘–π‘+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘Žπ‘–π΅π‘—π‘π΅π‘˜π›Ύ+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–π΅π‘˜π‘π»π‘π›Ύπ΅π‘Žπ‘–π΅π‘—π‘.(2.31) Next, using (2.18) in (2.29), we have πΆπ‘Žπ‘π›Ύ=π΅π‘Žπ‘–ξƒ©Μ‡πœ•π›Ύπ΅π‘–π‘+π΅π‘—π‘βˆ˜ξ„πΆπ‘–π‘—π›Ύξƒͺ+ξƒ¬πœ‘ξ…žξ…žξ‚€π›Ώπœ‘β€²π‘–π‘—βˆ˜π‘¦π‘˜+π›Ώπ‘–π‘˜βˆ˜π‘¦π‘—ξ‚+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‘˜ξƒ­π΅π‘˜π›Ύπ΅π‘Žπ‘–π΅π‘—π‘.(2.32) Taking βˆ˜πΆπ‘Žπ‘π›Ύ=π΅π‘Žπ‘–(Μ‡πœ•π›Ύπ΅π‘–π‘+π΅π‘—π‘βˆ˜ξ„πΆπ‘–π‘—π›Ύ) and using (1.9) and βˆ˜π‘¦π‘—π΅π‘—π‘=0, the last equation yields πΆπ‘Žπ‘π›Ύ=βˆ˜πΆπ‘Žπ‘π›Ύ+πœ‘ξ…žξ…žπ›Ώπœ‘β€²π‘Žπ‘βˆ˜π‘¦π‘˜π΅π‘˜π›Ύ+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–π΅π‘˜π›Ύπ΅π‘Žπ‘–π΅π‘—π‘.(2.33) Thus, we have the following.

Theorem 2.9. The coefficients of induced normal connection π·βŸ‚ for the subspace ξ‚πΏπ‘š of an APL space 𝐿𝑛 are given by (2.31) and (2.33).

Definition 2.10 (cf. [8]). The (mixed) derivative of a mixed d-tensor field π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘ is given by βˆ‡π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘=ξ‚€π›Ώπœ‚π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘+π‘‡π‘˜β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘ξƒ‚πΏπ‘–π‘˜πœ‚+β‹―+π‘‡π‘–β‹―π›Ύβ‹―π‘Žπ‘—β‹―π›½β‹―π‘πΏπ›Όπ›Ύπœ‚+β‹―+π‘‡π‘–β‹―π›Όβ‹―π‘π‘—β‹―π›½β‹―π‘πΏπ‘Žπ‘πœ‚βˆ’π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘˜β‹―π›½β‹―π‘ξ‚πΏπ‘˜π‘—πœ‚βˆ’β‹―βˆ’π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›Ύβ‹―π‘πΏπ›Ύπ›½πœ‚βˆ’β‹―βˆ’π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘ξ‚πΏπ‘π‘πœ‚ξ‚π‘‘π‘’πœ‚+ξ‚€Μ‡πœ•πœ‚π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘+π‘‡π‘˜β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘ξ‚πΆπ‘–π‘˜πœ‚+β‹―+π‘‡π‘–β‹―π›Ύβ‹―π‘Žπ‘—β‹―π›½β‹―π‘πΆπ›Όπ›Ύπœ‚+β‹―+π‘‡π‘–β‹―π›Όβ‹―π‘π‘—β‹―π›½β‹―π‘πΆπ‘Žπ‘πœ‚βˆ’π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘˜β‹―π›½β‹―π‘ξ‚πΆπ‘˜π‘—πœ‚βˆ’β‹―βˆ’π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›Ύβ‹―π‘πΆπ›Ύπ›½πœ‚βˆ’β‹―βˆ’π‘‡π‘–β‹―π›Όβ‹―π‘Žπ‘—β‹―π›½β‹―π‘ξ‚πΆπ‘π‘πœ‚ξ‚π›Ώπ‘£πœ‚.(2.34) The connection 1-forms, ξ„πœ”π‘–π‘—ξ‚πΏ=βˆΆπ‘–π‘—π›Όπ‘‘π‘’π›Ό+𝐢𝑖𝑗𝛼𝛿𝑣𝛼,(2.35)πœ”π›Όπ›½=βˆΆπΏπ›Όπ›½π›Ύπ‘‘π‘’π›Ύ+𝐢𝛼𝛽𝛾𝛿𝑣𝛾,(2.36)πœ”π‘Žπ‘=βˆΆπΏπ‘Žπ‘π›Ύπ‘‘π‘’π›Ύ+πΆπ‘Žπ‘π›Ύπ›Ώπ‘£π›Ύ,(2.37) are called the connection 1-forms of βˆ‡. We have the following structure equations of βˆ‡.

Theorem 2.11 (cf. [8]). The structure equations of βˆ‡ are as follows: 𝑑(𝑑𝑒𝛼)βˆ’π‘‘π‘’π›½βˆ§πœ”π›Όπ›½=βˆ’Ξ©π›Ό,𝑑(𝛿𝑒𝛼)βˆ’π›Ώπ‘’π›½βˆ§πœ”π›Όπ›½Μ‡Ξ©=βˆ’π›Ό,π‘‘ξ„πœ”π‘–π‘—βˆ’ξ„πœ”β„Žπ‘—βˆ§ξ„πœ”π‘–β„Žξ„Ξ©=βˆ’π‘–π‘—,π‘‘πœ”π›Όπ›½βˆ’πœ”π›Ύπ›½βˆ§πœ”π›Όπ›Ύ=βˆ’Ξ©π›Όπ›½,π‘‘πœ”π‘Žπ‘βˆ’πœ”π‘π‘βˆ§πœ”π‘Žπ‘=βˆ’Ξ©π‘Žπ‘,(2.38) where the 2-forms of torsions Ω𝛼,̇Ω𝛼 are given by Ω𝛼=12π‘‡π›Όπ›½π›Ύπ‘‘π‘’π›½βˆ§π‘‘π‘’π›Ύ+πΆπ›Όπ›½π›Ύπ‘‘π‘’π›½βˆ§π›Ώπ‘£π›Ύ,̇Ω𝛼=12π‘…π›Όπ›½π›Ύπ‘‘π‘’π›½βˆ§π‘‘π‘’π›Ύ+π‘ƒπ›Όπ›½π›Ύπ‘‘π‘’π›½βˆ§π›Ώπ‘£π›Ύ,(2.39) with 𝑃𝛼𝛽𝛾=Μ‡πœ•π›Ύξ‚π‘π›Όπ›½βˆ’πΏπ›Όπ›½π›Ύ, and the 2-forms of curvature Ω𝑖𝑗,Ω𝛼𝛽 and Ξ©π‘Žπ‘, are given by Ω𝑖𝑗=12ξƒ‚π‘…π‘–π‘—π›Όπ›½π‘‘π‘’π›Όβˆ§π‘‘π‘’π›½+ξƒ‚π‘ƒπ‘–π‘—π›Όπ›½π‘‘π‘’π›Όβˆ§π›Ώπ‘£π›½+12ξƒ‚π‘†π‘–π‘—π›Όπ›½π›Ώπ‘£π›Όβˆ§π›Ώπ‘£π›½,Ω𝛼𝛽=12π‘…π›Όπ›½π›Ύπ›Ώπ‘‘π‘’π›Ύβˆ§π‘‘π‘’π›Ώ+π‘ƒπ›Όπ›½π›Ύπ›Ώπ‘‘π‘’π›Ύβˆ§π›Ώπ‘£π›Ώ+12π‘†π›Όπ›½π›Ύπ›Ώπ›Ώπ‘£π›Ύβˆ§π›Ώπ‘£π›Ώ,Ξ©π‘Žπ‘=12π‘…π‘Žπ‘π›Όπ›½π‘‘π‘’π›Όβˆ§π‘‘π‘’π›½+π‘ƒπ‘Žπ‘π›Όπ›½π‘‘π‘’π›Όβˆ§π›Ώπ‘£π›½+12π‘†π‘Žπ‘π›Όπ›½π›Ώπ‘£π›Όβˆ§π›Ώπ‘£π›½.(2.40)

We will use the following notations in Section 4: (a)Ω𝑖𝑗=ξ„Ξ©β„Žπ‘–π‘Žβ„Žπ‘—,(b)Ω𝛼𝛽=Ξ©π›Ύπ›Όπ‘Žπ›Ύπ›½,(c)Ξ©π‘Žπ‘=Ξ©π‘π‘π›Ώπ‘Žπ‘.(2.41)

3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspace ξ‚πΏπ‘šξ‚ξ„=(𝑀,𝐿(𝑒,𝑣)) of a Lagrange space 𝐿𝑛 are given by (cf. [8]) βˆ‡π΅π‘–π›Ό=π΅π‘–π‘ŽΞ π‘Žπ›Ό,βˆ‡π΅π‘–π‘Ž=βˆ’π΅π‘–π›½Ξ π›½π‘Ž,(3.1) where Ξ π‘Žπ›Ό=π»π‘Žπ›Όπ›½π‘‘π‘’π›½+πΎπ‘Žπ›Όπ›½π›Ώπ‘£π›½,Ξ π›½π‘Ž=π‘”π›½π›Ύπ›Ώπ‘Žπ‘Ξ π‘π›Ύ,(3.2)(a)π»π‘Žπ›Όπ›½=π΅π‘Žπ‘–ξ‚€π›Ώπ›½π΅π‘–π›Ό+𝐡𝑗𝛼𝐿𝑖𝑗𝛽,(b)πΎπ‘Žπ›Όπ›½=π΅π‘Žπ‘–π΅π‘—π›Όξ‚πΆπ‘–π‘—π›½.(3.3) Using (2.6) and (2.16) in (3.3)(a), we have π»π‘Žπ›Όπ›½=π΅π‘Žπ‘–ξƒ©βˆ˜π›Ώπ›½π΅π‘–π›Ό+π΅π‘—π›Όβˆ˜ξ„πΏπ‘–π‘—π›½ξƒͺ+π΅π‘Žπ‘–π΅π›Ύπ‘π‘‰π‘π‘—π΅π‘—π›½π΅π‘–π›Όπ›Ύ+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘π»π‘π›½π΅π‘Žπ‘–π΅π‘—π›Ό.(3.4) If we take βˆ˜π»π‘Žπ›Όπ›½=π΅π‘Žπ‘–(βˆ˜π›Ώπ›½π΅π‘–π›Ό+π΅π‘—π›Όβˆ˜ξ„πΏπ‘–π‘—π›½), the last expression provides π»π‘Žπ›Όπ›½=βˆ˜π»π‘Žπ›Όπ›½+π΅π‘Žπ‘–π΅π›Ύπ‘π‘‰π‘π‘—π΅π‘—π›½π΅π‘–π›Όπ›Ύ+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘π»π‘π›½π΅π‘Žπ‘–π΅π‘—π›Ό.(3.5) Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find πΎπ‘Žπ›Όπ›½=βˆ˜πΎπ‘Žπ›Όπ›½+ξƒ©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’π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‘˜ξƒͺπ΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½,(3.6) where βˆ˜πΎπ‘Žπ›Όπ›½=π΅π‘Žπ‘–π΅π‘—π›Όβˆ˜ξ„πΆπ‘–π‘—π›½. Thus, we have the following.

Theorem 3.1. The following Gauss-Weingarten formulae for the subspace ξ‚πΏπ‘š of an APL space hold: βˆ‡π΅π‘–π›Ό=π΅π‘–π‘ŽΞ π‘Žπ›Ό,βˆ‡π΅π‘–π‘Ž=βˆ’π΅π‘–π›½Ξ π›½π‘Ž,(3.7) where Ξ π‘Žπ›Ό=π»π‘Žπ›Όπ›½π‘‘π‘’π›½+πΎπ‘Žπ›Όπ›½π›Ώπ‘£π›½,Ξ π›½π‘Ž=π‘”π›½π›Ύπ›Ώπ‘Žπ‘Ξ π‘π›Ύ,π»π‘Žπ›Όπ›½=βˆ˜π»π‘Žπ›Όπ›½+π΅π‘Žπ‘–π΅π›Ύπ‘π‘‰π‘π‘—π΅π‘—π›½π΅π‘–π›Όπ›Ύ+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½+ξ‚΅πœ‘ξ…žξ…žπœ‘β€²βˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘π»π‘π›½π΅π‘Žπ‘–π΅π‘—π›Ό,πΎπ‘Žπ›Όπ›½=βˆ˜πΎπ‘Žπ›Όπ›½+ξƒ©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‘ξ…žβˆ’πœ‘ξ…žξ…ž2ξ€Έπœ‘ξ…žξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘–βˆ˜π‘¦π‘—βˆ˜π‘¦π‘˜ξƒͺπ΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½.(3.8)

Remark 3.2. π»π‘Žπ›Όπ›½ and πΎπ‘Žπ›Όπ›½ given, respectively, by (3.5) and (3.6) are called the second fundamental 𝑑-tensor fields of immersion 𝑖.

The following consequences of Theorem 3.1 are straightforward.

Corollary 3.3. In a subspace ξ‚πΏπ‘š of an APL space, we have the following: (a)βˆ‡π‘Žπ›Όπ›½(=0,b)βˆ‡π΅π‘–π›Ό=0,(3.9) if and only if βˆ˜π»π‘Žπ›Όπ›½ξ‚Έπ΅=βˆ’π‘Žπ‘–π΅π›Ύπ‘π‘‰π‘π‘—π΅π‘—π›½π΅π‘–π›Όπ›Ύ+ξ‚΅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‘—βˆ’βˆ˜πΆπ‘–π‘—π‘Ÿπ΅π‘Ÿπ‘π΅π‘π‘π‘‰π‘π‘˜ξ‚Άπ΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½+ξ‚΅πœ‘ξ…žξ…žπœ‘ξ…žβˆ˜π‘¦π‘—π›Ώπ‘–π‘˜+πœ‘ξ…žξ…žπœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–ξ‚Άπ΅π‘˜π‘π»π‘π›½π΅π‘Žπ‘–π΅π‘—π›Όξ‚Ή,βˆ˜πΎπ‘Žπ›Όπ›½βŽ›βŽœβŽœβŽπœ‘=βˆ’ξ…žξ…žπœ‘β€²+2𝐹2πœ‘ξ…žξ…žπ‘”π‘—π‘˜π‘¦π‘–+2ξ‚€πœ‘ξ…žξ…žξ…žπœ‘ξ…žβˆ’πœ‘2ξ…žξ…žξ‚πœ‘ξ…žξ€·πœ‘ξ…ž+2𝐹2πœ‘ξ…žξ…žξ€Έπ‘¦π‘–βˆ˜π‘¦π‘—βˆ˜π‘¦π‘˜βŽžβŽŸβŽŸβŽ π΅π‘Žπ‘–π΅π‘—π›Όπ΅π‘˜π›½.(3.10)

4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspace ξ‚πΏπ‘šξ‚ξ„=(𝑀,𝐿(𝑒,𝑣)) of a Lagrange space 𝐿𝑛 are given by (cf. [8]) π΅π‘–π›Όπ΅π‘—π›½ξ„Ξ©π‘–π‘—βˆ’Ξ©π›Όπ›½=Ξ π›½π‘Žβˆ§Ξ π‘Žπ›Ό,(4.1)π΅π‘–π‘Žπ΅π‘—π‘ξ„Ξ©π‘–π‘—βˆ’Ξ©π‘Žπ‘=Ξ π›Ύπ‘βˆ§Ξ π›Ύπ‘Ž,(4.2)βˆ’π΅π‘–π›Όπ΅π‘—π‘Žξ„Ξ©π‘–π‘—=π›Ώπ‘Žπ‘ξ‚€π‘‘Ξ π‘π›Ό+Ξ π‘π›½βˆ§πœ”π›½π›Όβˆ’Ξ π‘π›Όβˆ§πœ”π‘π‘ξ‚,(4.3) where (a)Ξ π›Όπ‘Ž=π‘”π›Όπ›½Ξ π›½π‘Ž,(b)Π𝛾𝑏=𝛿𝑏𝑐Π𝑐𝛾.(4.4) Using (1.3) in (2.41)(a), we find that Ω𝑖𝑗Ω=πœ‘β€²β„Žπ‘–π‘”β„Žπ‘—+2πœ‘ξ…žξ…žξ„Ξ©β„Žπ‘–βˆ˜π‘¦β„Žβˆ˜π‘¦π‘—.(4.5) Applying π‘Žπ›Ύπ›½=π΅π‘–π›Ύπ΅π‘—π›½π‘Žπ‘–π‘— in (2.41)(b), we have Ω𝛼𝛽=π΅π‘–π›Ύπ΅π‘—π›½Ξ©π›Ύπ›Όπ‘Žπ‘–π‘—, which in view of (1.3) becomes Ω𝛼𝛽=πœ‘β€²π‘”π‘–π‘—π΅π‘–π›Ύπ΅π‘—π›½Ξ©π›Ύπ›Ό+2πœ‘βˆ˜ξ…žξ…žπ‘¦π‘–βˆ˜π‘¦π‘—π΅π‘–π›Ύπ΅π‘—π›½Ξ©π›Ύπ›Ό,(4.6) that is, Ω𝛼𝛽=πœ‘β€²π‘”π›Ύπ›½Ξ©π›Ύπ›Ό+2πœ‘βˆ˜ξ…žξ…žπ‘¦π‘–βˆ˜π‘¦π‘—π΅π‘–π›Ύπ΅π‘—π›½Ξ©π›Ύπ›Ό.(4.7) For the subspace ξ‚πΏπ‘š of an APL space, (4.4)(a) is of the form Ξ π›Όπ‘Ž=π‘Žπ›Όπ›½Ξ π›½π‘Ž, which in view of π‘Žπ›Όπ›½=π΅π‘–π›Όπ΅π‘—π›½π‘Žπ‘–π‘— and (1.3) becomes Ξ π›Όπ‘Ž=πœ‘β€²π΅π‘–π›Όπ΅π‘—π›½π‘Žπ‘–π‘—Ξ π›½π‘Ž+2πœ‘βˆ˜ξ…žξ…žπ‘¦π‘–βˆ˜π‘¦π‘—π΅π‘–π›Όπ΅π‘—π›½Ξ π›½π‘Ž, that is, Ξ π›Όπ‘Ž=πœ‘β€²π‘”π›Όπ›½Ξ π›½π‘Ž+2πœ‘βˆ˜ξ…žξ…žπ‘¦π‘–βˆ˜π‘¦π‘—π΅π‘–π›Όπ΅π‘—π›½Ξ π›½π‘Ž.(4.8) Thus, we have the following.

Theorem 4.1. The Gauss-Codazzi equations for a Lagrange subspace ξ‚πΏπ‘š of an APL space are given by (4.1)–(4.3) with Ξ π›Όπ‘Ž, Π𝛾𝑏, Ω𝑖𝑗, Ω𝛼𝛽, and πœ”π‘π‘, respectively, given by (4.8), (4.4)(b), (4.5), (4.7), and (2.37).

Acknowledgments

Authors are thankful to the reviewers for their valuable comments and suggestions. S. K. Shukla gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), India.

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Copyright © 2012 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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