On Subspaces of an Almost 𝝋-Lagrange Space

Abstract

We discuss the subspaces of an almost 𝜑-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 𝜑-Lagrangians and studied 𝜑-Lagrange manifolds. Generalizing the notion of a 𝜑-Lagrange manifold, the present authors recently studied the geometry of an almost 𝜑-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 𝐹^𝑛=(𝑀,𝐹(𝑥,𝑦)) be an 𝑛-dimensional Finsler space and 𝜑∶ℝ⁺→ℝ a smooth function. If the function 𝜑 has the following properties:(a)𝜑^(𝑡)≠0, (b)𝜑^(𝑡)+𝜑^(𝑡)≠0, for every 𝑡∈Im(𝐹²),then the Lagrangian given by 𝐿(𝑥,𝑦)=𝜑𝐹²+𝐴_𝑖(𝑥)𝑦^𝑖+𝑈(𝑥),(1.1) where 𝐴_𝑖(𝑥) is a covector and 𝑈(𝑥) is a smooth function, is a regular Lagrangian [7]. The space 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦)) is a Lagrange space. The present authors [7] called such space as an almost 𝜑-Lagrange space (shortly APL space) associated to the Finsler space 𝐹^𝑛. An APL space reduces to a 𝜑-Lagrange space if and only if 𝐴_𝑖(𝑥)=0 and 𝑈(𝑥)=0. We take 𝑔_𝑖𝑗=12𝜕̇^𝑖𝜕̇_𝑗𝐹²,𝑎_𝑖𝑗=12𝜕̇^𝑖𝜕̇_𝑗𝐿,where𝜕̇_𝑖≡𝜕𝜕𝑦^𝑖.(1.2) We indicate all the geometrical objects related to 𝐹^𝑛 by putting a small circle “∘” over them. Equations (1.2), in view of (1.1), provide the following expressions for 𝑎_𝑖𝑗 and its inverse (cf. [7]): 𝑎_𝑖𝑗=𝜑^⋅𝑔_𝑖𝑗+2𝜑^𝜑^∘𝑦𝑦_𝑖∘𝑗,𝑎^𝑖𝑗=1𝜑′𝑔^𝑖𝑗−2𝜑^𝜑′+2𝐹²𝜑^𝑦^𝑖𝑦^𝑗,(1.3) where 𝑔_𝑖𝑗𝑦^𝑗=𝑦^∘_𝑖.

Let 𝑀 be a smooth manifold of dimension 𝑚, 1<𝑚<𝑛, immersed in 𝑀 by immersion 𝑖∶𝑀→𝑀. The immersion 𝑖 induces an immersion 𝑇_𝑖∶𝑇𝑀→𝑇𝑀 making the following diagram commutative: 𝑀𝑇𝜋𝑀↓⟶⟶𝑖^𝑇𝑖↓𝑇𝑀𝜋𝑀.(1.4)

Let (𝑢^𝛼,𝑣^𝛼) (throughout the paper, the Greek indices 𝛼,𝛽,𝛾,… run from 1 to 𝑚) be local coordinates on 𝑇𝑀. The restriction of the Lagrangian 𝐿 on 𝑇𝑀 is 𝐿(𝑢,𝑣)=𝐿(𝑥(𝑢),𝑦(𝑢,𝑣)). Let 𝑎_𝛼𝛽=(1/2)(𝜕²𝐿/𝜕𝑢^𝛼𝜕𝑢^𝛽). Then, we have (cf. [8]) 𝑎_𝛼𝛽=𝐵^𝑖𝛼𝐵^𝑗𝛽𝑎_𝑖𝑗 where 𝐵^𝑖𝛼(𝑢)=𝜕𝑥^𝑖/𝜕𝑢^𝛼 are the projection factors. The pair 𝐿^𝑚=(𝑀,𝐿(𝑢,𝑣)) is also a Lagrange space, called the subspace of 𝐿^𝑛. For the natural bases (𝜕/𝜕𝑥^𝑖,𝜕/𝜕𝑦^𝑖) on TM and (𝜕/𝜕𝑢^𝛼,𝜕/𝜕𝑣^𝛼) on 𝑇𝑀, we have [8] 𝜕𝜕𝑢^𝛼=𝐵^𝑖𝛼𝜕𝜕𝑥^𝑖+𝐵^𝑖0𝛼𝜕𝜕𝑦^𝑖,𝜕𝜕𝑣^𝛼=𝐵^𝑖𝛼𝜕𝜕𝑦^𝑖,(1.5) where 𝐵^𝑖0𝛼=𝐵^𝑖𝛽𝛼𝑣^𝛽,𝐵^𝑖𝛽𝛼=𝜕²𝑥^𝑖/𝜕𝑢^𝛼𝜕𝑢^𝛽.

For the bases (𝑑𝑥^𝑖,𝑑𝑦^𝑖) and (𝑑𝑢^𝛼,𝑑𝑣^𝛼), we have 𝑑𝑥^𝑖=𝐵^𝑖𝛼𝑑𝑢^𝛼,𝑑𝑦^𝑖=𝐵^𝑖𝛼𝑑𝑣^𝛼+𝐵^𝑖0𝛼𝑑𝑢^𝛼.(1.6) Since (𝐵^𝑖𝛼) are 𝑚 linearly independent vector fields tangent to 𝑀, a vector field 𝜉^𝑖(𝑥,𝑦) is normal to 𝑀 along 𝑇𝑀 if on 𝑇𝑀, we have 𝑎_𝑖𝑗(𝑥(𝑢),𝑦(𝑢,𝑣))𝐵^𝑖𝛼𝜉^𝑗=0,∀𝛼=1,2,…,𝑚.(1.7) There are, at least locally, (𝑛−𝑚) unit vector fields 𝐵^𝑖𝑎(𝑢,𝑣)(𝑎=𝑚+1,𝑚+2,…,𝑛) normal to 𝑀 and mutually orthonormal, that is, 𝑎_𝑖𝑗𝐵^𝑖𝛼𝐵^𝑗𝑏=0,𝑎_𝑖𝑗𝐵^𝑖𝑎𝐵^𝑗𝑏=𝛿_𝑎𝑏,(𝑎,𝑏=𝑚+1,𝑚+2,…,𝑛).(1.8) Thus, at every point (𝑢,𝑣)∈𝑇𝑀, we have a moving frame ℜ=((𝑢,𝑣),𝐵^𝑖𝛼(𝑢,𝑣),𝐵^𝑖𝑎(𝑢,𝑣)). Using (1.3) in the first expression of (1.8) and keeping 𝑦^∘_𝑖𝐵^𝑖𝑎=0 (this fact is clear from 𝑔_𝑖𝑗𝑦^𝑖𝐵^𝑗𝑎=0) in view, we observe that 𝐵^𝑖𝑎's are normal to 𝑀 with respect to 𝐿^𝑛 if and only if they are so with respect to 𝐹^𝑛. The dual frame of ℜ is ℜ^∗=((𝑢,𝑣),𝐵^𝛼𝑖(𝑢,𝑣),𝐵^𝑎𝑖(𝑢,𝑣)) with the following duality conditions: 𝐵^𝑖𝛼𝐵^𝛽𝑖=𝛿^𝛽𝛼,𝐵^𝑖𝑎𝐵^𝛽𝑖=0,𝐵^𝑖𝛼𝐵^𝑏𝑖=0,𝐵^𝑖𝑎𝐵^𝑏𝑖=𝛿^𝑏𝑎,𝐵^𝑖𝑎𝐵^𝑎𝑗+𝐵^𝑖𝛼𝐵^𝛼𝑗=𝛿^𝑖𝑗.(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 𝐿^𝑛 has the local coefficients given by 𝑁^𝑖𝑗=𝑁^∘𝑖𝑗−𝑉^𝑖𝑗,(1.10) where 𝑉^𝑖𝑗=(1/2)𝐹^𝑖𝑗−𝑆^𝑗𝑖𝑟(2𝐹_𝑟𝑘𝑦^𝑘+𝜕_𝑟𝑈), 𝑆^𝑗𝑖𝑟=12𝜑′^∘𝐶^𝑖𝑞𝑗𝑔^𝑞𝑟+12𝜑𝜑′^2𝑔^∘𝑖𝑟𝑦_𝑗+𝜑^𝛿^𝑟𝑗𝑦^𝑖+𝛿^𝑖𝑗𝑦^𝑟2𝜑^𝜑^+2𝐹²𝜑^+𝜑′²𝜑^−2𝜑^3𝐹²−4𝜑′𝜑^22𝜑′²𝜑^+2𝐹²𝜑^²𝐹_𝑟𝑘𝑦𝑦^𝑖∘_𝑗𝑦^𝑟,(𝑥)=12𝜕^𝑟𝐴_𝑘−𝜕_𝑘𝐴_𝑟,𝐹^𝑖𝑗=𝑎^𝑖𝑘𝐹_𝑘𝑗.(1.11)

Theorem 1.2 (cf. [7]). The coefficients of the canonical metrical d-connection 𝐶Γ(𝑁) of an APL space 𝐿^𝑛 are given by 𝐶^𝑖𝑗𝑘=𝐶^{∘𝑖𝑗𝑘}+𝜑^𝜑^𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−2𝜑^2𝜑^𝜑^+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘,(1.12)𝐿^𝑖𝑗𝑘=𝐿^{∘𝑖𝑗𝑘}+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗.(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 𝑁=(𝑁^𝛼𝛽(𝑢,𝑣)) be a nonlinear connection for 𝐿^𝑚=(𝑀,𝐿(𝑢,𝑣)). The adapted basis of 𝑇_(𝑢,𝑣)𝑇𝑀 induced by 𝑁 is (𝛿/𝛿𝑢^𝛼=𝛿_𝛼,𝜕/𝜕𝑣^𝛼=𝜕̇_𝛼), where 𝛿_𝛼=𝜕_𝛼−𝑁^𝛽𝛼𝜕̇_𝛽.(2.1) The dual basis (cobasis) of the adapted basis (𝛿_𝛼,𝜕̇_𝛼) is (𝑑𝑢^𝛼,𝛿𝑣^𝛼=𝑑𝑣^𝛼+𝑁^𝛼𝛽𝑑𝑢^𝛽).

Definition 2.1 (cf. [8]). A nonlinear connection 𝑁=(𝑁^𝛼𝛽(𝑢,𝑣)) of 𝐿^𝑚 is said to be induced by the canonical nonlinear connection 𝑁 if the following equation holds good: 𝛿𝑣^𝛼=𝐵^𝛼𝑖𝛿𝑦^𝑖.(2.2) The local coefficients of the induced nonlinear connection 𝑁=(𝑁^𝛼𝛽(𝑢,𝑣)) for the subspace 𝐿^𝑚=(𝑀,𝐿(𝑢,𝑣)) of a Lagrange space 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦)) are given by (cf. [8]) 𝑁^𝛼𝛽=𝐵^𝛼𝑖𝑁^𝑖𝑗𝐵^𝑗𝛽+𝐵^𝑖0𝛽,(2.3)𝑁^𝑖𝑗 being the local coefficients of canonical nonlinear connection 𝑁 of the Lagrange space 𝐿^𝑛=(𝑀,𝐿(𝑥,𝑦)). Now using (1.10) in (2.3), we get 𝑁^𝛼𝛽=𝐵^𝛼𝑖^∘𝑁^𝑖𝑗𝐵^𝑗𝛽+𝐵^𝑖0𝛽−𝐵^𝛼𝑖𝑉^𝑖𝑗𝐵^𝑗𝛽.(2.4) If we take 𝑁^∘𝛼𝛽=𝐵^𝛼𝑖(𝑁^∘𝑖𝑗𝐵^𝑗𝛽+𝐵^𝑖0𝛽), it follows from (2.4) that 𝑁^𝛼𝛽=𝑁^∘𝛼𝛽−𝐵^𝛼𝑖𝑉^𝑖𝑗𝐵^𝑗𝛽.(2.5) Thus, we have the following.

Theorem 2.2. The local coefficients of the induced nonlinear connection 𝑁 of the subspace 𝐿^𝑚 of an APL space 𝐿^𝑛 are given by (2.5).

In view of (2.5), (2.1) takes the following form, for the subspace 𝐿^𝑚 of an APL space 𝐿^𝑛: 𝛿_𝛽=𝛿^∘_𝛽+𝐵^𝛼𝑝𝑉^𝑝𝑗𝐵^𝑗𝛽𝜕̇_𝛼,(2.6) where 𝛿^∘_𝛽=𝜕_𝛽−𝑁^∘𝛼𝛽𝜕̇_𝛼.

We can put (𝑑𝑥^𝑖,𝛿𝑦^𝑖) as (cf. [8]) 𝑑𝑥^𝑖=𝐵^𝑖𝛼𝑑𝑢^𝛼,𝛿𝑦^𝑖=𝐵^𝑖𝛼𝛿𝑦^𝛼+𝐵^𝑖𝑎𝐻^𝑎𝛼𝑑𝑢^𝛼,(2.7) where 𝐻^𝑎𝛼=𝐵^𝑎𝑖𝑁^𝑖𝑗𝐵^𝑗𝛼+𝐵^𝑖0𝛼.(2.8) Using (1.10) in (2.8) and simplifying, we get 𝐻^𝑎𝛼=𝐵^𝑎𝑖^∘𝑁^𝑖𝑗𝐵^𝑗𝛼+𝐵^𝑖0𝛼−𝐵^𝑎𝑖𝑉^𝑖𝑗𝐵^𝑗𝛼.(2.9) Taking 𝐻^∘𝑎𝛼=𝐵^𝑎𝑖(𝑁^∘𝑖𝑗𝐵^𝑗𝛼+𝐵^𝑖0𝛼), in (2.9), it follows that 𝐻^𝑎𝛼=𝐻^∘𝑎𝛼−𝐵^𝑎𝑖𝑉^𝑖𝑗𝐵^𝑗𝛼.(2.10) Now, 𝑑𝑥^𝑖=𝐵^𝑖𝛼𝑑𝑢^𝛼,𝛿𝑦^𝑖=𝐵^𝑖𝛼𝛿𝑦^𝛼 if and only if 𝐻^𝑎𝛼=0, that is, if and only if 𝐻^∘𝑎𝛼=𝐵^𝑎𝑖𝑉^𝑖𝑗𝐵^𝑗𝛼. Thus, we have the following.

Theorem 2.3. The adapted cobasis (𝑑𝑥^𝑖,𝛿𝑦^𝑖) of the basis (𝜕/𝜕𝑥^𝑖,𝜕/𝜕𝑦^𝑖) induced by the nonlinear connection 𝑁 of an APL space 𝐿^𝑛 is of the form 𝑑𝑥^𝑖=𝐵^𝑖𝛼𝑑𝑢^𝛼,𝛿𝑦^𝑖=𝐵^𝑖𝛼𝛿𝑦^𝛼 if and only if 𝐻^∘𝑎𝛼=𝐵^𝑎𝑖𝑉^𝑖𝑗𝐵^𝑗𝛼.

Definition 2.4 (cf. [8]). Let 𝐷=𝐷Γ(𝑁) be the canonical metrical 𝑑-connection of 𝐿^𝑛. An operator 𝐷 is said to be a coupling of 𝐷 with 𝑁 if 𝐷𝑋^𝑖=𝑋^𝑖|𝛼𝑑𝑢^𝛼+𝑋^𝑖|_𝛼𝛿𝑣^𝛼,(2.11) where 𝑋^𝑖|𝛼=𝛿_𝛼𝑋^𝑖+𝑋^𝑗𝐿^𝑖𝑗𝛼,𝑋^𝑖|_𝛼=𝜕̇_𝛼𝑋^𝑖+𝑋^𝑗𝐶^𝑖𝑗𝛼.
The coefficients (𝐿^𝑖𝑗𝛼,𝐶^𝑖𝑗𝛼) of coupling 𝐷 of 𝐷 with 𝑁 are given by 𝐿^𝑖𝑗𝛼=𝐿^𝑖𝑗𝑘𝐵^𝑘𝛼+𝐶^𝑖𝑗𝑘𝐵^𝑘𝑎𝐻^𝑎𝛼,(2.12)𝐶^𝑖𝑗𝛼=𝐶^𝑖𝑗𝑘𝐵^𝑘𝛼.(2.13) Using (1.12) and (1.13) in (2.12), we have 𝐿^𝑖𝑗𝛽=𝐿^{∘𝑖𝑗𝑘}+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗𝐵^𝑘𝛽+𝐶^{∘𝑖𝑗𝑘}+𝜑^𝜑^𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑^−2𝜑^2𝜑^𝜑^+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑘𝑎𝐻^𝑎𝛽.(2.14) In view of (2.10) and 𝑦^∘_𝑖𝐵^𝑖𝑎=0, (2.14) becomes 𝐿^𝑖𝑗𝛽=𝐿^{∘𝑖𝑗𝑘}𝐵^𝑘𝛽+𝐶^{∘𝑖𝑗𝑘}𝐵𝐻^{𝑘𝑎∘𝑎𝛽}+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑘𝛽+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑎𝐻^𝑎𝛽,(2.15) that is, 𝐿^𝑖𝑗𝛽=𝐿^{∘𝑖𝑗𝛽}+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑘𝛽+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑎𝐻^𝑎𝛽,(2.16) where 𝐿^{∘𝑖𝑗𝛽}=𝐿^{∘𝑖𝑗𝑘}𝐵^𝑘𝛽+𝐶^{∘𝑖𝑗𝑘}𝐵𝐻^{𝑘𝑎∘𝑎𝛽}.
Using (1.12) in (2.13), we find that 𝐶^𝑖𝑗𝛽=𝐶^{∘𝑖𝑗𝑘}𝐵^𝑘𝛽+𝜑^𝜑′𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−2𝜑^2𝜑′𝜑′+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑘𝛽,(2.17) that is, 𝐶^𝑖𝑗𝛽=𝐶^{∘𝑖𝑗𝛽}+𝜑^𝜑′𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−2𝜑^2𝜑′𝜑′+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑘𝛽,(2.18) where 𝐶^{∘𝑖𝑗𝛽}=𝐶^{∘𝑖𝑗𝑘}𝐵^𝑘𝛽. Thus, we have the following.

Theorem 2.5. The coefficients of coupling for the subspace 𝐿^𝑚 of an APL space 𝐿^𝑛 are given by (2.16) and (2.18).

Definition 2.6 (cf. [8]). An operator 𝐷^𝑇 given by 𝐷^𝑇𝑋^𝛼=𝑋^𝛼|𝛽𝑑𝑢^𝛽+𝑋^𝛼|_𝛽𝛿𝑣^𝛽,(2.19) where 𝑋^𝛼|𝛽=𝛿_𝛽𝑋^𝛼+𝑋^𝛾𝐿^𝛼𝛾𝛽,𝑋^𝛼|_𝛽=𝜕̇_𝛽𝑋^𝛼+𝑋^𝛾𝐶^𝛼𝛾𝛽, is called the induced tangent connection by 𝐷. This defines an 𝑁-linear connection for 𝐿^𝑚.
The coefficients (𝐿^𝛼𝛾𝛽,𝐶^𝛼𝛾𝛽) of 𝐷^𝑇 are given by 𝐿^𝛼𝛽𝛾=𝐵^𝛼𝑖𝐵^𝑖𝛽𝛾+𝐵^𝑗𝛽𝐿^𝑖𝑗𝛾,(2.20)𝐶^𝛼𝛽𝛾=𝐵^𝛼𝑖𝐵^𝑗𝛽𝐶^𝑖𝑗𝛾.(2.21) Using (2.16) in (2.20), we get 𝐿^𝛼𝛽𝛾=𝐵^𝛼𝑖𝐵^𝑖𝛽𝛾+𝐵^𝑗𝛽𝐵^𝛼𝑖^∘𝐿^𝑖𝑗𝛾+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑘𝛾+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑎𝐻^𝑎𝛾,(2.22) that is, 𝐿^𝛼𝛽𝛾=𝐵^𝛼𝑖𝐵^𝑖𝛽𝛾+𝐿^{∘𝑖𝑗𝛾}𝐵^𝑗𝛽+𝐵^𝛼𝑖𝐵^𝑗𝛽𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑘𝛾+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑎𝐻^𝑎𝛾.(2.23) If we take 𝐿^{∘𝛼𝛽𝛾}=𝐵^𝛼𝑖(𝐵^𝑖𝛽𝛾+𝐿^{∘𝑖𝑗𝛾}𝐵^𝑗𝛽), the last expression gives 𝐿^𝛼𝛽𝛾=𝐿^{∘𝛼𝛽𝛾}+𝐵^𝛼𝑖𝐵^𝑗𝛽𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑘𝛾+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑎𝐻^𝑎𝛾.(2.24) Next, using (2.18) in (2.21), we obtain 𝐶^𝛼𝛽𝛾=𝐵^𝛼𝑖𝐵𝐶^{𝑗𝛽∘𝑖𝑗𝛾}+𝜑^𝜑′𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−2𝜑^2𝜑′𝜑′+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑘𝛾𝐵^𝛼𝑖𝐵^𝑗𝛽.(2.25) If we take 𝐶^{∘𝛼𝛽𝛾}=𝐵^𝛼𝑖𝐵𝐶^{𝑗𝛽∘𝑖𝑗𝛾}, (2.25) becomes 𝐶^𝛼𝛽𝛾=𝐶^{∘𝛼𝛽𝛾}+𝜑^𝜑′𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−2𝜑^2𝜑′𝜑′+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑘𝛾𝐵^𝛼𝑖𝐵^𝑗𝛽.(2.26) Thus, we have the following.

Theorem 2.7. The coefficients of the induced tangent connection 𝐷^𝑇 for the subspace 𝐿^𝑚 of an APL space are given by (2.24) and (2.26).

Remarks 2. The torsion 𝑇^𝛼𝛽𝛾=𝐿^𝛼𝛽𝛾−𝐿^𝛼𝛾𝛽 does not vanish, in general, while 𝑆^𝛼𝛽𝛾=𝐶^𝛼𝛽𝛾−𝐶^𝛼𝛾𝛽=0. These facts may be observed from (2.24) and (2.26).

Definition 2.8 (cf. [8]). An operator 𝐷^⟂ given by 𝐷^⟂𝑋^𝑎=𝑋^𝑎|𝛼𝑑𝑢^𝛼+𝑋^𝑎|_𝛼𝛿𝑣^𝛼,(2.27) where 𝑋^𝑎|𝛼=𝛿_𝛼𝑋^𝑎+𝑋^𝑏𝐿^𝑎𝑏𝛼,𝑋^𝑎|_𝛼=𝜕̇_𝛼𝑋^𝑎+𝑋^𝑏𝐶^𝑎𝑏𝛼, is called the induced normal connection by 𝐷.
The coefficients (𝐿^𝑎𝑏𝛾,𝐶^𝑎𝑏𝛾) of 𝐷^⟂ are given by 𝐿^𝑎𝑏𝛾=𝐵^𝑎𝑖𝛿^𝛾𝐵^𝑖𝑏+𝐵^𝑗𝑏𝐿^𝑖𝑗𝛾,(2.28)𝐶^𝑎𝑏𝛾=𝐵^𝑎𝑖𝜕̇^𝛾𝐵^𝑖𝑏+𝐵^𝑗𝑏𝐶^𝑖𝑗𝛾.(2.29) Using (2.6) and (2.16) in (2.28), we find 𝐿^𝑎𝑏𝛾=𝐵𝛿^𝑎𝑖∘_𝛾𝐵^𝑖𝑏+𝐵^𝑎𝑖𝐵^𝛼𝑝𝑉^𝑝𝑗𝐵^𝑗𝛾𝜕̇_𝛼𝐵^𝑖𝑏+𝐵^𝑗𝑏𝐵^𝑎𝑖^∘𝐿^𝑖𝑗𝛾+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑐𝐵^𝑐𝑝𝑉^𝑝𝑘𝐵^𝑘𝛾+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑐𝐻^𝑐𝛾.(2.30) Taking 𝐿^{∘𝑎𝑏𝛾}=𝐵^𝑎𝑖(𝛿^∘_𝛾𝐵^𝑖𝑏+𝐵𝐿^{𝑗𝑏∘𝑖𝑗𝛾}) and using 𝑦^∘_𝑗𝐵^𝑗𝑏=0, (2.30) reduces to 𝐿^𝑎𝑏𝛾=𝐿^{∘𝑎𝑏𝛾}+𝐵^𝑎𝑖𝐵^𝛼𝑝𝑉^𝑝𝑗𝐵^𝑗𝛾𝜕̇_𝛼𝐵^𝑖𝑏+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑎𝑖𝐵^𝑗𝑏𝐵^𝑘𝛾+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑐𝐻^𝑐𝛾𝐵^𝑎𝑖𝐵^𝑗𝑏.(2.31) Next, using (2.18) in (2.29), we have 𝐶^𝑎𝑏𝛾=𝐵^𝑎𝑖𝜕̇^𝛾𝐵^𝑖𝑏+𝐵𝐶^{𝑗𝑏∘𝑖𝑗𝛾}+𝜑𝜑′^𝛿^𝑖𝑗∘𝑦_𝑘+𝛿𝑦^𝑖𝑘∘_𝑗+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−2𝜑^2𝜑′𝜑′+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑘𝛾𝐵^𝑎𝑖𝐵^𝑗𝑏.(2.32) Taking 𝐶^{∘𝑎𝑏𝛾}=𝐵^𝑎𝑖(𝜕̇_𝛾𝐵^𝑖𝑏+𝐵𝐶^{𝑗𝑏∘𝑖𝑗𝛾}) and using (1.9) and 𝑦^∘_𝑗𝐵^𝑗𝑏=0, the last equation yields 𝐶^𝑎𝑏𝛾=𝐶^{∘𝑎𝑏𝛾}+𝜑^𝜑′𝛿^𝑎𝑏∘𝑦_𝑘𝐵^𝑘𝛾+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝛾𝐵^𝑎𝑖𝐵^𝑗𝑏.(2.33) Thus, we have the following.

Theorem 2.9. The coefficients of induced normal connection 𝐷^⟂ for the subspace 𝐿^𝑚 of an APL space 𝐿^𝑛 are given by (2.31) and (2.33).

Definition 2.10 (cf. [8]). The (mixed) derivative of a mixed d-tensor field 𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑏} is given by ∇𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑏}=𝛿_𝜂𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑏}+𝑇^{𝑘⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑏}𝐿^𝑖𝑘𝜂+⋯+𝑇^{𝑖⋯𝛾⋯𝑎𝑗⋯𝛽⋯𝑏}𝐿^𝛼𝛾𝜂+⋯+𝑇^{𝑖⋯𝛼⋯𝑐𝑗⋯𝛽⋯𝑏}𝐿^𝑎𝑐𝜂−𝑇^{𝑖⋯𝛼⋯𝑎𝑘⋯𝛽⋯𝑏}𝐿^𝑘𝑗𝜂−⋯−𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛾⋯𝑏}𝐿^𝛾𝛽𝜂−⋯−𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑐}𝐿^𝑐𝑏𝜂𝑑𝑢^𝜂+𝜕̇_𝜂𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑏}+𝑇^{𝑘⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑏}𝐶^𝑖𝑘𝜂+⋯+𝑇^{𝑖⋯𝛾⋯𝑎𝑗⋯𝛽⋯𝑏}𝐶^𝛼𝛾𝜂+⋯+𝑇^{𝑖⋯𝛼⋯𝑐𝑗⋯𝛽⋯𝑏}𝐶^𝑎𝑐𝜂−𝑇^{𝑖⋯𝛼⋯𝑎𝑘⋯𝛽⋯𝑏}𝐶^𝑘𝑗𝜂−⋯−𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛾⋯𝑏}𝐶^𝛾𝛽𝜂−⋯−𝑇^{𝑖⋯𝛼⋯𝑎𝑗⋯𝛽⋯𝑐}𝐶^𝑐𝑏𝜂𝛿𝑣^𝜂.(2.34) The connection 1-forms, 𝜔^𝑖𝑗=∶𝐿^𝑖𝑗𝛼𝑑𝑢^𝛼+𝐶^𝑖𝑗𝛼𝛿𝑣^𝛼,(2.35)𝜔^𝛼𝛽=∶𝐿^𝛼𝛽𝛾𝑑𝑢^𝛾+𝐶^𝛼𝛽𝛾𝛿𝑣^𝛾,(2.36)𝜔^𝑎𝑏=∶𝐿^𝑎𝑏𝛾𝑑𝑢^𝛾+𝐶^𝑎𝑏𝛾𝛿𝑣^𝛾,(2.37) are called the connection 1-forms of ∇. We have the following structure equations of ∇.

Theorem 2.11 (cf. [8]). The structure equations of ∇ are as follows: 𝑑𝑑((𝑑𝛿𝑢𝑢^𝛼_𝛼)−𝑑𝑢^𝛽∧𝜔^𝛼𝛽=−Ω^𝛼,𝑑𝜔_𝑖𝑗)−𝛿𝑢^𝛽∧𝜔^𝛼𝛽=−Ω̇^𝛼,𝑑𝜔_𝛼𝛽−𝜔^ℎ𝑗∧𝜔^𝑖ℎ=−Ω^𝑖𝑗,𝑑𝜔_𝑎𝑏−𝜔^𝛾𝛽∧𝜔^𝛼𝛾=−Ω^𝛼𝛽,−𝜔^𝑐𝑏∧𝜔^𝑎𝑐=−Ω^𝑎𝑏,(2.38) where the 2-forms of torsions Ω^𝛼,Ω̇^𝛼 are given by Ω^𝛼=12𝑇^𝛼𝛽𝛾Ω̇_𝛼𝑑𝑢^𝛽∧𝑑𝑢^𝛾+𝐶^𝛼𝛽𝛾𝑑𝑢^𝛽∧𝛿𝑣^𝛾,=12𝑅^𝛼𝛽𝛾𝑑𝑢^𝛽∧𝑑𝑢^𝛾+𝑃^𝛼𝛽𝛾𝑑𝑢^𝛽∧𝛿𝑣^𝛾,(2.39) with 𝑃^𝛼𝛽𝛾=𝜕̇_𝛾𝑁^𝛼𝛽−𝐿^𝛼𝛽𝛾, and the 2-forms of curvature Ω^𝑖𝑗,Ω^𝛼𝛽 and Ω^𝑎𝑏, are given by Ω^𝑖𝑗=12𝑅^{𝑖𝑗𝛼𝛽}𝑑𝑢^𝛼∧𝑑𝑢^𝛽+𝑃^{𝑖𝑗𝛼𝛽}𝑑𝑢^𝛼∧𝛿𝑣^𝛽+12𝑆^{𝑖𝑗𝛼𝛽}Ω_𝛼𝛽𝛿𝑣^𝛼∧𝛿𝑣^𝛽,=12𝑅^{𝛼𝛽𝛾𝛿}𝑑𝑢^𝛾∧𝑑𝑢^𝛿+𝑃^{𝛼𝛽𝛾𝛿}𝑑𝑢^𝛾∧𝛿𝑣^𝛿+12𝑆^{𝛼𝛽𝛾𝛿}Ω_𝑎𝑏𝛿𝑣^𝛾∧𝛿𝑣^𝛿,=12𝑅^{𝑎𝑏𝛼𝛽}𝑑𝑢^𝛼∧𝑑𝑢^𝛽+𝑃^{𝑎𝑏𝛼𝛽}𝑑𝑢^𝛼∧𝛿𝑣^𝛽+12𝑆^{𝑎𝑏𝛼𝛽}𝛿𝑣^𝛼∧𝛿𝑣^𝛽.(2.40)

We will use the following notations in Section 4: (a)Ω_𝑖𝑗=Ω^ℎ𝑖𝑎_ℎ𝑗,(b)Ω_𝛼𝛽=Ω^𝛾𝛼𝑎_𝛾𝛽,(c)Ω_𝑎𝑏=Ω^𝑐𝑏𝛿_𝑎𝑐.(2.41)

3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspace 𝐿^𝑚=(𝑀,𝐿(𝑢,𝑣)) of a Lagrange space 𝐿^𝑛 are given by (cf. [8]) ∇𝐵^𝑖𝛼=𝐵^𝑖𝑎Π^𝑎𝛼,∇𝐵^𝑖𝑎=−𝐵^𝑖𝛽Π^𝛽𝑎,(3.1) where Π^𝑎𝛼=Π𝐻^𝑎𝛼𝛽_𝛽𝑎𝑑𝑢^𝛽+𝐾^𝑎𝛼𝛽𝛿𝑣^𝛽,=𝑔^𝛽𝛾𝛿_𝑎𝑏Π^𝑏𝛾,(3.2)(a)𝐻^𝑎𝛼𝛽=𝐵^𝑎𝑖𝛿^𝛽𝐵^𝑖𝛼+𝐵^𝑗𝛼𝐿^𝑖𝑗𝛽,(b)𝐾^𝑎𝛼𝛽=𝐵^𝑎𝑖𝐵^𝑗𝛼𝐶^𝑖𝑗𝛽.(3.3) Using (2.6) and (2.16) in (3.3)(a), we have 𝐻^𝑎𝛼𝛽=𝐵^𝑎𝑖^∘𝛿_𝛽𝐵^𝑖𝛼+𝐵𝐿^{𝑗𝛼∘𝑖𝑗𝛽}+𝐵^𝑎𝑖𝐵^𝛾𝑝𝑉^𝑝𝑗𝐵^𝑗𝛽𝐵^𝑖𝛼𝛾+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑏𝐻^𝑏𝛽𝐵^𝑎𝑖𝐵^𝑗𝛼.(3.4) If we take 𝐻^{∘𝑎𝛼𝛽}=𝐵^𝑎𝑖(𝛿^∘_𝛽𝐵^𝑖𝛼+𝐵𝐿^{𝑗𝛼∘𝑖𝑗𝛽}), the last expression provides 𝐻^𝑎𝛼𝛽=𝐻^{∘𝑎𝛼𝛽}+𝐵^𝑎𝑖𝐵^𝛾𝑝𝑉^𝑝𝑗𝐵^𝑗𝛽𝐵^𝑖𝛼𝛾+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑏𝐻^𝑏𝛽𝐵^𝑎𝑖𝐵^𝑗𝛼.(3.5) Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find 𝐾^𝑎𝛼𝛽=𝐾^{∘𝑎𝛼𝛽}+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑′−𝜑^2𝜑′𝜑′+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽,(3.6) where 𝐾^{∘𝑎𝛼𝛽}=𝐵^𝑎𝑖𝐵𝐶^{𝑗𝛼∘𝑖𝑗𝛽}. Thus, we have the following.

Theorem 3.1. The following Gauss-Weingarten formulae for the subspace 𝐿^𝑚 of an APL space hold: ∇𝐵^𝑖𝛼=𝐵^𝑖𝑎Π^𝑎𝛼,∇𝐵^𝑖𝑎=−𝐵^𝑖𝛽Π^𝛽𝑎,(3.7) where 𝐻_𝑎𝛼𝛽Π^𝑎𝛼=𝐻^𝑎𝛼𝛽𝑑𝑢^𝛽+𝐾^𝑎𝛼𝛽𝛿𝑣^𝛽,Π^𝛽𝑎=𝑔^𝛽𝛾𝛿_𝑎𝑏Π^𝑏𝛾,=𝐻^{∘𝑎𝛼𝛽}+𝐵^𝑎𝑖𝐵^𝛾𝑝𝑉^𝑝𝑗𝐵^𝑗𝛽𝐵^𝑖𝛼𝛾+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽+𝜑^𝜑′^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑏𝐾_𝑎𝛼𝛽𝐻^𝑏𝛽𝐵^𝑎𝑖𝐵^𝑗𝛼,=𝐾^{∘𝑎𝛼𝛽}+𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑^−𝜑^2𝜑^𝜑^+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘𝐵^𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽.(3.8)

Remark 3.2. 𝐻^𝑎𝛼𝛽 and 𝐾^𝑎𝛼𝛽 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 𝐿^𝑚 of an APL space, we have the following: ((a)∇𝑎_𝛼𝛽=0,b)∇𝐵^𝑖𝛼=0,(3.9) if and only if 𝐻^{∘𝑎𝛼𝛽}=−𝐵^𝑎𝑖𝐵^𝛾𝑝𝑉^𝑝𝑗𝐵^𝑗𝛽𝐵^𝑖𝛼𝛾+𝑉^𝑟𝑘𝐶^𝑖𝑗𝑟+𝑉^𝑟𝑗𝐶^𝑖𝑘𝑟+𝑉^𝑟𝑝𝑎^𝑖𝑝𝐶_𝑟𝑘𝑗−𝐶^{∘𝑖𝑗𝑟}𝐵^𝑟𝑏𝐵^𝑏𝑝𝑉^𝑝𝑘𝐵^𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽+𝜑^𝜑^∘𝑦_𝑗𝛿^𝑖𝑘+𝜑^𝜑^+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖𝐵^𝑘𝑏𝐻^𝑏𝛽𝐵^𝑎𝑖𝐵^𝑗𝛼,_∘𝐾^𝑎𝛼𝛽=−⎛⎜⎜⎝𝜑^𝜑′+2𝐹²𝜑^𝑔_𝑗𝑘𝑦^𝑖+2𝜑^𝜑^−𝜑^2𝜑^𝜑^+2𝐹²𝜑^𝑦^𝑖∘𝑦𝑦_𝑗∘𝑘⎞⎟⎟⎠𝐵_𝑎𝑖𝐵^𝑗𝛼𝐵^𝑘𝛽.(3.10)

4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspace 𝐿^𝑚=(𝑀,𝐿(𝑢,𝑣)) of a Lagrange space 𝐿^𝑛 are given by (cf. [8]) 𝐵^𝑖𝛼𝐵^𝑗𝛽Ω_𝑖𝑗−Ω_𝛼𝛽=Π_𝛽𝑎∧Π^𝑎𝛼,(4.1)𝐵^𝑖𝑎𝐵^𝑗𝑏Ω_𝑖𝑗−Ω_𝑎𝑏=Π_𝛾𝑏∧Π^𝛾𝑎,(4.2)−𝐵^𝑖𝛼𝐵^𝑗𝑎Ω_𝑖𝑗=𝛿_𝑎𝑏𝑑Π^𝑏𝛼+Π^𝑏𝛽∧𝜔^𝛽𝛼−Π^𝑐𝛼∧𝜔^𝑏𝑐,(4.3) where (a)Π_𝛼𝑎=𝑔_𝛼𝛽Π^𝛽𝑎,(b)Π_𝛾𝑏=𝛿_𝑏𝑐Π^𝑐𝛾.(4.4) Using (1.3) in (2.41)(a), we find that Ω_𝑖𝑗=𝜑′Ω^ℎ𝑖𝑔_ℎ𝑗+2𝜑^Ω𝑦^ℎ𝑖∘𝑦_ℎ∘𝑗.(4.5) Applying 𝑎_𝛾𝛽=𝐵^𝑖𝛾𝐵^𝑗𝛽𝑎_𝑖𝑗 in (2.41)(b), we have Ω_𝛼𝛽=𝐵^𝑖𝛾𝐵^𝑗𝛽Ω^𝛾𝛼𝑎_𝑖𝑗, which in view of (1.3) becomes Ω_𝛼𝛽=𝜑′𝑔_𝑖𝑗𝐵^𝑖𝛾𝐵^𝑗𝛽Ω^𝛾𝛼+2𝜑^∘𝑦𝑦_𝑖∘𝑗𝐵^𝑖𝛾𝐵^𝑗𝛽Ω^𝛾𝛼,(4.6) that is, Ω_𝛼𝛽=𝜑′𝑔_𝛾𝛽Ω^𝛾𝛼+2𝜑^∘𝑦𝑦_𝑖∘𝑗𝐵^𝑖𝛾𝐵^𝑗𝛽Ω^𝛾𝛼.(4.7) For the subspace 𝐿^𝑚 of an APL space, (4.4)(a) is of the form Π_𝛼𝑎=𝑎_𝛼𝛽Π^𝛽𝑎, which in view of 𝑎_𝛼𝛽=𝐵^𝑖𝛼𝐵^𝑗𝛽𝑎_𝑖𝑗 and (1.3) becomes Π_𝛼𝑎=𝜑′𝐵^𝑖𝛼𝐵^𝑗𝛽𝑎_𝑖𝑗Π^𝛽𝑎+2𝜑^∘𝑦𝑦_𝑖∘𝑗𝐵^𝑖𝛼𝐵^𝑗𝛽Π^𝛽𝑎, that is, Π_𝛼𝑎=𝜑′𝑔_𝛼𝛽Π^𝛽𝑎+2𝜑^∘𝑦𝑦_𝑖∘𝑗𝐵^𝑖𝛼𝐵^𝑗𝛽Π^𝛽𝑎.(4.8) Thus, we have the following.

Theorem 4.1. The Gauss-Codazzi equations for a Lagrange subspace 𝐿^𝑚 of an APL space are given by (4.1)–(4.3) with Π_𝛼𝑎, Π_𝛾𝑏, Ω_𝑖𝑗, Ω_𝛼𝛽, and 𝜔^𝑏𝑐, 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.