On Subspaces of an Almost φ-Lagrange Space

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 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:


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 F 2 , then the Lagrangian given by L x, y ϕ F 2 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 ∂i ∂j L, 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 : where g ij y 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 : T M → TM making the following diagram commutative:
For the bases dx i , dy i and du α , dv α , we have Since B i α are m linearly independent vector fields tangent to M, a vector field ξ i x, y is normal to M along T M if on T M, we have There are, at least locally, n − m unit vector fields B i a u, v a m 1, m 2, . . ., n normal to M and mutually orthonormal, that is, 1.9 We will make use of the following results due to the present authors 7 , during further discussion.
Theorem 1.1 cf.7 .The canonical nonlinear connection of an APL space L n has the local coefficients given by 10

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

Induced Nonlinear Connection
Let Ň Ňα β u, v be a nonlinear connection for Ľm M, Ľ u, v .The adapted basis of where The dual basis cobasis of the adapted basis δ α , ∂α is du α , δv α dv α Ňα β du β .
Definition 2.1 cf. 8 .A nonlinear connection Ň Ňα β u, v of Ľm is said to be induced by the canonical nonlinear connection N if the following equation holds good: The local coefficients of the induced nonlinear connection Ň Ňα β u, v for the subspace Ľm M, Ľ u, v of a Lagrange space L n M, L x, y are given by cf. 8 Ňα

2.4
If we take Thus, we have the following.
Theorem 2.2.The local coefficients of the induced nonlinear connection Ň of the subspace Ľm 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 Ľm of an APL space L n : where We can put dx i , δy i as cf.8 where Using 1.10 in 2.8 and simplifying, we get Taking

2.13
International Journal of Mathematics and Mathematical Sciences Using 1.12 and 1.13 in 2.12 , we have Ľi

2.14
In view of 2.10 and

2.22
that is,

2.23
If we take

2.24
International Journal of Mathematics and Mathematical Sciences Next, using 2.18 in 2.21 , we obtain

2.25
If we take

2.31
Next, using 2.18 in 2.29 , we have

2.32
Taking • Či jγ and using 1.9 and 0, the last equation yields

2.33
Thus, we have the following.
Theorem 2.9.The coefficients of induced normal connection D ⊥ for the subspace Ľm 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

2.34
The

2.38
where the 2-forms of torsions Ω α , Ωα are given by

2.40
We will use the following notations in Section 4: 2.41

The Gauss-Weingarten Formulae
The Gauss-Weingarten formulae for the subspace Ľm M, Ľ u, v of a Lagrange space L n are given by cf. 8

3.4
If we take

3.5
Next, using 2.18 in 3.3 b and keeping 1.9 in view, we find Thus, we have the following.
Theorem 3.1.The following Gauss-Weingarten formulae for the subspace Ľm of an APL space hold:

3.8
Remark 3.2.H a αβ and K a αβ 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 Ľm of an APL space, we have the following: 3.10

The Gauss-Codazzi Equations
The Gauss-Codazzi Equations for the subspace Ľm M, Ľ u, v of a Lagrange space L n are given by cf. 8 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 Definition 2.4 cf. 8 .Let D DΓ N be the canonical metrical d-connection of L n .An operator Ď is said to be a coupling of D with Ň if The coefficients of coupling for the subspace Ľm of an APL space L n are given by 2.16 and 2.18 .
is called the induced tangent connection by D. This defines an N-linear connection for Ľm .