#### Abstract

We discuss the subspaces of an almost -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 -Lagrangians and studied -Lagrange manifolds. Generalizing the notion of a -Lagrange manifold, the present authors recently studied the geometry of an almost -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 be an -dimensional Finsler space and a smooth function. If the function has the following properties:(a), (b), for every ,then the Lagrangian given by where is a covector and is a smooth function, is a regular Lagrangian [7]. The space is a Lagrange space. The present authors [7] called such space as an almost -Lagrange space (shortly APL space) associated to the Finsler space . An APL space reduces to a -Lagrange space if and only if and . We take We indicate all the geometrical objects related to by putting a small circle “” over them. Equations (1.2), in view of (1.1), provide the following expressions for and its inverse (cf. [7]): where .

Let be a smooth manifold of dimension , , immersed in by immersion . The immersion induces an immersion making the following diagram commutative:

Let (throughout the paper, the Greek indices run from 1 to ) be local coordinates on . The restriction of the Lagrangian on is . Let . Then, we have (cf. [8]) where are the projection factors. The pair is also a Lagrange space, called the subspace of . For the natural bases on TM and on , we have [8] where .

For the bases and , we have Since are linearly independent vector fields tangent to , a vector field is normal to along if on , we have There are, at least locally, unit vector fields normal to and mutually orthonormal, that is, Thus, at every point , we have a moving frame . Using (1.3) in the first expression of (1.8) and keeping (this fact is clear from ) in view, we observe that 's are normal to with respect to if and only if they are so with respect to . The dual frame of is with the following duality conditions: 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 has the local coefficients given by
**
where ,
*

Theorem 1.2 (cf. [7]). *The coefficients of the canonical metrical d-connection of an APL space are given by
*

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 be a nonlinear connection for . The adapted basis of induced by is , where The dual basis (cobasis) of the adapted basis is .

*Definition 2.1 (cf. [8]). * A nonlinear connection of is said to be induced by the canonical nonlinear connection if the following equation holds good:
The local coefficients of the induced nonlinear connection for the subspace of a Lagrange space are given by (cf. [8])
being the local coefficients of canonical nonlinear connection of the Lagrange space . Now using (1.10) in (2.3), we get
If we take , it follows from (2.4) that
Thus, we have the following.

Theorem 2.2. *The local coefficients of the induced nonlinear connection of the subspace of an APL space are given by (2.5).*

In view of (2.5), (2.1) takes the following form, for the subspace of an APL space : where .

We can put as (cf. [8]) where Using (1.10) in (2.8) and simplifying, we get Taking , in (2.9), it follows that Now, if and only if , that is, if and only if . Thus, we have the following.

Theorem 2.3. *The adapted cobasis of the basis induced by the nonlinear connection of an APL space is of the form if and only if .*

*Definition 2.4 (cf. [8]). *Let be the canonical metrical -connection of . An operator is said to be a coupling of with if
where .

The coefficients of coupling of with are given by
Using (1.12) and (1.13) in (2.12), we have
In view of (2.10) and , (2.14) becomes
that is,
where .

Using (1.12) in (2.13), we find that
that is,
where . Thus, we have the following.

Theorem 2.5. *The coefficients of coupling for the subspace of an APL space are given by (2.16) and (2.18).*

*Definition 2.6 (cf. [8]). *An operator given by
where , is called the induced tangent connection by . This defines an -linear connection for .

The coefficients of are given by
Using (2.16) in (2.20), we get
that is,
If we take , the last expression gives
Next, using (2.18) in (2.21), we obtain
If we take , (2.25) becomes
Thus, we have the following.

Theorem 2.7. *The coefficients of the induced tangent connection for the subspace of an APL space are given by (2.24) and (2.26).*

*Remarks 2. *The torsion does not vanish, in general, while . These facts may be observed from (2.24) and (2.26).

*Definition 2.8 (cf. [8]). * An operator given by
where , is called the induced normal connection by .

The coefficients of are given by
Using (2.6) and (2.16) in (2.28), we find
Taking and using , (2.30) reduces to
Next, using (2.18) in (2.29), we have
Taking and using (1.9) and , the last equation yields
Thus, we have the following.

Theorem 2.9. *The coefficients of induced normal connection for the subspace of an APL space are given by (2.31) and (2.33).*

*Definition 2.10 (cf. [8]). *The (mixed) derivative of a mixed d-tensor field is given by
The connection 1-forms,
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:
**
where the 2-forms of torsions are given by
**
with , and the 2-forms of curvature and , are given by
*

We will use the following notations in Section 4:

#### 3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspace of a Lagrange space are given by (cf. [8]) where Using (2.6) and (2.16) in (3.3)(a), we have If we take , the last expression provides Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find where . Thus, we have the following.

Theorem 3.1. *The following Gauss-Weingarten formulae for the subspace of an APL space hold:
**
where
*

*Remark 3.2. * and given, respectively, by (3.5) and (3.6) are called the second fundamental -tensor fields of immersion .

The following consequences of Theorem 3.1 are straightforward.

Corollary 3.3. *In a subspace of an APL space, we have the following:
**
if and only if
*

#### 4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspace of a Lagrange space are given by (cf. [8]) where Using (1.3) in (2.41)(a), we find that Applying in (2.41)(b), we have , which in view of (1.3) becomes that is, For the subspace of an APL space, (4.4)(a) is of the form , which in view of and (1.3) becomes , that is, Thus, we have the following.

Theorem 4.1. *The Gauss-Codazzi equations for a Lagrange subspace of an APL space are given by (4.1)–(4.3) with , , , , and , 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.