IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 981059 10.1155/2012/981059 981059 Research Article On Subspaces of an Almost φ-Lagrange Space Pandey P. N. Shukla Suresh K. Shen Zhongmin 1 Department of Mathematics University of Allahabad Allahabad 211002 India allduniv.ac.in 2012 8 8 2012 2012 29 03 2012 04 06 2012 2012 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.

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 . He developed the theory of subspaces of a Lagrange space together with Bejancu . Miron and Anastasiei  and Sakaguchi  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 . This paper is devoted to the subspaces of an APL space.

Let Fn=(M,F(x,y)) be an n-dimensional Finsler space and φ:+ a smooth function. If the function φ has the following properties:

φ(t)0,

φ(t)+φ′′(t)0, for every t  Im(F2),

then the Lagrangian given by (1.1)L(x,y)=φ(F2)+Ai(x)yi+U(x), where Ai(x) is a covector and U(x) is a smooth function, is a regular Lagrangian . The space Ln=(M,L(x,y)) is a Lagrange space. The present authors  called such space as an almost φ-Lagrange space (shortly APL space) associated to the Finsler space Fn. An APL space reduces to a φ-Lagrange space if and only if Ai(x)=0 and U(x)=0. We take (1.2)gij=12˙i˙jF2,aij=12˙i˙jL,where˙iyi. We indicate all the geometrical objects related to Fn by putting a small circle “” over them. Equations (1.2), in view of (1.1), provide the following expressions for aij and its inverse (cf. ): (1.3)aij=φ(gij+2φ′′φyiyj),aij=1φ(gij-2φ′′φ+2F2φ′′yi  yj),       where gijyj=yi.

Let Mˇ be a smooth manifold of dimension m, 1<m<n, immersed in M by immersion i:MˇM. The immersion i induces an immersion Ti:TMˇTM making the following diagram commutative: (1.4)TMˇTiTMπˇπMˇiM.

Let (uα,vα) (throughout the paper, the Greek indices α,β,γ, run from 1 to m) be local coordinates on TMˇ. The restriction of the Lagrangian L on TMˇ is L(u,v)=L(x(u),y(u,v)). Let aαβ=(1/2)(2Lˇ/uαuβ). Then, we have (cf. ) aαβ=BαiBβjaij where Bαi(u)=xi/uα are the projection factors. The pair Lˇm=(Mˇ,Lˇ(u,v)) is also a Lagrange space, called the subspace of Ln. For the natural bases (/xi,/yi) on TM and (/uα,/vα) on TMˇ, we have  (1.5)uα=Bαixi+B0αiyi,vα=Bαiyi, where B0αi=Bβαivβ,  Bβαi=2xi/uαuβ.

For the bases (dxi,dyi) and (duα,dvα), we have (1.6)dxi=Bαiduα,dyi=Bαidvα+B0αiduα. Since (Bαi) are m linearly independent vector fields tangent to Mˇ, a vector field ξi(x,y) is normal to Mˇ along TMˇ if on TMˇ, we have (1.7)aij(x(u),y(u,v))Bαiξj=0,α=1,2,,m. There are, at least locally, (n-m) unit vector fields Bai(u,v)(a=m+1,m+2,,n) normal to Mˇ and mutually orthonormal, that is, (1.8)aijBαiBbj=0,aijBaiBbj=δab,(a,b=m+1,m+2,,n). Thus, at every point (u,v)TMˇ, we have a moving frame =((u,v),Bαi(u,v),Bai(u,v)). Using (1.3) in the first expression of (1.8) and keeping yiBai=0 (this fact is clear from gijyiBaj  =  0) in view, we observe that Bai's are normal to Mˇ with respect to Ln if and only if they are so with respect to Fn. The dual frame of is *=((u,v),Biα(u,v),Bia(u,v)) with the following duality conditions: (1.9)BαiBiβ=δαβ,BaiBiβ=0,BαiBib=0,BaiBib=δab,BaiBja+BαiBjα=δji. We will make use of the following results due to the present authors , during further discussion.

Theorem 1.1 <xref ref-type="statement" rid="thm1.1">1.1</xref> (cf. [<xref ref-type="bibr" rid="B2">7</xref>]).

The canonical nonlinear connection of an APL space Ln has the local coefficients given by (1.10)Nji=Nji-Vji, where Vji=(1/2)Fji-Sjir(2Frkyk+rU), (1.11)Sjir=12φCqjigqr+12φ′′φ2giryj+φ′′(δjryi+δjiyr)2φ(φ+2F2φ′′)+φ2φ′′′-2φ′′3F2-4φφ′′22φ2(φ+2F2φ′′)2yiyjyr,Frk(x)=12(rAk-kAr),Fji=aikFkj.

Theorem 1.2 (cf. [<xref ref-type="bibr" rid="B2">7</xref>]).

The coefficients of the canonical metrical d-connection CΓ(N) of an APL space Ln are given by (1.12)Cjki=Cjki+φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk,(1.13)Ljki=Ljki+VkrCjri+VjrCkri+VpraipCrkj.

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 Nˇ=(Nˇβα(u,v)) be a nonlinear connection for Lˇm=(Mˇ,Lˇ(u,v)). The adapted basis of T(u,v)TMˇ induced by Nˇ is (δ/δuα=δα,/vα=˙α), where (2.1)δα=α-Nˇαβ˙β. The dual basis (cobasis) of the adapted basis (δα,˙α) is (duα,δvα=dvα+Nˇβαduβ).

Definition 2.1 (cf. [<xref ref-type="bibr" rid="B1">8</xref>]).

A nonlinear connection Nˇ=(Nˇβα(u,v)) of Lˇm is said to be induced by the canonical nonlinear connection N if the following equation holds good: (2.2)δvα=Biαδyi. The local coefficients of the induced nonlinear connection Nˇ=(Nˇβα(u,v)) for the subspace Lˇm=(Mˇ,Lˇ(u,v)) of a Lagrange space Ln=(M,L(x,y)) are given by (cf. ) (2.3)Nˇβα=Biα(NjiBβj+B0βi),Nji being the local coefficients of canonical nonlinear connection N of the Lagrange space Ln=(M,L(x,y)). Now using (1.10) in (2.3), we get (2.4)Nˇβα=Biα(NjiBβj+B0βi)-BiαVjiBβj. If we take Nˇβα=Biα(NjiBβj+B0βi), it follows from (2.4) that (2.5)Nˇβα=Nˇβα-BiαVjiBβj. Thus, we have the following.

Theorem 2.2.

The local coefficients of the induced nonlinear connection Nˇ of the subspace Lˇm of an APL space Ln are given by (2.5).

In view of (2.5), (2.1) takes the following form, for the subspace Lˇm of an APL space Ln: (2.6)δβ=δβ+BpαVjpBβj˙α, where δβ=β-Nˇβα˙α.

We can put (dxi,δyi) as (cf. ) (2.7)dxi=Bαiduα,δyi=Bαiδyα+BaiHαaduα, where (2.8)Hαa=Bia(NjiBαj+B0αi). Using (1.10) in (2.8) and simplifying, we get (2.9)Hαa=Bia(NjiBαj+B0αi)-BiaVjiBαj. Taking Hαa=Bia(NjiBαj+B0αi), in (2.9), it follows that (2.10)Hαa=Hαa-BiaVjiBαj. Now, dxi=Bαiduα,δyi=Bαiδyα if and only if Hαa=0, that is, if and only if Hαa=BiaVjiBαj. Thus, we have the following.

Theorem 2.3.

The adapted cobasis (dxi,δyi) of the basis (/xi,/yi) induced by the nonlinear connection N of an APL space Ln is of the form dxi=Bαiduα,δyi=Bαiδyα if and only if Hαa=BiaVjiBαj.

Definition 2.4 (cf. [<xref ref-type="bibr" rid="B1">8</xref>]).

Let D=DΓ(N) be the canonical metrical d-connection of Ln. An operator Dˇ is said to be a coupling of D with Nˇ if (2.11)DˇXi=X|αiduα+Xi|αδvα, where X|αi=δαXi+XjLˇjαi,Xi|α=˙αXi+XjCˇjαi.

The coefficients (Lˇjαi,Cˇjαi) of coupling Dˇ of D with Nˇ are given by (2.12)Lˇjαi=LjkiBαk+CjkiBakHαa,(2.13)Cˇjαi=CjkiBαk. Using (1.12) and (1.13) in (2.12), we have (2.14)Lˇjβi=(Ljki+VkrCjri+VjrCkri+VpraipCrkj)Bβk+[Cjki+φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk]BakHβa. In view of (2.10) and yiBai=0, (2.14) becomes (2.15)Lˇjβi=(LjkiBβk+CjkiBakHβa)+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)Bβk+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BakHβa, that is, (2.16)Lˇjβi=Lˇjβi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)Bβk+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BakHβa, where Lˇjβi=LjkiBβk+CjkiBakHβa.

Using (1.12) in (2.13), we find that (2.17)Cˇjβi=CjkiBβk+(2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyi  CjkiBβk+Bβk+2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk)Bβk, that is, (2.18)Cˇjβi=Cˇjβi+(2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyiC˘jβi+C˘jβi  +2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk)Bβk, where Cˇjβi=CjkiBβk. Thus, we have the following.

Theorem 2.5.

The coefficients of coupling for the subspace Lˇm of an APL space Ln are given by (2.16) and (2.18).

Definition 2.6 (cf. [<xref ref-type="bibr" rid="B1">8</xref>]).

An operator DT given by (2.19)DTXα=X|βαduβ+Xα|βδvβ, where X|βα=δβXα+XγLγβα,Xα|β=˙βXα+XγCγβα, is called the induced tangent connection by D. This defines an N-linear connection for Lˇm.

The coefficients (Lγβα,Cγβα) of DT are given by (2.20)Lβγα=Biα(Bβγi+BβjLˇjγi),(2.21)Cβγα=BiαBβjCˇjγi. Using (2.16) in (2.20), we get (2.22)Lβγα=BiαBβγi+BβjBiα[Lˇjγi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)Bγk  BiαBβγi+BβjBiαBβγi+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BakHγa(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)], that is, (2.23)Lβγα=Biα(Bβγi+LˇjγiBβj)+BiαBβj[(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)BγkBiα(Bβγi+L˘jγiBβj)BiαBβjBβγi  +(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BakHγa(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)]. If we take Lβγα=Biα(Bβγi+LˇjγiBβj), the last expression gives (2.24)Lβγα=Lβγα+BiαBβj[(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)Bγk  LβγαBiαBβjBβγi+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BakHγa(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)]. Next, using (2.18) in (2.21), we obtain (2.25)Cβγα=BiαBβjCˇjγi+(φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk)BγkBiαBβj. If we take Cβγα=BiαBβjCˇjγi, (2.25) becomes (2.26)Cβγα=Cβγα+(φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk)BγkBiαBβj. Thus, we have the following.

Theorem 2.7.

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

Remarks 2.

The torsion Tβγα=Lβγα-Lγβα does not vanish, in general, while Sβγα=Cβγα-Cγβα=0. These facts may be observed from (2.24) and (2.26).

Definition 2.8 (cf. [<xref ref-type="bibr" rid="B1">8</xref>]).

An operator D given by (2.27)DXa=X|αaduα+Xa|αδvα, where X|αa=δαXa+XbLbαa,Xa|α=˙αXa+XbCbαa, is called the induced normal connection by D.

The coefficients (Lbγa,Cbγa) of D are given by (2.28)Lbγa=Bia(δγBbi+BbjLˇjγi),(2.29)Cbγa=Bia(˙γBbi+BbjCˇjγi). Using (2.6) and (2.16) in (2.28), we find (2.30)Lbγa=BiaδγBbi+BiaBpαVjpBγj˙αBbi+BbjBia[Lˇjγi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBcrBpcVkp)Bγk  +BbjBiaBβγi+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BckHγc(VkrCjri+VjrCkri+VpraipCrkj-CjriBcrBpcVkp)]. Taking Lbγa=Bia(δγBbi+BbjLˇjγi) and using yjBbj=0, (2.30) reduces to (2.31)Lbγa=Lbγa+BiaBpαVjpBγj˙αBbi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)BiaBbjBγk+φ′′φ+2F2φ′′gjkyiBckHγcBiaBbj. Next, using (2.18) in (2.29), we have (2.32)Cbγa=Bia(˙γBbi+BbjCˇjγi)+[2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)φ′′φ(δjiyk+δkiyj)+φ′′φ+2F2φ′′gjkyiBia(˙γBbi+BbjC˘jγi)+C˘jγi  +2(φ′′′φ-2φ′′2)φ(φ+2F2φ′′)yiyjyk]BγkBiaBbj. Taking Cbγa=Bia(˙γBbi+BbjCˇjγi) and using (1.9) and yjBbj=0, the last equation yields (2.33)Cbγa=Cbγa+φ′′φδbaykBγk+φ′′φ+2F2φ′′gjkyiBγkBiaBbj. Thus, we have the following.

Theorem 2.9.

The coefficients of induced normal connection D for the subspace Lˇm of an APL space Ln are given by (2.31) and (2.33).

Definition 2.10 (cf. [<xref ref-type="bibr" rid="B1">8</xref>]).

The (mixed) derivative of a mixed d-tensor field Tjβbiαa is given by (2.34)Tjβbiαa=(δηTjβbiαa+TjβbkαaLˇkηi++TjβbiγaLγηα++TjβbiαcLcηaδη-TkβbiαaLˇjηk--TjγbiαaLβηγ--TjβciαaLˇbηc)duη+(˙ηTjβbiαa+TjβbkαaCˇkηi++TjβbiγaCγηα++TjβbiαcCcηaδηδη  -TkβbiαaCˇjηk--TjγbiαaCβηγ--TjβciαaCˇbηc)δvη. The connection 1-forms, (2.35)ωˇji=:Lˇjαiduα+Cˇjαiδvα,(2.36)ωβα=:Lβγαduγ+Cβγαδvγ,(2.37)ωba=:Lbγaduγ+Cbγaδvγ, are called the connection 1-forms of . We have the following structure equations of .

Theorem 2.11 (cf. [<xref ref-type="bibr" rid="B1">8</xref>]).

The structure equations of are as follows: (2.38)d(duα)-duβωβα=-Ωα,d(δuα)-δuβωβα=-Ω˙α,dωˇji-ωˇjhωˇhi=-Ωˇji,dωβα-ωβγωγα=-Ωβα,dωba-ωbcωca=-Ωba, where the 2-forms of torsions Ωα,Ω˙α are given by (2.39)Ωα=12Tβγαduβduγ+Cβγαduβδvγ,Ω˙α=12Rβγαduβduγ+Pβγαduβδvγ, with Pβγα=˙γNˇβα-Lβγα, and the 2-forms of curvature Ωˇji,Ωβα and Ωba, are given by (2.40)Ωˇji=12Rˇjαβiduαduβ+Pˇjαβiduαδvβ+12Sˇjαβiδvαδvβ,Ωβα=12Rβγδαduγduδ+Pβγδαduγδvδ+12Sβγδαδvγδvδ,Ωba=12Rbαβaduαduβ+Pbαβaduαδvβ+12Sbαβaδvαδvβ.

We will use the following notations in Section 4: (2.41)(a)  Ωˇij=Ωˇihahj,(b)  Ωαβ=Ωαγaγβ,(c)  Ωab=Ωbcδac.

3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspace Lˇm=(Mˇ,Lˇ(u,v)) of a Lagrange space Ln are given by (cf. ) (3.1)Bαi=BaiΠαa,Bai=-BβiΠaβ, where (3.2)Παa=Hαβaduβ+Kαβaδvβ,Πaβ=gβγδabΠγb,(3.3)(a)  Hαβa=Bia(δβBαi+BαjLˇjβi),(b)  Kαβa=BiaBαjCˇjβi. Using (2.6) and (2.16) in (3.3)(a), we have (3.4)Hαβa=Bia(δβBαi+BαjLˇjβi)+BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)BiaBαjBβk+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BbkHβbBiaBαj. If we take Hαβa=Bia(δβBαi+BαjLˇjβi), the last expression provides (3.5)Hαβa=Hαβa+BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)BiaBαjBβk+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BbkHβbBiaBαj. Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find (3.6)Kαβa=Kαβa+(φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-φ′′2)φ(φ+2F2φ′′)yiyjyk)BiaBαjBβk, where Kαβa=BiaBαjCˇjβi. Thus, we have the following.

Theorem 3.1.

The following Gauss-Weingarten formulae for the subspace Lˇm of an APL space hold: (3.7)Bαi=BaiΠαa,Bai=-BβiΠaβ, where (3.8)Παa=Hαβaduβ+Kαβaδvβ,Πaβ=gβγδabΠγb,Hαβa=Hαβa+BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)BiaBαjBβk+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BbkHβbBiaBαj,Kαβa=Kαβa+(φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-φ′′2)φ(φ+2F2φ′′)yiyjyk)BiaBαjBβk.

Remark 3.2.

H α β a and Kαβa 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 Lˇm of an APL space, we have the following: (3.9)(a)  aαβ=0,(b)  Bαi=0, if and only if (3.10)Hαβa=-[BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)BiaBαjBβk  -C˘jγi+(φ′′φyjδki+φ′′φ+2F2φ′′gjkyi)BbkHβbBiaBαj(VkrCjri+VjrCkri+VpraipCrkj-CjriBbrBpbVkp)],Kαβa=-(φ′′φ+2F2φ′′gjkyi+2(φ′′′φ-φ′′2)φ(φ+2F2φ′′)yiyjyk)BiaBαjBβk.

4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspace Lˇm=(Mˇ,Lˇ(u,v)) of a Lagrange space Ln are given by (cf. ) (4.1)BαiBβjΩˇij-Ωαβ=ΠβaΠαa,(4.2)BaiBbjΩˇij-Ωab=ΠγbΠaγ,(4.3)-BαiBajΩˇij=δab(dΠαb+Πβbωαβ-Παcωcb), where (4.4)(a)  Παa=gαβΠaβ,(b)  Πγb  =δbcΠγc. Using (1.3) in (2.41)(a), we find that (4.5)Ωˇij=φΩˇihghj+2φ′′Ωˇihyhyj. Applying aγβ=BγiBβjaij in (2.41)(b), we have Ωαβ=BγiBβjΩαγaij, which in view of (1.3) becomes (4.6)Ωαβ=φgijBγiBβjΩαγ+2φ′′yiyjBγiBβjΩαγ, that is, (4.7)Ωαβ=φgγβΩαγ+2φ′′yiyjBγiBβjΩαγ. For the subspace Lˇm of an APL space, (4.4)(a) is of the form Παa=aαβΠaβ, which in view of aαβ=BαiBβjaij and (1.3) becomes Παa=φBαiBβjaijΠaβ+2φ′′yiyjBαiBβjΠaβ, that is, (4.8)Παa=φgαβΠaβ+2φ′′yiyjBαiBβjΠaβ. Thus, we have the following.

Theorem 4.1.

The Gauss-Codazzi equations for a Lagrange subspace Lˇm of an APL space are given by (4.1)–(4.3) with Παa, Πγb, Ωˇij, Ωαβ, and ωcb, 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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