# On Subspaces of an Almostφ-Lagrange Space

Authors: P. N. Pandey; Suresh K. Shukla
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We discuss the subspaces of an almostφ-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.


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## Body

## 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.LetFn=(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
(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 [7]. The space Ln=(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 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∂˙i≡∂∂yi.
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. [7]):
(1.3)aij=φ′⋅(gij+2φ′′φ′yi∘yj∘),aij=1φ′(gij-2φ′′φ′+2F2φ′′yiyj),
where gijyj=yi∘.LetMˇ be a smooth manifold of dimension m, 1<m<n, immersed in M by immersion i:Mˇ→M. The immersion i induces an immersion Ti:TMˇ→TM making the following diagram commutative:
(1.4)TMˇ⟶TiTMπˇ↓↓πMˇi⟶M.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/∂uα∂uβ). Then, we have (cf. [8]) 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 [8]
(1.5)∂∂uα=Bαi∂∂xi+B0αi∂∂yi,∂∂vα=Bαi∂∂yi,
where B0αi=Bβαivβ,Bβαi=∂2xi/∂uα∂uβ.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 ℜ=((u,v),Bαi(u,v),Bai(u,v)). Using (1.3) in the first expression of (1.8) and keeping yi∘Bai=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 ℜ is ℜ*=((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 [7], during further discussion.Theorem 1.11.1 (cf. [7]).
The canonical nonlinear connection of an APL spaceLn has the local coefficients given by
(1.10)Nji=N∘ji-Vji,
where Vji=(1/2)Fji-Sjir(2Frkyk+∂rU),
(1.11)Sjir=12φ′C∘qjigqr+12φ′′φ′2giryj∘+φ′′(δjryi+δjiyr)2φ′(φ′+2F2φ′′)+φ′2φ′′′-2φ′′3F2-4φ′φ′′22φ′2(φ′+2F2φ′′)2yiyj∘yr,Frk(x)=12(∂rAk-∂kAr),Fji=aikFkj.Theorem 1.2 (cf. [7]).
The coefficients of the canonical metrical d-connectionCΓ(N) of an APL space Ln are given by
(1.12)Cjki=C∘jki+φ′′φ′(δjiyk∘+δkiyj∘)+φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘,(1.13)Ljki=L∘jki+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

LetNˇ=(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. [8]).
A nonlinear connectionNˇ=(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. [8])
(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(N∘jiBβj+B0βi)-BiαVjiBβj.
If we take Nˇ∘βα=Biα(N∘jiBβj+B0βi), it follows from (2.4) that
(2.5)Nˇβα=Nˇ∘βα-BiαVjiBβj.
Thus, we have the following.Theorem 2.2.
The local coefficients of the induced nonlinear connectionNˇ 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-Nˇ∘βα∂˙α.We can put(dxi,δyi) as (cf. [8])
(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(N∘jiBαj+B0αi)-BiaVjiBαj.
Taking H∘αa=Bia(N∘jiBαj+B0αi), in (2.9), it follows that
(2.10)Hαa=H∘αa-BiaVjiBαj.
Now, dxi=Bαiduα,δyi=Bαiδyα if and only if Hαa=0, that is, if and only if H∘αa=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 H∘αa=BiaVjiBαj.Definition 2.4 (cf. [8]).
LetD=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=(L∘jki+VkrCjri+VjrCkri+VpraipCrkj)Bβk+[C∘jki+φ′′φ′(δjiyk∘+δkiyj∘)+φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘]BakHβa.
In view of (2.10) and yi∘Bai=0, (2.14) becomes
(2.15)Lˇjβi=(L∘jkiBβk+C∘jkiBakH∘βa)+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)Bβk+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BakHβa,
that is,
(2.16)Lˇjβi=Lˇ∘jβi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)Bβk+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BakHβa,
where Lˇ∘jβi=L∘jkiBβk+C∘jkiBakH∘βa.
Using (1.12) in (2.13), we find that
(2.17)Cˇjβi=C∘jkiBβk+(2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)φ′′φ′(δjiyk∘+δkiyj∘)+φ′′φ′+2F2φ′′gjkyiCjki∘Bβk+Bβk+2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘)Bβk,
that is,
(2.18)Cˇjβi=Cˇ∘jβi+(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′′)yiyj∘yk∘)Bβk,
where Cˇ∘jβi=C∘jkiBβk. Thus, we have the following.Theorem 2.5.
The coefficients of coupling for the subspaceLˇm of an APL space Ln are given by (2.16) and (2.18).Definition 2.6 (cf. [8]).
An operatorDT 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[Lˇ∘jγi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BγkBiαBβγi+BβjBiαBβγi+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BakHγa(VkrCjri+VjrCkri+VpraipCrkj-Cjri∘BbrBpbVkp)],
that is,
(2.23)Lβγα=Biα(Bβγi+Lˇ∘jγiBβj)+BiαBβj[(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BγkBiα(Bβγi+L˘jγi∘Bβj)BiαBβjBβγi+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BakHγa(VkrCjri+VjrCkri+VpraipCrkj-Cjri∘BbrBpbVkp)].
If we take L∘βγα=Biα(Bβγi+Lˇ∘jγiBβj), the last expression gives
(2.24)Lβγα=L∘βγα+BiαBβj[(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BγkLβγα∘BiαBβjBβγi+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BakHγa(VkrCjri+VjrCkri+VpraipCrkj-Cjri∘BbrBpbVkp)].
Next, using (2.18) in (2.21), we obtain
(2.25)Cβγα=BiαBβjCˇ∘jγi+(φ′′φ′(δjiyk∘+δkiyj∘)+φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘)BγkBiαBβj.
If we take C∘βγα=BiαBβjCˇ∘jγi, (2.25) becomes
(2.26)Cβγα=C∘βγα+(φ′′φ′(δjiyk∘+δkiyj∘)+φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘)BγkBiαBβj.
Thus, we have the following.Theorem 2.7.
The coefficients of the induced tangent connectionDT for the subspace Lˇm of an APL space are given by (2.24) and (2.26).Remarks 2.
The torsionTβγα=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. [8]).
An operatorD⊥ given by
(2.27)D⊥Xa=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=Biaδγ∘Bbi+BiaBpαVjpBγj∂˙αBbi+BbjBia[Lˇ∘jγi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBcrBpcVkp)Bγk+BbjBiaBβγi+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BckHγc(VkrCjri+VjrCkri+VpraipCrkj-Cjri∘BcrBpcVkp)].
Taking L∘bγa=Bia(δγ∘Bbi+BbjLˇ∘jγi) and using yj∘Bbj=0, (2.30) reduces to
(2.31)Lbγa=L∘bγa+BiaBpαVjpBγj∂˙αBbi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BiaBbjBγk+φ′′φ′+2F2φ′′gjkyiBckHγcBiaBbj.
Next, using (2.18) in (2.29), we have
(2.32)Cbγa=Bia(∂˙γBbi+BbjCˇ∘jγi)+[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′′)yiyj∘yk∘]BγkBiaBbj.
Taking C∘bγa=Bia(∂˙γBbi+BbjCˇ∘jγi) and using (1.9) and yj∘Bbj=0, the last equation yields
(2.33)Cbγa=C∘bγa+φ′′φ′δbayk∘Bγk+φ′′φ′+2F2φ′′gjkyiBγkBiaBbj.
Thus, we have the following.Theorem 2.9.
The coefficients of induced normal connectionD⊥ for the subspace Lˇm 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 fieldTj⋯β⋯bi⋯α⋯a is given by
(2.34)∇Tj⋯β⋯bi⋯α⋯a=(δηTj⋯β⋯bi⋯α⋯a+Tj⋯β⋯bk⋯α⋯aLˇkηi+⋯+Tj⋯β⋯bi⋯γ⋯aLγηα+⋯+Tj⋯β⋯bi⋯α⋯cLcηaδη-Tk⋯β⋯bi⋯α⋯aLˇjηk-⋯-Tj⋯γ⋯bi⋯α⋯aLβηγ-⋯-Tj⋯β⋯ci⋯α⋯aLˇbηc)duη+(∂˙ηTj⋯β⋯bi⋯α⋯a+Tj⋯β⋯bk⋯α⋯aCˇkηi+⋯+Tj⋯β⋯bi⋯γ⋯aCγηα+⋯+Tj⋯β⋯bi⋯α⋯cCcηaδηδη-Tk⋯β⋯bi⋯α⋯aCˇjηk-⋯-Tj⋯γ⋯bi⋯α⋯aCβηγ-⋯-Tj⋯β⋯ci⋯α⋯aCˇbηc)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. [8]).
The structure equations of∇ are as follows:
(2.38)d(duα)-duβ∧ωβα=-Ωα,d(δuα)-δuβ∧ωβα=-Ω˙α,dωˇji-ωˇjh∧ωˇhi=-Ωˇji,dωβα-ωβγ∧ωγα=-Ωβα,dωba-ωbc∧ωca=-Ωba,
where the 2-forms of torsions Ωα,Ω˙α are given by
(2.39)Ωα=12Tβγαduβ∧duγ+Cβγαduβ∧δvγ,Ω˙α=12Rβγαduβ∧duγ+Pβγαduβ∧δvγ,
with Pβγα=∂˙γNˇβα-Lβγα, and the 2-forms of curvature Ωˇji,Ωβα and Ωba, are given by
(2.40)Ωˇji=12Rˇjαβiduα∧duβ+Pˇjαβiduα∧δvβ+12Sˇjαβiδvα∧δvβ,Ωβα=12Rβγδαduγ∧duδ+Pβγδαduγ∧δvδ+12Sβγδαδvγ∧δvδ,Ωba=12Rbαβaduα∧duβ+Pbαβaduα∧δvβ+12Sbαβaδvα∧δvβ.We will use the following notations in Section4:
(2.41)(a)Ωˇij=Ωˇihahj,(b)Ωαβ=Ωαγaγβ,(c)Ωab=Ωbcδac.


## 3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspaceLˇm=(Mˇ,Lˇ(u,v)) of a Lagrange space Ln are given by (cf. [8])
(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(δβ∘Bαi+BαjLˇ∘jβi)+BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BiaBαjBβk+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BbkHβbBiaBαj.
If we take H∘αβa=Bia(δβ∘Bαi+BαjLˇ∘jβi), the last expression provides
(3.5)Hαβa=H∘αβa+BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BiaBαjBβk+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BbkHβbBiaBαj.
Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find
(3.6)Kαβa=K∘αβa+(φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘)BiaBαjBβk,
where K∘αβa=BiaBαjCˇ∘jβi. Thus, we have the following.Theorem 3.1.
The following Gauss-Weingarten formulae for the subspaceLˇm of an APL space hold:
(3.7)∇Bαi=BaiΠαa,∇Bai=-BβiΠaβ,
where
(3.8)Παa=Hαβaduβ+Kαβaδvβ,Πaβ=gβγδabΠγb,Hαβa=H∘αβa+BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BiaBαjBβk+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BbkHβbBiaBαj,Kαβa=K∘αβa+(φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘)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 Theorem3.1 are straightforward.Corollary 3.3.
In a subspaceLˇm of an APL space, we have the following:
(3.9)(a)∇aαβ=0,(b)∇Bαi=0,
if and only if
(3.10)H∘αβa=-[BiaBpγVjpBβjBαγi+(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)BiaBαjBβk-C˘jγi∘+(φ′′φ′yj∘δki+φ′′φ′+2F2φ′′gjkyi)BbkHβbBiaBαj(VkrCjri+VjrCkri+VpraipCrkj-C∘jriBbrBpbVkp)],K∘αβa=-(φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-φ′′2)φ′(φ′+2F2φ′′)yiyj∘yk∘)BiaBαjBβk.


## 4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspaceLˇm=(Mˇ,Lˇ(u,v)) of a Lagrange space Ln are given by (cf. [8])
(4.1)BαiBβjΩˇij-Ωαβ=Πβa∧Παa,(4.2)BaiBbjΩˇij-Ωab=Πγb∧Πaγ,(4.3)-BαiBajΩˇij=δab(dΠαb+Πβb∧ωαβ-Παc∧ωcb),
where
(4.4)(a)Παa=gαβΠaβ,(b)Πγb=δbcΠγc.
Using (1.3) in (2.41)(a), we find that
(4.5)Ωˇij=φ′Ωˇihghj+2φ′′Ωˇihyh∘yj∘.
Applying aγβ=BγiBβjaij in (2.41)(b), we have Ωαβ=BγiBβjΩαγaij, which in view of (1.3) becomes
(4.6)Ωαβ=φ′gijBγiBβjΩαγ+2φ′′yi∘yj∘BγiBβjΩαγ,
that is,
(4.7)Ωαβ=φ′gγβΩαγ+2φ′′yi∘yj∘BγiBβjΩαγ.
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=φ′BαiBβjaijΠaβ+2φ′′y∘iy∘jBαiBβjΠaβ, that is,
(4.8)Παa=φ′gαβΠaβ+2φ′′yi∘yj∘BαiBβjΠaβ.
Thus, we have the following.Theorem 4.1.
The Gauss-Codazzi equations for a Lagrange subspaceLˇm 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).



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*Source: 981059-2012-08-08.xml*