# On Almostφ-Lagrange Spaces

Authors: P. N. Pandey; Suresh K. Shukla
Journal: ISRN Geometry
 (2011)

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## Abstract

We initiate a study on the geometry of an almostφ-Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and h- and v-deflection tensors in an APL-space. Corresponding expressions in a φ-Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.


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## Body

## 1. Introduction

In the last three decades, various meaningful generalizations of Finsler spaces have been considered. These generalizations have been found much applicable to mechanics, theoretical physics, variational calculus, optimal control, complex analysis, biology, ecology, and so forth. The geometry of Lagrange spaces is one such generalization of the geometry of Finsler spaces which was introduced and studied by Miron [1, 2]. He [1, 2] introduced the most natural generalization of Lagrange spaces named as generalized Lagrange space. Since the introduction of Lagrange spaces and generalized Lagrange spaces, many geometers and physicists have been engaged in the exploration, development, and application of these concepts [3–13]. Antonelli and Hrimiuc [14, 15] introduced a special type of regular Lagrangian called φ-Lagrangian. Applications of such Lagrangian have been discussed by Antonelli et al. in the monograph [16]. In the present paper, we generalize the notion of φ-Lagrangian and introduce the concept of almost φ-Lagrange spaces. We hope that the results obtained in the paper will be interesting for the researchers working on the application of Lagrange spaces in various fields of science.LetFn=(M,F(x,y)) be an n-dimensional Finsler space, and let φ:ℝ+→ℝ be a smooth function. The composition L:=φ(F2) defines a differentiable Lagrangian. This was regarded by Antonelli and Hrimiuc [14, 15] as φ-Lagrangian associated to the Finsler space Fn. They [14] proved that if the function φ has the following properties:(1.1)(a)φ′(t)≠0,(b)φ′(t)+φ′′(t)≠0,foreveryt∈Im⁡(F2),
then L is a regular Lagrangian and thus Ln=(M,L(x,y)) is a Lagrange space, called a φ-Lagrange space.In this paper, we consider a more general Lagrangian as follows:(1.2)L(x,y)=φ(F2)+Ai(x)yi+U(x),
where φ is the same as discussed earlier, Ai(x) is a covector, and U(x) is a smooth function.In Section2, we show that if the function φ has the properties (1.1), then L(x,y) is a regular Lagrangian and thus the pair Ln=(M,L(x,y)) is a Lagrange space. We call this space as an almost φ-Lagrange space (shortly APL-space).An APL-space reduces to aφ-Lagrange space if and only if Ai(x)=0 and U(x)=0.Ifφ(t)=t,forallt∈Im(F2), then the Lagrangian in (1.2) takes the form(1.3)L(x,y)=F2+Ai(x)yi+U(x).
This defines a regular Lagrangian, and the pair Ln=(M,L(x,y)) is called an almost Finsler Lagrange space (shortly AFL-space). Such Lagrange space was introduced by Miron and Anastasiei (vide Chapter IX of [17]).We take(1.4)gij=12∂̇i∂̇jF2,aij=12∂̇i∂̇jL,∂̇i≡∂∂yi.
Henceforth, we will indicate all the geometrical objects related to Fn by a small circle “∘” put over them.In a Finsler space, the geodesics, parameterized by arc length (the extremals of the length integral), coincide with the extremals of action integral or with the autoparallel curves of the Cartan nonlinear connection [16]:(1.5)Nji∘=γ∘j0i-C∘jkiγ∘00k,
where(1.6)γ∘jki=12gih(∂jghk+∂kgjh-∂hgjk);∂j≡∂∂xj,C∘jki=12gih∂̇hgjk,γ∘j0i=γ∘jkiyk,γ∘00i=γ∘jkiyjyk.
These geodesics are the integral curves of the spray [16] (i.e., (2) p-homogeneous):(1.7)G∘i=14gij(yk∂̇j∂kF2-∂jF2),
that is, solutions of the differential equations(1.8)d2xids2+2G∘i(x(s),dxds)=0.
We have the following equalities:(1.9)(a)G∘i=12γ∘00i,(b)Nji∘=∂̇jG∘i.
In a general Lagrange space Ln=(M,L(x,y)), the geodesics are the extremals of the action integral and coincide with the integral curves of the semispray [17, 18] (i.e., may not be a spray):(1.10)Gi=14aij(yk∂̇j∂kL-∂jL).
As in a Finsler space, a remarkable nonlinear connection can be considered in a Lagrange space:(1.11)Nji=∂̇jGi.
Such nonlinear connection is a canonical nonlinear connection [17, 18] as it depends only on the fundamental function L(x,y) of the Lagrange space.In general, the autoparallel curves of(Nji) are different from the geodesics of Ln=(M,L(x,y)) (cf. [17]).Given a nonlinear connection(Nji) on a Lagrange space Ln=(M,L(x,y)), there is a unique h- and v-metrical d-connection (cf. [17, 19]) CΓ(N)=(Nji,Ljki,Cjki) with torsions Tjki=0 and Sjki=0, called the canonical metrical d-connection. This connection is linear and its coefficients are given by(1.12)Ljki=12aih(δjahk+δkajh-δhajk),(1.13)Cjki=12aih(∂̇jahk+∂̇kajh-∂̇hajk),
where δi=∂i-Nir∂̇r is the Lagrange differentiation operator.IfC∘Γ(N∘)=(N∘ji,L∘jki,C∘jki) is the Cartan connection of the Finsler space Fn=(M,F(x,y)), then its coefficients are given by(1.14)L∘jki=12gih(δ∘jghk+δ∘kgjh-δ∘hgjk),(1.15)C∘jki=12gih(∂̇jghk+∂̇kgjh-∂̇hgjk),
where δ∘i=∂i-N∘ir∂̇r.Theh- and v-deflection tensor fields Dji and dji, respectively, of a Lagrange space Ln are defined by (cf. [19])(1.16)Dji:=y|ji=ysLsji-Nji,(1.17)dji:=yi|j=δji+ysCsji,
where | and |, respectively, denote the h- and v-covariant derivatives with respect to CΓ.IfD∘ji is the h-deflection tensor field and d∘ji is the v-deflection tensor field of the Finsler space Fn, then(1.18)D∘ji:=y|∘ji=ysL∘sji-N∘ji=0,(1.19)d∘ji:=yi|∘j=δji,
where |∘ and |∘, respectively, denote the h- and v-covariant derivatives with respect to C∘Γ.For basic terminology and notations related to a Finsler space and a Lagrange space, we refer to the books [17, 20].


## 2. Almostφ-Lagrange Spaces

As discussed earlier, we consider the Lagrangian given by (1.2) in which the function φ satisfies (1.1). We prove that it is a regular Lagrangian and the pair Ln=(M,L(x,y)) is a Lagrange space which we term as an almost φ-Lagrange space (APL-space in short).Theorem 2.1.
If the functionφ satisfies the conditions (1.1), then  L(x,y), given by (1.2), is a regular Lagrangian and  Ln=(M,L(x,y)) is a Lagrange space.Proof.
Differentiating (1.2) partially with respect to yi, we get
(2.1)∂̇iL=φ′(F2)∂̇iF2+Ai(x).
Again differentiating (2.1) partially with respect to yj, we obtain
(2.2)∂̇j∂̇iL=φ′′(F2)∂̇iF2∂̇jF2+φ′(F2)∂̇i∂̇jF2,
which, in view of (1.4), provides
(2.3)aij=2F2φ′′(F2)∂̇iF∂̇jF+φ′(F2)gij.
Now
(2.4)F∂̇iF=12∂̇iF2=12∂̇i(gjkyjyk)=gikyk:=y∘i.
In view of (2.4), (2.3) takes the form
(2.5)aij=φ′⋅(gij+2φ′′φ′y∘iy∘j).
Under the hypothesis, the matrix (aij) is invertible and its inverse is (see Lemma 6.2.2.1, page 891 in [20])
(2.6)aij=1φ′(gij-2φ′′φ′+2F2φ′′yiyj).
This proves the theorem.Remarks 1.
(i) IfAi(x)=0 and U(x)=0 in (1.2), then expression (2.5) remains unchanged. Hence, the symmetric metric tensor of a φ-Lagrange space is the same as that of an APL-space.
(ii) Ifφ(F2)=F2, then φ′=1 and φ′′=0. Hence, the symmetric metric tensor of an AFL-space coincides with that of the associated Finsler space.


## 3. Semispray, Integral Curves of Euler-Lagrange Equations

In this section, we obtain the coefficients of the canonical semispray of the APL-spaceLn=(M,L(x,y)) and deduce corresponding expressions for a φ-Lagrange space and an AFL-space. Next, we obtain the differential equations whose solution curves are the integral curves of Euler-Lagrange equations in an APL-space. We deduce corresponding differential equations for a φ-Lagrange space and an AFL-space.If we differentiate (1.2) partially with respect to xk, we have(3.1)∂kL=φ′(F2)∂kF2+yi∂kAi(x)+∂kU(x).
Differentiating (3.1) partially with respect to yj, we obtain(3.2)∂̇j∂kL=2φ′′(F2)F∂̇jF∂kF2+φ′(F2)∂̇j∂kF2+∂kAj(x),
which, in view of (2.4), takes the form(3.3)∂̇j∂kL=2φ′′(F2)y∘j∂kF2+φ′(F2)∂̇j∂kF2+∂kAj(x).
Using (3.1) and (3.3) in (1.10), we have(3.4)Gi=14aij{2φ′′(F2)y∘jyk∂kF2+φ′(F2)(yk∂̇j∂kF2-∂jF2)-2ykFjk-∂jU},
where(3.5)Fjk(x)=12(∂jAk-∂kAj)
is electromagnetic tensor field of the potentials Ai(x).Applying (2.6) in (3.4) and using y∘iyi=F2,gijy∘j=yi, and yj∂̇j∂kF2=2∂kF2 (by Euler’s theorem on homogeneous functions), we obtain(3.6)Gi=12φ′′φ′(1-2φ′′F2φ′+2F2φ′′)yiyk∂kF2+14{gij(yk∂̇j∂kF2-∂jF2)-2φ′′F2φ′+2F2φ′′yiyk∂kF2}-14aij(2Fjkyk+∂jU).
Using (1.7) in (3.6) and simplifying, we get(3.7)Gi=G∘i-14aij(2Fjkyk+∂jU).
Thus, we have the following.Theorem 3.1.
The canonical semispray of an APL-space has the local coefficients given by(3.8)Gi=G∘i-14aij(2Fjkyk+∂jU),
where G∘i are the local coefficients of the spray of Fn.For aφ-Lagrange space, Ai(x)=0 and U(x)=0. Hence, from (3.5), we have Fjk=0. Therefore, (3.7) reduces to(3.9)Gi=G∘i.
Thus, we may state the following.Corollary 3.2 (see [14]).
The canonical semispray of aφ-Lagrange space becomes a spray and coincides with that of the associated Finsler space.For an AFL-space,aij=gij (see Remark (ii)). Hence, (3.7) takes the form(3.10)Gi=G∘i-14gij(2Fjkyk+∂jU).
Thus, we have the following.Corollary 3.3 (see [17, 20]).
The canonical semispray of an AFL-space has the local coefficients given by (3.10).In a Lagrange space, the integral curves of the Euler-Lagrange equations:(3.11)Ei(L):=∂iL-ddt(∂̇iL)=0
are the solution curves of the equations [20](3.12)d2xidt2+2Gi(x,y)=0.
Using (3.7) in (3.12), we obtain(3.13)d2xidt2+2G∘i=12(2Fkiyk+aij∂jU),
where Fki=aijFjk.Using (1.9) (a) in (3.13), we have(3.14)d2xidt2+γ∘00i=12(2Fkiyk+aij∂jU).
Thus, we have the following.Theorem 3.4.
In an APL-spaceLn=(M,L(x,y)), the integral curves of the Euler-Lagrange equations  Ei(L)=0 are the solution curves of (3.14).For aφ-Lagrange space, equations (3.14) take the following simple form:(3.15)d2xidt2+γ∘00i=0.
This enables us to state the following.Corollary 3.5 (see [14]).
In aφ-Lagrange space, the integral curves of the Euler-Lagrange equations are the solution curves of (3.15).For an AFL-space,aij=gij. Therefore, equations (3.14) become(3.16)d2xidt2+γ∘00i=Fkiyk+12gij∂jU,
where Fki=gijFjk.Thus, we have the following.Corollary 3.6 (see [17, 20]).
In an AFL-space, the integral curves of the Euler-Lagrange equationsEi(L)=0 are the solution curves of (3.16).


## 4. Nonlinear Connection, Autoparallel Curves

In this section, we find the coefficients of the nonlinear connection of an APL-space and obtain the differential equations of the autoparallel curves of the nonlinear connection. Corresponding results have been deduced for aφ-Lagrange space and an AFL-space.Partial differentiation of (2.5) with respect to yk yields(4.1)∂̇kaij=:2Cijk=2φ′C∘ijk+2φ′′(gijy∘k+gjky∘i+gkiy∘j)+4φ′′′y∘iy∘jy∘k.
Using (3.7) in (1.11) and taking (1.9) (b), (2.6), (4.1), C∘pqjyj=0, yiy∘i=F2, and gijy∘j=yi into account, we obtain(4.2)Nji=N∘ji-12Fji+[12φ′C∘qjigqr+12φ′′φ′2giry∘j+φ′′2φ′(φ′+2F2φ′′)(δjryi+δjiyr)+φ′2φ′′′-2φ′′3F2-4φ′φ′′22φ′2(φ′+2F2φ′′)2yiy∘jyr](2Frkyk+∂rU).
If we take(4.3)Sjir=12φ′C∘qjigqr+12φ′′φ′2giry∘j+φ′′(δjryi+δjiyr)2φ′(φ′+2F2φ′′)+φ′2φ′′′-2φ′′3F2-4φ′φ′′22φ′2(φ′+2F2φ′′)2yiy∘jyr,
the last expression becomes(4.4)Nji=N∘ji-12Fji+Sjir(2Frkyk+∂rU),
that is,(4.5)Nji=N∘ji-Vji,
where(4.6)Vji=12Fji-Sjir(2Frkyk+∂rU).
Thus, we have the following.Theorem 4.1.
The canonical nonlinear connection of an APL-spaceLn has the local coefficients given by (4.5).For aφ-Lagrange space, we have Frk=0,Fji=0 and U=0 and hence Vji=0. Therefore, (4.5) reduces to(4.7)Nji=N∘ji.
Thus, we have the following.Corollary 4.2 (see [14]).
The canonical nonlinear connection of aφ-Lagrange space coincides with the nonlinear connection of the associated Finsler space.For an AFL-space, (4.3) reduces to(4.8)Sjir=12C∘qjigqr
and hence (4.6) gives(4.9)Vji=12Fji-C∘qjiFkqyk-12C∘qjigqr∂rU:=Bji.
Therefore, (4.5) takes the form(4.10)Nji=N∘ji-Bji.
Thus, we have the following.Corollary 4.3 (see [17, 20]).
The canonical nonlinear connection of an AFL-spaceLn has the local coefficients given by (4.10).Transvecting (4.5) by yi and using N∘jiyj=γ∘00i, we obtain(4.11)Njiyj=γ∘00i-V0i,
where V0i=Vjiyj.The autoparallel curves of the canonical nonlinear connectionN=(Nji) of a Lagrange space are given by the following system of differential equations (vide [20]):(4.12)d2xidt2+Nji(x,y)yj=0.
Equations (4.12), in view of (4.11), take the form(4.13)d2xidt2+γ∘00i=V0i.
Thus, we have the following.Theorem 4.4.
The autoparallel curves of the canonical nonlinear connectionN=(Nji) of an APL-space  Ln=(M,L(x,y)) are given by the system of differential equations (4.13).For aφ-Lagrange space, Vji=0 and hence V0i=0. Therefore, (4.13) reduces to(4.14)d2xidt2+γ∘00i=0.
Thus, we have the following.Corollary 4.5 (see [14]).
The autoparallel curves of the canonical nonlinear connection of aφ-Lagrange space  Ln=(M,L(x,y)) are given by the system of differential equations (4.14).For an AFL-space,(4.15)Vji=Bji=:12Fji-C∘qjiFkqyk-12C∘qjigqr∂rU
and hence, by virtue of C∘qjiyj=0, we have V0i=(1/2)Fjiyj. Therefore, equations (4.12) take the form(4.16)d2xidt2+γ∘00i=12Fjiyj.
Thus, we deduce the following.Corollary 4.6 (see [17, 20]).
The autoparallel curves of the nonlinear connectionN=(Nji) of an AFL-space  Ln=(M,L(x,y)) are given by the system of differential equations (4.16).If we compare (3.14), (3.15), and (3.16), respectively, with (4.13), (4.14), and (4.16), we observe that, in an APL-space as well as in an AFL-space, solution curves of Euler-Lagrange equations do not coincide with the autoparallel curves of the canonical nonlinear connection whereas in a φ-Lagrange space they do. Therefore, in a φ-Lagrange space, geodesics are autoparallel curves whereas in an APL-space and in an AFL-space they are not so.


## 5. Canonical Metricald-Connection

LetCΓ(N)=(Nji,Ljki,Cjki) be the canonical metrical d-connection of the APL-space Ln=(M,L(x,y)), and let C∘Γ(N∘)=(N∘ji,L∘jki,C∘jki) be the Cartan connection of the associated Finsler space Fn=(M,F(x,y)). In this section, we obtain the expressions for the coefficients of CΓ(N) and we investigate some properties of CΓ(N). We deduce corresponding results for a φ-Lagrange space and an AFL-space.Using (4.1) in (1.13) and taking (1.15) into account, we find(5.1)Cjki=C∘jki+φ′′φ′(δjiy∘k+δkiy∘j)+φ′′φ′+2F2φ′′gjkyi+2(φ′′′φ′-2φ′′2)φ′(φ′+2F2φ′′)yiy∘jy∘k.For anyC∞-class function ψ:ℝ+→ℝ, taking f(x,y)=ψ(F2(x,y)), we have(5.2)δ∘kf=f′F|∘k2
which, in view of  F|∘k2=0 (see proposition 9.4, page 1037 of [20]), gives(5.3)δ∘kf=0.
Since 0=yi|∘k:=δ∘ky∘i-L∘ikry∘r (see proposition 9.4, page 1037 of [20]), we have(5.4)δ∘ky∘i=L∘ikry∘r.
If we operate δ∘k on (2.5) and utilize (5.3) and (5.4), it follows that(5.5)δ∘kaij=φ′δ∘kgij+2φ′′y∘r(L∘ikry∘j+L∘jkry∘i).
In view of δi=∂i-Nir∂̇r, (4.5), and δ∘i=∂i-N∘ir∂̇r, we get(5.6)δkaij=δ∘kaij+Vkr∂̇raij,
which, on account of (4.1) and (5.5), becomes(5.7)δkaij=φ′δ∘kgij+2φ′′y∘r(L∘ikry∘j+L∘jkry∘i)+2VkrCijr.
Using (5.7) in (1.12) and taking (1.14) and ailCjkl=Cjki into account, we obtain(5.8)Ljki=L∘jki+VkrCjri+VjrCkri+VpraipCrkj.
Equations (5.1) and (5.8) enable us to state the following.Theorem 5.1.
The coefficients of the canonical metricald-connection  CΓ(N) of an APL-space  Ln are given by (5.1) and (5.8).For aφ-Lagrange space, Vji=0. Hence, (5.1) remains unchanged whereas (5.8) reduces to(5.9)Ljki=L∘jki.
Thus, we have the following.Corollary 5.2 (see [14]).
The coefficients of the canonical metricald-connection  CΓ(N) of a  φ-Lagrange space  Ln are given by (5.1) and (5.9).For an AFL-space,φ(F2)=F2,φ′(F2)=1,φ′′(F2)=0, and aij=gij. Therefore, we have Cijk=C∘ijk and Vjr=Bjr.In view of these facts, (5.1) reduces to(5.10)Cjki=C∘jki,
whereas (5.8) gives the following:(5.11)Ljki=L∘jki+BkrC∘jri+BjrC∘kri+BprgipC∘rkj,
where Bkr is given by (4.9). Thus, we have the following.Corollary 5.3 (see [17, 20]).
The coefficients of the canonical metricald-connection  CΓ(N) of an AFL-space  Ln are given by (5.10) and (5.11).Now, we investigate some properties of the canonical metricald-connection CΓ(N) of an APL-space and deduce the corresponding properties for a φ-Lagrange space and an AFL-space.Theorem 5.4.
The canonical metricald-connection  CΓ(N) of an APL-space has the following properties:
(5.12)(1)Dki:=y|ki=Vki+VkpCpriyr+VrpCkpiyr+VspaisCpkryr,(5.13)yi|k=Vks(asi+Csijyj)-VisCkspyp-VpsCskiyp,
where yi=:aijyj,
(5.14)(2)dki:=yi|k=φ′+φ′′F2φ′δki+By∘kyi,yi|k=aik+Cikjyj,
where B=2{φ′φ′′+F2(φ′′′φ′-φ′′2)}/φ′(φ′+2F2φ′′),
(5.15)(3)L|k=Xk+2φ′(φ′+2F2φ′′)Vkryr,L|k=2φ′(φ′+2F2φ′′)yk+Ak,
where Xk=yr∂kAr-NkpAp+∂kU.Proof.
(1) Using (5.8) and (4.5) in (1.16), we have
(5.16)Dki=yr(L∘rki+VkpCrpi+VrpCkpi+VspaisCpkr)-N∘ki+Vki,
which, in view of (1.18), reduces to
(5.17)Dki=Vki+yr(VkpCrpi+VrpCkpi+VspaisCpkr).
Next, if we use (2.5) in yi=aijyj, then it follows that
(5.18)yi=(φ′+2F2φ′′)y∘i.
Now, applying successively δi=∂i-Nir∂̇r, (4.5), and δ∘i=∂i-N∘ir∂̇r in yi|k=δkyi-yrLikr and keeping (5.8) and (5.18) in view, we have
(5.19)yi|k=δ∘k{(φ′+2F2φ′′)y∘i}-(φ′+2F2φ′′)y∘rL∘ikr+Vkr∂̇ryi-yr(VksCisr+VisCksr+VpsarpCski).
Differentiating yi=aijyj partially with respect to yr, we have
(5.20)∂̇ryi=air+2Cirjyj.
Also,
(5.21)ylCjkl=alhyhCjkl=yhChjk.
In view of (5.3), we have
(5.22)δ∘k(φ′+2F2φ′′)=0.
Using (5.20), (5.21), and (5.22) in (5.19), we obtain
(5.23)yi|k=(φ′+2F2φ′′)(δ∘ky∘i-y∘rL∘ikr)+Vks(asi+Csijyj)-(VisCksp+VpsCksi)yp,
which, in view of (5.4), gives the desired result.
(2) Using (5.1) in (1.17), we get
(5.24)dki=φ′+φ′′F2φ′δki+By∘kyi,
where B=2{φ′φ′′+F2(φ′′′φ′-φ′′2)}/φ′(φ′+2F2φ′′).
In view of (5.20) and (5.21), it follows, from yi|k=∂̇kyi-yrCikr, that
(5.25)yi|k=aik+2Cikjyj-Cijkyj,
that is, yi|k=aik+Cikjyj as Cijk is totally symmetric.
(3) Utilizing successivelyδi=∂i-Nir∂̇r, (4.5), and δ∘i=∂i-N∘ir∂̇r in L|k=δkL, we get
(5.26)L|k=δ∘kL+Vkr∂̇rL.
Using (1.2) and (2.1) in (5.26), we have
(5.27)L|k=δ∘k(φ+Aryr+U)+Vkr(2φ′y∘r+Ar),
which, in view of (5.3), gives
(5.28)L|k=δ∘k(Aryr+U)+Vkr(2φ′y∘r+Ar).
Using δ∘k=∂k-N∘kr∂̇r and (5.18) in (5.28) and keeping (4.5) in view, we find
(5.29)L|k=yr∂kAr-NkpAp+∂kU+2φ′φ′+2F2φ′′Vkryr.
If we take Xk=yr∂kAr-NkpAp+∂kU, then the last expression takes the form
(5.30)L|k=Xk+2φ′φ′+2F2φ′′Vkryr.
Next, using (2.1) in L|k=∂̇kL, we get
(5.31)L|k=2φ′y∘k+Ak,
which, in view of (5.18), gives the required result.Corollary 5.5 (see [14]).
The canonical metricald-connection  CΓ(N) of a  φ-Lagrange space has the following properties:
(5.32)(1)Dki:=y|ki=0,yi|k=0,(5.33)(2)dki:=yi|k=φ′+φ′′F2φ′δki+By∘kyi,yi|k=aik+Cikjyj,
where B=2{φ′φ′′+F2(φ′′′φ′-φ′′2)}/φ′(φ′+2F2φ′′),
(5.34)(3)L|k=0,L|k=2φ′(φ′+2F2φ′′)yk.Proof.
ApplyingAi(x)=0, U(x)=0, and Vji=0 in Theorem 5.4, we have the corollary.Corollary 5.6.
The canonical metricald-connection  CΓ(N) of an AFL-space has the following properties:
(5.35)(1)Dki=Bki+BrpC∘kpiyr,yi|k=gsi(Bks-BplypC∘lks),
where yi=gijyj,
(5.36)(2)dki=δki,yi|k=gik,(5.37)(3)L|k=yr∂kAr-NkpAp+∂kU+2Bkryr,L|k=2yk+Ak.Proof.
Usingφ(F2)=F2,φ′(F2)=1,φ′′(F2)=0=φ′′′(F2),aij=gij,Cijk=C∘ijk,C∘ijkyj=0,C∘jkiyk=0, and Vjr=Bjr in Theorem 5.4, we have the corollary.



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*Source: 505161-2011-12-27.xml*