On Subspaces of an Almost bold0mu mumu -Lagrange Space · Hindawi Publishing Corporation...

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2012, Article ID 981059, 14 pagesdoi:10.1155/2012/981059

Research ArticleOn Subspaces of an Almost ϕ-Lagrange Space

P. N. Pandey and Suresh K. Shukla

Department of Mathematics, University of Allahabad, Allahabad 211002, India

Correspondence should be addressed to Suresh K. Shukla, [email protected]

Received 29 March 2012; Accepted 4 June 2012

Academic Editor: Zhongmin Shen

Copyright q 2012 P. N. Pandey and S. K. Shukla. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction 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 theinduced nonlinear connection, coefficients of coupling, coefficients of induced tangent and inducednormal connections, the Gauss-Weingarten formulae, and the Gauss-Codazzi equations for asubspace of an APL-space. Some consequences of the Gauss-Weingarten formulae have also beendiscussed.

1. Introduction

The credit for introducing the geometry of Lagrange spaces and their subspaces goes to thefamous Romanian geometer Miron [1]. He developed the theory of subspaces of a Lagrangespace together with Bejancu [2]. Miron and Anastasiei [3] and Sakaguchi [4] studied thesubspaces of generalized Lagrange spaces (GL spaces in short). Antonelli and Hrimiuc [5, 6]introduced the concept of ϕ-Lagrangians and studied ϕ-Lagrange manifolds. Generalizingthe notion of a ϕ-Lagrange manifold, the present authors recently studied the geometry of analmost ϕ-Lagrange space (APL space briefly) and obtained the fundamental entities relatedto such space [7]. 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 : 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

L(x, y)= ϕ(F2)+Ai(x)yi +U(x), (1.1)

2 International Journal of Mathematics and Mathematical Sciences

where Ai(x) is a covector and U(x) is a smooth function, is a regular Lagrangian [7]. Thespace Ln = (M,L(x, y)) is a Lagrange space. The present authors [7] called such space asan almost ϕ-Lagrange space (shortly APL space) associated to the Finsler space Fn. An APLspace reduces to a ϕ-Lagrange space if and only if Ai(x) = 0 and U(x) = 0. We take

gij =12∂i∂jF

2, aij =12∂i∂jL, where ∂i ≡ ∂

∂yi. (1.2)

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]):

aij = ϕ′ ·(gij +

2ϕ′′

ϕ′◦yi

◦yj

), aij =

1ϕ′

(gij − 2ϕ′′

ϕ′ + 2F2ϕ′′yi yj

), (1.3)

where gijyj =◦yi.

Let M be a smoothmanifold of dimensionm, 1 < m < n, immersed inM by immersioni : M → M. The immersion i induces an immersion Ti : TM → TM making the followingdiagram commutative:

TMTi−→ TM

π ↓ ↓ π

M−→i M.

(1.4)

Let (uα, vα) (throughout the paper, the Greek indices α, β, γ, . . . run from 1 tom) be localcoordinates on TM. The restriction of the Lagrangian L on TM is L(u, v) = L(x(u), y(u, v)).Let aαβ = (1/2)(∂2L/∂uα∂uβ). Then, we have (cf. [8]) aαβ = Bi

αBj

βaij where Bi

α(u) = ∂xi/∂uα

are the projection factors. The pair Lm = (M, L(u, v)) is also a Lagrange space, called thesubspace of Ln. For the natural bases (∂/∂xi, ∂/∂yi) on TM and (∂/∂uα, ∂/∂vα) on TM, wehave [8]

∂

∂uα= Bi

α

∂

∂xi+ Bi

0α∂

∂yi,

∂

∂vα= Bi

α

∂

∂yi, (1.5)

where Bi0α = Bi

βαvβ, Bi

βα = ∂2xi/∂uα∂uβ.For the bases (dxi, dyi) and (duα, dvα), we have

dxi = Biαdu

α, dyi = Biαdv

α + Bi0αdu

α. (1.6)

Since (Biα) are m linearly independent vector fields tangent to M, a vector field ξi(x, y) is

normal to M along TM if on TM, we have

aij

(x(u), y(u, v)

)Biαξ

j = 0, ∀α = 1, 2, . . . , m. (1.7)

International Journal of Mathematics and Mathematical Sciences 3

There are, at least locally, (n −m) unit vector fields Bia(u, v) (a = m + 1, m + 2, . . . , n) normal

to M and mutually orthonormal, that is,

aijBiαB

j

b= 0, aijB

iaB

j

b= δab, (a, b = m + 1, m + 2, . . . , n). (1.8)

Thus, at every point (u, v) ∈ TM, we have a moving frame = ((u, v), Biα(u, v), B

ia(u, v)).

Using (1.3) in the first expression of (1.8) and keeping◦yiB

ia = 0 (this fact is clear from

gijyiB

ja = 0) in view, we observe that Bi

a’s are normal to M with respect to Ln if and only ifthey are so with respect to Fn. The dual frame of is ∗ = ((u, v), Bα

i (u, v), Bai (u, v))with the

following duality conditions:

BiαB

β

i = δβα, Bi

aBβ

i = 0, BiαB

bi = 0, Bi

aBbi = δb

a, BiaB

aj + Bi

αBαj = δi

j . (1.9)

We will make use of the following results due to the present authors [7], during furtherdiscussion.

Theorem 1.1 (cf. [7]). The canonical nonlinear connection of an APL space Ln has the localcoefficients given by

Nij =

◦N

i

j − V ij ,

(1.10)

where V ij = (1/2)Fi

j − Sirj (2Frky

k + ∂rU),

Sirj =

12ϕ′

◦C

i

qjgqr +

12ϕ′′

ϕ′2 gir ◦yj +

ϕ′′(δrj y

i + δijy

r)

2ϕ′(ϕ′ + 2F2ϕ′′) +ϕ′2ϕ′′′ − 2ϕ′′3F2 − 4ϕ′ϕ′′2

2ϕ′2(ϕ′ + 2F2ϕ′′)2yi ◦yjy

r,

Frk(x) =12(∂rAk − ∂kAr), Fi

j = aikFkj .

(1.11)

Theorem 1.2 (cf. [7]). The coefficients of the canonical metrical d-connection CΓ(N) of an APLspace Ln are given by

Cijk =

◦C

i

jk +ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′

ϕ′ + 2F2ϕ′′ gjkyi +

2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk, (1.12)

Lijk =

◦Li

jk + V rkC

ijr + V r

j Cikr + V r

p aipCrkj . (1.13)

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 N = (Nαβ(u, v)) be a nonlinear connection for Lm = (M, L(u, v)). The adapted basis of

T(u,v)TM induced by N is (δ/δuα = δα, ∂/∂vα = ∂α), where

δα = ∂α − Nβα∂β. (2.1)

The dual basis (cobasis) of the adapted basis (δα, ∂α) is (duα, δvα = dvα + Nαβduβ).

Definition 2.1 (cf. [8]). A nonlinear connection N = (Nαβ(u, v)) of Lm is said to be induced by

the canonical nonlinear connection N if the following equation holds good:

δvα = Bαi δy

i. (2.2)

The local coefficients of the induced nonlinear connection N = (Nαβ (u, v)) for the subspace

Lm = (M, L(u, v)) of a Lagrange space Ln = (M,L(x, y)) are given by (cf. [8])

Nαβ = Bα

i

(Ni

jBj

β + Bi0β

), (2.3)

Nij 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

Nαβ = Bα

i

(◦N

i

jBj

β+ Bi

0β

)

− Bαi V

ij B

j

β. (2.4)

If we take◦N

α

β = Bαi (

◦N

i

jBj

β + Bi0β), it follows from (2.4) that

Nαβ =

◦N

α

β − Bαi V

ij B

j

β. (2.5)

Thus, we have the following.

Theorem 2.2. The local coefficients of the induced nonlinear connection N of the subspace Lm of anAPL space Ln are given by (2.5).

In view of (2.5), (2.1) takes the following form, for the subspace Lm of an APL spaceLn:

δβ =◦δβ + Bα

pVp

j Bj

β∂α, (2.6)

where◦δβ = ∂β −

◦N

α

β∂α.


We can put (dxi, δyi) as (cf. [8])

dxi = Biαdu

α, δyi = Biαδy

α + BiaH

aαdu

α, (2.7)

where

Haα = Ba

i

(Ni

jBjα + Bi

0α

). (2.8)

Using (1.10) in (2.8) and simplifying, we get

Haα = Ba

i

(◦N

i

jBjα + Bi

0α

)

− Bai V

ij B

jα. (2.9)

Taking◦H

a

α = Bai (

◦N

i

jBjα + Bi

0α), in (2.9), it follows that

Haα =

◦H

a

α − Bai V

ij B

jα. (2.10)

Now, dxi = Biαdu

α, δyi = Biαδy

α if and only if Haα = 0, that is, if and only if

◦H

a

α = Bai V

ij B

jα.


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 = Biαdu

α, δyi = Biαδy

α if and only if◦H

a

α =Bai V

ij B

jα.

Definition 2.4 (cf. [8]). Let D = DΓ(N) be the canonical metrical d-connection of Ln. Anoperator D is said to be a coupling of D with N if

DXi = Xi|αdu

α +Xi|αδvα, (2.11)

where Xi|α = δαX

i +XjLijα, X

i|α = ∂αXi +XjCi

jα.The coefficients (Li

jα, Cijα) of coupling D of D with N are given by

Lijα = Li

jkBkα + Ci

jkBkaH

aα , (2.12)

Cijα = Ci

jkBkα. (2.13)


Using (1.12) and (1.13) in (2.12), we have

Lijβ =

(◦Li

jk + V rkC

ijr + V r

j Cikr + V r

p aipCrkj

)

Bkβ

+

[◦C

i

jk +ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′

ϕ′ + 2F2ϕ′′ gjkyi

+2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk

]

BkaH

aβ .

(2.14)

In view of (2.10) and◦yiB

ia = 0, (2.14) becomes

Lijβ =

(◦Li

jkBkβ +

◦C

i

jkBka

◦H

a

β

)

+

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bkβ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkaH

aβ ,

(2.15)

that is,

Lijβ =

◦Li

jβ +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bkβ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkaH

aβ ,

(2.16)

where◦Li

jβ =◦Li

jkBkβ+

◦C

i

jkBka

◦H

a

β .Using (1.12) in (2.13), we find that

Cijβ =

◦C

i

jkBkβ +

(ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′


+2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk

)

Bkβ ,

(2.17)

that is,

Cijβ =

◦C

i

jβ +

(ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′


+2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk

)

Bkβ ,

(2.18)


where◦C

i

jβ =◦C

i

jkBkβ . Thus, we have the following.

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

Definition 2.6 (cf. [8]). An operator DT given by

DTXα = Xα|βdu

β +Xα|βδvβ, (2.19)

where Xα|β = δβX

α +XγLαγβ, X

α|β = ∂βXα +XγCα

γβ, is called the induced tangent connection by

D. This defines an N-linear connection for Lm.The coefficients (Lα

γβ, Cαγβ) of D

T are given by

Lαβγ = Bα

i

(Biβγ + B

j

βLijγ

), (2.20)

Cαβγ = Bα

i Bj

βCi

jγ . (2.21)

Using (2.16) in (2.20), we get

Lαβγ = Bα

i Biβγ + B

j

βBαi

[ ◦Li

jγ +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bkγ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkaH

aγ

]

,

(2.22)

that is,

Lαβγ = Bα

i

(

Biβγ +

◦Li

jγBj

β

)

+ Bαi B

j

β

[(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bkγ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkaH

aγ

]

.

(2.23)

If we take◦Lα

βγ = Bαi (B

iβγ +

◦Li

jγBj

β), the last expression gives

Lαβγ =

◦Lα

βγ + Bαi B

j

β

[(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bkγ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkaH

aγ

]

.

(2.24)


Next, using (2.18) in (2.21), we obtain

Cαβγ = Bα

i Bj

β

◦C

i

jγ +

(ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′


2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk

)

Bkγ B

αi B

j

β.

(2.25)

If we take◦C

α

βγ = Bαi B

j

β

◦C

i

jγ , (2.25) becomes

Cαβγ =

◦C

α

βγ +

(ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′


2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk

)

Bkγ B

αi B

j

β. (2.26)


Theorem 2.7. The coefficients of the induced tangent connection DT for the subspace Lm of an APLspace are given by (2.24) and (2.26).

Remarks. The torsion Tαβγ

= Lαβγ

− Lαγβ

does not vanish, in general, while Sαβγ

= Cαβγ

− Cαγβ

= 0.These facts may be observed from (2.24) and (2.26).

Definition 2.8 (cf. [8]). An operator D⊥ given by

D⊥Xa = Xa|αdu

α +Xa|αδvα, (2.27)

where Xa|α = δαX

a +XbLabα, Xa|α = ∂αX

a +XbCabα, is called the induced normal connection by

D.The coefficients (La

bγ , Cabγ) of D

⊥ are given by

Labγ = Ba

i

(δγB

ib + B

j

bLijγ

), (2.28)

Cabγ = Ba

i

(∂γB

ib + B

j

bCijγ

). (2.29)

Using (2.6) and (2.16) in (2.28), we find

Labγ = Ba

i

◦δγB

ib + Ba

i BαpV

p

j Bjγ ∂αB

ib

+ Bj

bBai

[ ◦Li

jγ +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrcB

cpV

p

k

)

Bkγ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkcH

cγ

]

.

(2.30)


Taking◦La

bγ = Bai (

◦δγB

ib + B

j

b

◦Li

jγ) and using◦yjB

j

b = 0, (2.30) reduces to

Labγ =

◦La

bγ + Bai B

αpV

p

j Bjγ ∂αB

ib +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bai B

j

bBkγ

+ϕ′′

ϕ′ + 2F2ϕ′′ gjkyiBk

cHcγB

ai B

j

b.

(2.31)

Next, using (2.18) in (2.29), we have

Cabγ = Ba

i

(

∂γBib + B

j

b

◦C

i

jγ

)

+

[ϕ′′

ϕ′(δij

◦yk + δi

k

◦yj

)+

ϕ′′


+2(ϕ′′′ϕ′ − 2ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′) yi ◦yj

◦yk

]

Bkγ B

ai B

j

b.

(2.32)

Taking◦C

a

bγ = Bai (∂γB

ib+ B

j

b

◦C

i

jγ) and using (1.9) and◦yjB

j

b= 0, the last equation yields

Cabγ =

◦C

a

bγ +ϕ′′

ϕ′ δab

◦ykB

kγ +

ϕ′′

ϕ′ + 2F2ϕ′′ gjkyiBk

γ Bai B

j

b. (2.33)


Theorem 2.9. The coefficients of induced normal connectionD⊥ for the subspace Lm of an APL spaceLn are given by (2.31) and (2.33).

Definition 2.10 (cf. [8]). The (mixed) derivative of a mixed d-tensor field Ti···α···aj···β···b is given by

∇Ti···α···aj···β···b =

(δηT

i···α···aj···β···b + Tk···α···a

j···β···b Likη + · · · + T

i···γ ···aj···β···bL

αγη + · · · + Ti···α···c

j···β···bLacη

−Ti···α···ak···β···bL

kjη − · · · − Ti···α···a

j···γ ···b Lγ

βη − · · · − Ti···α···aj···β···c L

cbη

)duη

+(∂ηT

i···α···aj···β···b + Tk···α···a

j···β···b Cikη + · · · + T

i···γ ···aj···β···bC

αγη + · · · + Ti···α···c

j···β···bCacη

−Ti···α···ak···β···bC

kjη − · · · − Ti···α···a

j···γ ···bCγ

βη − · · · − Ti···α···aj···β···c C

cbη

)δvη.

(2.34)

The connection 1-forms,

ωij =: L

ijαdu

α + Cijαδv

α, (2.35)

ωαβ =: Lα

βγduγ + Cα

βγδvγ , (2.36)

ωab =: La

bγduγ + Ca

bγδvγ , (2.37)


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:

d(duα) − duβ ∧ωαβ = −Ωα,

d(δuα) − δuβ ∧ωαβ = −Ωα,

dωij − ωh

j ∧ ωih = −Ωi

j ,

dωαβ −ω

γ

β∧ωα

γ = −Ωαβ,

dωab −ωc

b ∧ωac = −Ωa

b ,

(2.38)

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

Ωα =12Tαβγdu

β ∧ duγ + Cαβγdu

β ∧ δvγ ,

Ωα =12Rα

βγduβ ∧ duγ + Pα

βγduβ ∧ δvγ ,

(2.39)

with Pαβγ

= ∂γ Nαβ− Lα

βγ, and the 2-forms of curvature Ωi

j ,Ωαβand Ωa

b, are given by

Ωij =

12Ri

jαβduα ∧ duβ + P i

jαβduα ∧ δvβ +

12Sijαβδv

α ∧ δvβ,

Ωαβ =

12Rα

βγδduγ ∧ duδ + Pα

βγδduγ ∧ δvδ +

12Sαβγδδv

γ ∧ δvδ,

Ωab =

12Ra

bαβduα ∧ duβ + Pa

bαβduα ∧ δvβ +

12Sabαβδv

α ∧ δvβ.

(2.40)

We will use the following notations in Section 4:

(a) Ωij = Ωhi ahj , (b) Ωαβ = Ωγ

αaγβ, (c) Ωab = Ωcbδac. (2.41)

3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspace Lm = (M, L(u, v)) of a Lagrange space Ln

are given by (cf. [8])

∇Biα = Bi

aΠaα, ∇Bi

a = −BiβΠ

βa, (3.1)


where

Πaα = Ha

αβduβ +Ka

αβδvβ,

Πβa = gβγδabΠb

γ ,(3.2)

(a) Haαβ = Ba

i

(δβB

iα + B

jαL

ijβ

), (b) Ka

αβ = Bai B

jαC

ijβ. (3.3)

Using (2.6) and (2.16) in (3.3)(a), we have

Haαβ = Ba

i

(◦δβB

iα + B

jα

◦Li

jβ

)

+ Bai B

γpV

p

j Bj

βBiαγ

+

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bai B

jαB

kβ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkbH

bβB

ai B

jα.

(3.4)

If we take◦H

a

αβ = Bai (

◦δβB

iα + B

jα

◦Li

jβ), the last expression provides

Haαβ =

◦H

a

αβ + Bai B

γpV

p

j Bj

βBiαγ +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bai B

jαB

kβ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkbH

bβB

ai B

jα.

(3.5)

Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find

Kaαβ =

◦K

a

αβ +

(ϕ′′


2(ϕ′′′ϕ′ − ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′)yi ◦yj

◦yk

)

Bai B

jαB

kβ , (3.6)

where◦K

a

αβ = Bai B

jα

◦C

i

jβ. Thus, we have the following.

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

∇Biα = Bi

aΠaα, ∇Bi

a = −BiβΠ

βa, (3.7)


where

Πaα = Ha

αβduβ +Ka

αβδvβ, Πβ

a = gβγδabΠbγ ,

Haαβ =

◦H

a

αβ + Bai B

γpV

p

j Bj

βBiαγ +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bai B

jαB

kβ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkbH

bβB

ai B

jα,

Kaαβ =

◦K

a

αβ +

(ϕ′′


2(ϕ′′′ϕ′ − ϕ′′2)

ϕ′(ϕ′ + 2F2ϕ′′)yi ◦yj

◦yk

)

Bai B

jαB

kβ .

(3.8)

Remark 3.2. Haαβ

and Kaαβ

given, respectively, by (3.5) and (3.6) are called the secondfundamental d-tensor fields of immersion i.

The following consequences of Theorem 3.1 are straightforward.

Corollary 3.3. In a subspace Lm of an APL space, we have the following:

(a) ∇aαβ = 0,

(b) ∇Biα = 0,

(3.9)

if and only if

◦H

a

αβ = −[

Bai B

γpV

p

j Bj

βBiαγ +

(

V rkC

ijr + V r

j Cikr + V r

p aipCrkj −

◦C

i

jrBrbB

bpV

p

k

)

Bai B

jαB

kβ

+(ϕ′′

ϕ′◦yjδ

ik +

ϕ′′


)BkbH

bβB

ai B

jα

]

,

◦K

a

αβ = −

⎛

⎜⎝

ϕ′′


2(ϕ′′′ϕ′ − ϕ′′2

)

ϕ′(ϕ′ + 2F2ϕ′′)yi ◦yj

◦yk

⎞

⎟⎠Ba

i BjαB

kβ .

(3.10)

4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspace Lm = (M, L(u, v)) of a Lagrange space Ln aregiven by (cf. [8])

BiαB

j

βΩij −Ωαβ = Πβa ∧Πa

α, (4.1)

BiaB

j

bΩij −Ωab = Πγb ∧Πγ

a, (4.2)

−BiαB

jaΩij = δab

(dΠb

α + Πbβ ∧ω

βα −Πc

α ∧ωbc

), (4.3)


where

(a) Παa = gαβΠβa, (b) Πγb = δbcΠc

γ . (4.4)

Using (1.3) in (2.41)(a), we find that

Ωij = ϕ′Ωhi ghj + 2ϕ′′Ωh

i

◦yh

◦yj. (4.5)

Applying aγβ = BiγB

j

βaij in (2.41)(b), we have Ωαβ = BiγB

j

βΩγαaij , which in view of (1.3)

becomes

Ωαβ = ϕ′gijBiγB

j

βΩγ

α + 2ϕ′′ ◦yi◦yjB

iγB

j

βΩγ

α, (4.6)

that is,

Ωαβ = ϕ′gγβΩγα + 2ϕ′′ ◦yi

◦yjB

iγB

j

βΩγ

α. (4.7)

For the subspace Lm of an APL space, (4.4)(a) is of the form Παa = aαβΠβa, which in view of

aαβ = BiαB

j

βaij and (1.3) becomes Παa = ϕ′Bi

αBj

βaijΠ

βa + 2ϕ′′ ◦yi

◦yjB

iαB

j

βΠβ

a, that is,

Παa = ϕ′gαβΠβa + 2ϕ′′ ◦yi

◦yjB

iαB

j

βΠβ

a. (4.8)


Theorem 4.1. The Gauss-Codazzi equations for a Lagrange subspace Lm of an APL space are givenby (4.1)–(4.3) with Παa, Πγb, Ωij , Ωαβ, and ωb

c , 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 Scientificand Industrial Research (CSIR), India.

References

[1] R. Miron, “Lagrange geometry,”Mathematical and Computer Modelling, vol. 20, no. 4-5, pp. 25–40, 1994.[2] R. Miron and A. Bejancu, “A nonstandard theory of Finsler subspaces,” in Topics in Differential Geo-

metry, vol. 46 of Colloquia Mathematica Societatis Janos Bolyai, pp. 815–851, 1984.[3] R. Miron and M. Anastasiei, Vector Bundles and Lagrange Spaces with Applications to Relativity, vol. 1,

Geometry Balkan Press, Bucharest, Romania, 1997.[4] T. Sakaguchi, Subspaces in Lagrange spaces [Ph.D. thesis], University ”Al. I. Cuza”, Iasi, Romania, 1986.[5] P. L. Antonelli and D. Hrimiuc, “A new class of spray-generating Lagrangians,” in Lagrange and

Finsler Geometry, vol. 76 of Applications to Physics and Biology, pp. 81–92, Kluwer Academic Publishers,Dordrecht, The Netherlands, 1996.


[6] P. L. Antonelli and D. Hrimiuc, “On the theory of ϕ-Lagrange manifolds with applications in biologyand physics,” Nonlinear World, vol. 3, no. 3, pp. 299–333, 1996.

[7] P. N. Pandey and S. K. Shukla, “On almost ϕ-Lagrange spaces,” ISRN Geometry, vol. 2011, Article ID505161, 16 pages, 2011.

[8] P. L. Antonelli, Ed.,Handbook of Finsler Geometry, Kluwer Academic Publishers, Dordrecht, The Nether-lands, 2001.

[9] R. Miron andM. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluwer AcademicPublishers, 1994.

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