Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 981059, 14 pages
doi:10.1155/2012/981059
Research Article
On 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, shuklasureshk@gmail.com
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 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 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.
Let F n M, Fx, y be an n-dimensional Finsler space and ϕ : R → R a smooth
function. If the function ϕ has the following properties:
0,
a ϕ t /
0, for every t ∈ ImF 2 ,
b ϕ t ϕ t /
then the Lagrangian given by
L x, y ϕ F 2 Ai xyi Ux,
1.1
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where Ai x is a covector and Ux is a smooth function, is a regular Lagrangian 7. The
space Ln M, Lx, 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 F n . An APL
space reduces to a ϕ-Lagrange space if and only if Ai x 0 and Ux 0. We take
gij 1˙ ˙ 2
∂i ∂j F ,
2
aij 1˙ ˙
∂i ∂j L,
2
∂
where ∂˙ i ≡
.
∂yi
1.2
We indicate all the geometrical objects related to F n 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:
2ϕ ◦ ◦
aij ϕ · gij yi yj ,
ϕ
aij 2ϕ
1
ij
i j
−
y
y
g
,
ϕ
ϕ 2F 2 ϕ
1.3
◦
where gij yj yi .
Let M̌ be a smooth manifold of dimension m, 1 < m < n, immersed in M by immersion
i : M̌ → M. The immersion i induces an immersion Ti : T M̌ → T M making the following
diagram commutative:
Ti
T M̌ −→ T M
π̌ ↓
↓π
M̌
−→
i
1.4
M.
Let uα , vα throughout the paper, the Greek indices α, β, γ, . . . run from 1 to m be local
coordinates on T M̌. The restriction of the Lagrangian L on T M̌ is Lu, v Lxu, yu, v.
j
Let aαβ 1/2∂2 Ľ/∂uα ∂uβ . Then, we have cf. 8 aαβ Bαi Bβ aij where Bαi u ∂xi /∂uα
are the projection factors. The pair Ľm M̌, Ľu, v is also a Lagrange space, called the
subspace of Ln . For the natural bases ∂/∂xi , ∂/∂yi on TM and ∂/∂uα , ∂/∂vα on T M̌, we
have 8
∂
∂
∂
i
Bαi i B0α
,
∂uα
∂x
∂yi
∂
∂
Bαi i ,
∂vα
∂y
1.5
i
i
i
where B0α
Bβα
vβ , Bβα
∂2 xi /∂uα ∂uβ .
For the bases dxi , dyi and duα , dvα , we have
dxi Bαi duα ,
i
dyi Bαi dvα B0α
duα .
1.6
Since Bαi are m linearly independent vector fields tangent to M̌, a vector field ξi x, y is
normal to M̌ along T M̌ if on T M̌, we have
aij xu, yu, v Bαi ξj 0,
∀α 1, 2, . . . , m.
1.7
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3
There are, at least locally, n − m unit vector fields Bai u, v a m 1, m 2, . . . , n normal
to M̌ and mutually orthonormal, that is,
j
aij Bαi Bb 0,
j
aij Bai Bb δab ,
a, b m 1, m 2, . . . , n.
1.8
Thus, at every point u, v ∈ T M̌, 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 yi Bai 0 this fact is clear from
j
gij yi Ba 0 in view, we observe that Bai ’s are normal to M̌ with respect to Ln if and only if
they are so with respect to F n . The dual frame of is ∗ u, v, Biα u, v, Bia u, v with the
following duality conditions:
β
β
Bαi Bi δα ,
β
Bai Bi 0,
Bαi Bib 0,
Bai Bib δab ,
Bai Bja Bαi Bjα δji .
1.9
We will make use of the following results due to the present authors 7, during further
discussion.
Theorem 1.1 cf. 7. The canonical nonlinear connection of an APL space Ln has the local
coefficients given by
◦ i
Nji N j − Vji ,
1.10
k
where Vji 1/2Fji − Sir
j 2Frk y ∂r U,
Sir
j ◦i
1
C g qr
2ϕ qj
r i
i r
δ
y
δ
y
ϕ
ϕ 2 ϕ − 2ϕ3 F 2 − 4ϕ ϕ2 i ◦ r
j
j
1 ϕ ir ◦
g
y
j
2 y yj y ,
2 ϕ 2
2ϕ ϕ 2F 2 ϕ
2ϕ 2 ϕ 2F 2 ϕ
Frk x 1
∂r Ak − ∂k Ar ,
2
1.11
Fji aik Fkj .
Theorem 1.2 cf. 7. The coefficients of the canonical metrical d-connection CΓN of an APL
space Ln are given by
i
Cjk
2 ϕ ϕ − 2ϕ2 i ◦ ◦
ϕ
ϕ i ◦
i ◦
i
Cjk δj yk δk yj gjk y y yj yk ,
ϕ
ϕ 2F 2 ϕ
ϕ ϕ 2F 2 ϕ
◦i
◦i
i
i
Vjr Ckr
Vpr aip Crkj .
Lijk Ljk Vkr Cjr
1.12
1.13
For basic notations related to a Finsler space, a Lagrange space, and their subspaces,
we refer to the books 8, 9.
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2. Induced Nonlinear Connection
Let Ň Ňβα u, v be a nonlinear connection for Ľm M̌, Ľu, v. The adapted basis of
Tu,v T M̌ induced by Ň is δ/δuα δα , ∂/∂vα ∂˙ α , where
β
δα ∂α − Ňα ∂˙ β .
2.1
The dual basis cobasis of the adapted basis δα , ∂˙ α is duα , δvα dvα Ňβα duβ .
Definition 2.1 cf. 8. A nonlinear connection Ň Ňβα u, v of Ľm is said to be induced by
the canonical nonlinear connection N if the following equation holds good:
2.2
δvα Biα δyi .
The local coefficients of the induced nonlinear connection Ň Ňβα u, v for the subspace
Ľm M̌, Ľu, v of a Lagrange space Ln M, Lx, y are given by cf. 8
j
i
,
Ňβα Biα Nji Bβ B0β
2.3
Nji being the local coefficients of canonical nonlinear connection N of the Lagrange space
Ln M, Lx, y. Now using 1.10 in 2.3, we get
Ňβα
◦ α
◦ i
Biα
◦ i
j
N j Bβ
i
B0β
j
− Biα Vji Bβ .
2.4
j
i
If we take Ň β Biα N j Bβ B0β
, it follows from 2.4 that
◦ α
j
Ňβα Ň β − Biα Vji Bβ .
2.5
Thus, we have the following.
Theorem 2.2. The local coefficients of the induced nonlinear connection Ň of the subspace Ľm of an
APL space Ln are given by 2.5.
In view of 2.5, 2.1 takes the following form, for the subspace Ľm of an APL space
n
L :
◦
p j
δβ δβ Bpα Vj Bβ ∂˙ α ,
◦
◦ α
where δβ ∂β − Ň β ∂˙ α .
2.6
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5
We can put dxi , δyi as cf. 8
dxi Bαi duα ,
2.7
δyi Bαi δyα Bai Hαa duα ,
where
j
i
.
Hαa Bia Nji Bα B0α
2.8
Using 1.10 in 2.8 and simplifying, we get
Hαa
◦ a
◦ i
Bia
◦ i
j
N j Bα
i
B0α
j
− Bia Vji Bα .
2.9
j
i
, in 2.9, it follows that
Taking H α Bia N j Bα B0α
◦ a
2.10
j
Hαa H α − Bia Vji Bα .
◦ a
j
Now, dxi Bαi duα , δyi Bαi δyα if and only if Hαa 0, that is, if and only if H α Bia Vji Bα .
Thus, we have the following.
Theorem 2.3. The adapted cobasis dxi , δyi of the basis ∂/∂xi , ∂/∂yi induced by the nonlinear
◦ a
connection N of an APL space Ln is of the form dxi Bαi duα , δyi Bαi δyα if and only if H α j
Bia Vji Bα .
Definition 2.4 cf. 8. Let D DΓN be the canonical metrical d-connection of Ln . An
operator Ď is said to be a coupling of D with Ň if
i
duα X i |α δvα ,
ĎX i X|α
2.11
i
i
δα X i X j Ľijα , X i |α ∂˙ α X i X j Čjα
.
where X|α
i
The coefficients Ľijα , Čjα
of coupling Ď of D with Ň are given by
i
Ľijα Lijk Bαk Cjk
Bak Hαa ,
2.12
i
i
Čjα
Cjk
Bαk .
2.13
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International Journal of Mathematics and Mathematical Sciences
Using 1.12 and 1.13 in 2.12, we have
Ľijβ
◦i
Ljk i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
Bβk
ϕ
ϕ i ◦
◦
δj yk δki yj gjk yi
ϕ
ϕ 2F 2 ϕ
2 ϕ ϕ − 2ϕ2 i ◦ ◦
y yj yk Bak Hβa .
ϕ ϕ 2F 2 ϕ
◦i
Cjk 2.14
◦
In view of 2.10 and yi Bai 0, 2.14 becomes
Ľijβ
◦i
Ljk Bβk
◦i
◦ a
Cjk Bak H β
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
Bβk
2.15
ϕ ◦ i
ϕ
i
y
δ
g
y
Bak Hβa ,
j
jk
k
ϕ
ϕ 2F 2 ϕ
that is,
Ľijβ
◦i
Ľjβ ◦i
◦i
◦i
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
ϕ ◦ i
ϕ
i
y
δ
g
y
Bak Hβa ,
j
jk
k
ϕ
ϕ 2F 2 ϕ
Bβk
2.16
◦ a
where Ľjβ Ljk Bβk Cjk Bak H β .
Using 1.12 in 2.13, we find that
i
Čjβ
◦i
Cjk Bβk
ϕ
ϕ i ◦
i ◦
δ
y
δ
y
gjk yi
k
j
k
ϕ j
ϕ 2F 2 ϕ
2 ϕ ϕ − 2ϕ2 i ◦ ◦
y yj yk Bβk ,
ϕ ϕ 2F 2 ϕ
2.17
that is,
i
Čjβ
◦i
Čjβ ϕ
ϕ i ◦
i ◦
δ
y
δ
y
gjk yi
k
j
j
k
ϕ
ϕ 2F 2 ϕ
2 ϕ ϕ − 2ϕ2 i ◦ ◦
y yj yk Bβk ,
ϕ ϕ 2F 2 ϕ
2.18
International Journal of Mathematics and Mathematical Sciences
◦i
7
◦i
where Čjβ Cjk Bβk . Thus, we have the following.
Theorem 2.5. The coefficients of coupling for the subspace Ľm of an APL space Ln are given by 2.16
and 2.18.
Definition 2.6 cf. 8. An operator DT given by
α
duβ X α |β δvβ ,
DT X α X|β
2.19
α
α
where X|β
δβ X α X γ Lαγβ , X α |β ∂˙ β X α X γ Cγβ
, is called the induced tangent connection by
D. This defines an N-linear connection for Ľm .
α
of DT are given by
The coefficients Lαγβ , Cγβ
j
i
Lαβγ Biα Bβγ
Bβ Ľijγ ,
2.20
j
α
i
Cβγ
Biα Bβ Čjγ
.
2.21
Using 2.16 in 2.20, we get
Lαβγ
i
Biα Bβγ
j
Bβ Biα
◦i
Ľjγ i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
Bγk
2.22
ϕ ◦ i
ϕ
i
k a
yj δk gjk y Ba Hγ ,
ϕ
ϕ 2F 2 ϕ
that is,
Lαβγ
Biα
i
Bβγ
◦i
j
Ľjγ Bβ
j
Biα Bβ
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
2.23
ϕ ◦ i
ϕ
i
k a
yj δk gjk y Ba Hγ .
ϕ
ϕ 2F 2 ϕ
◦i
◦α
Bγk
j
i
If we take Lβγ Biα Bβγ
Ľjγ Bβ , the last expression gives
Lαβγ
◦α
Lβγ j
Biα Bβ
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
ϕ ◦ i
ϕ
i
k a
yj δk gjk y Ba Hγ .
ϕ
ϕ 2F 2 ϕ
Bγk
2.24
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International Journal of Mathematics and Mathematical Sciences
Next, using 2.18 in 2.21, we obtain
α
Cβγ
◦i
j
Biα Bβ Čjγ
◦α
j
2 ϕ ϕ − 2ϕ2 i ◦ ◦
ϕ i ◦
ϕ
j
i ◦
i
δj yk δk yj gjk y y yj yk Bγk Biα Bβ .
2
2
ϕ
ϕ 2F ϕ
ϕ ϕ 2F ϕ
2.25
◦i
If we take Cβγ Biα Bβ Čjγ , 2.25 becomes
α
Cβγ
◦α
Cβγ 2 ϕ ϕ − 2ϕ2 i ◦ ◦
ϕ
ϕ i ◦
j
i ◦
i
δ yk δk yj gjk y y yj yk Bγk Biα Bβ . 2.26
ϕ j
ϕ 2F 2 ϕ
ϕ ϕ 2F 2 ϕ
Thus, we have the following.
Theorem 2.7. The coefficients of the induced tangent connection DT for the subspace Ľm of an APL
space are given by 2.24 and 2.26.
α
α
α
Lαβγ − Lαγβ does not vanish, in general, while Sαβγ Cβγ
− Cγβ
0.
Remarks. The torsion Tβγ
These facts may be observed from 2.24 and 2.26.
Definition 2.8 cf. 8. An operator D⊥ given by
a
D⊥ X a X|α
duα X a |α δvα ,
2.27
a
a
where X|α
δα X a X b Labα , X a |α ∂˙ α X a X b Cbα
, is called the induced normal connection by
D.
a
of D⊥ are given by
The coefficients Labγ , Cbγ
j
Labγ Bia δγ Bbi Bb Ľijγ ,
2.28
j i
a
.
Cbγ
Bia ∂˙ γ Bbi Bb Čjγ
2.29
Using 2.6 and 2.16 in 2.28, we find
◦
p j
Labγ Bia δγ Bbi Bia Bpα Vj Bγ ∂˙ α Bbi
j
Bb Bia
◦i
Ľjγ i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bcr Bpc Vk
ϕ ◦ i
ϕ
i
k c
yj δk gjk y Bc Hγ .
ϕ
ϕ 2F 2 ϕ
Bγk
2.30
International Journal of Mathematics and Mathematical Sciences
◦
◦a
j
◦i
◦
9
j
Taking Lbγ Bia δγ Bbi Bb Ľjγ and using yj Bb 0, 2.30 reduces to
Labγ
◦a
Lbγ p j
Bia Bpα Vj Bγ ∂˙ α Bbi
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
j
Bia Bb Bγk
2.31
ϕ
j
gjk yi Bck Hγc Bia Bb .
ϕ 2F 2 ϕ
Next, using 2.18 in 2.29, we have
a
Cbγ
Bia
◦a
◦i
j
∂˙ γ Bbi Bb Čjγ
j
◦i
ϕ i ◦
ϕ
i ◦
δ
y
δ
y
gjk yi
k
j
k
ϕ j
ϕ 2F 2 ϕ
2 ϕ ϕ − 2ϕ2 i ◦ ◦
j
y yj yk Bγk Bia Bb .
ϕ ϕ 2F 2 ϕ
◦
2.32
j
Taking Cbγ Bia ∂˙ γ Bbi Bb Čjγ and using 1.9 and yj Bb 0, the last equation yields
◦a
a
Cbγ
Cbγ ϕ a ◦ k
ϕ
j
δ
y
B
gjk yi Bγk Bia Bb .
k γ
ϕ b
ϕ 2F 2 ϕ
2.33
Thus, we have the following.
Theorem 2.9. The coefficients of induced normal connection D⊥ for the subspace Ľm of an APL space
Ln are given by 2.31 and 2.33.
i···α···a
Definition 2.10 cf. 8. The mixed derivative of a mixed d-tensor field Tj···β···b
is given by
i···γ···a
i···α···a
i···α···a
k···α···a i
i···α···c a
Ľkη · · · Tj···β···b Lαγη · · · Tj···β···b
δη Tj···β···b
Tj···β···b
Lcη
∇Tj···β···b
i···α···a k
i···α···a γ
i···α···a c
Ľjη − · · · − Tj···γ···b
Ľbη duη
−Tk···β···b
Lβη − · · · − Tj···β···c
i···γ···a α
i···α···a
k···α···a i
i···α···c a
Čkη · · · Tj···β···b Cγη
∂˙ η Tj···β···b
Tj···β···b
· · · Tj···β···b
Ccη
2.34
i···α···a k
i···α···a γ
i···α···a c
Čjη − · · · − Tj···γ···b
Čbη δvη .
−Tk···β···b
Cβη − · · · − Tj···β···c
The connection 1-forms,
i
ω̌ji : Ľijα duα Čjα
δvα ,
2.35
α
ωβα : Lαβγ duγ Cβγ
δvγ ,
2.36
a
ωba : Labγ duγ Cbγ
δvγ ,
2.37
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International Journal of Mathematics and Mathematical Sciences
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:
dduα − duβ ∧ ωβα −Ωα ,
dδuα − δuβ ∧ ωβα −Ω̇α ,
2.38
dω̌ji − ω̌jh ∧ ω̌hi −Ω̌ij ,
γ
dωβα − ωβ ∧ ωγα −Ωαβ ,
dωba − ωbc ∧ ωca −Ωab ,
where the 2-forms of torsions Ωα , Ω̇α are given by
Ωα 1 α β
α
T du ∧ duγ Cβγ
duβ ∧ δvγ ,
2 βγ
1
α
duβ ∧ δvγ ,
Ω̇ Rαβγ duβ ∧ duγ Pβγ
2
2.39
α
α
with Pβγ
∂˙ γ Ňβα − Lαβγ , and the 2-forms of curvature Ω̌ij , Ωαβ and Ωab , are given by
Ω̌ij 1 i
1
i
Ř duα ∧ duβ P̌jαβ
duα ∧ δvβ Šijαβ δvα ∧ δvβ ,
2 jαβ
2
Ωαβ 1 α
1
α
R duγ ∧ duδ Pβγδ
duγ ∧ δvδ Sαβγδ δvγ ∧ δvδ ,
2 βγδ
2
Ωab 1 a
1
a
Rbαβ duα ∧ duβ Pbαβ
duα ∧ δvβ Sabαβ δvα ∧ δvβ .
2
2
2.40
We will use the following notations in Section 4:
a Ω̌ij Ω̌hi ahj ,
γ
b Ωαβ Ωα aγβ ,
c Ωab Ωcb δac .
2.41
3. The Gauss-Weingarten Formulae
The Gauss-Weingarten formulae for the subspace Ľm M̌, Ľu, v of a Lagrange space Ln
are given by cf. 8
∇Bαi Bai Πaα ,
β
∇Bai −Bβi Πa ,
3.1
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11
where
a
a
Πaα Hαβ
duβ Kαβ
δvβ ,
3.2
β
a
a Hαβ
Πa g βγ δab Πbγ ,
j
j i
a
Bia δβ Bαi Bα Ľijβ ,
Bia Bα Čjβ
.
b Kαβ
3.3
Using 2.6 and 2.16 in 3.3a, we have
a
Hαβ
Bia
◦
δβ Bαi
◦i
j
Bα Ľjβ
γ
p
j
i
Bia Bp Vj Bβ Bαγ
i
Vkr Cjr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
3.4
j
Bia Bα Bβk
ϕ ◦ i
ϕ
j
i
Bbk Hβb Bia Bα .
y
δ
g
y
j k
jk
ϕ
ϕ 2F 2 ϕ
◦
◦ a
i
Vjr Ckr
j
◦i
If we take H αβ Bia δβ Bαi Bα Ľjβ , the last expression provides
a
Hαβ
◦ a
H αβ γ p j i
Bia Bp Vj Bβ Bαγ
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
j
Bia Bα Bβk
3.5
ϕ ◦ i
ϕ
j
i
y
δ
g
y
Bbk Hβb Bia Bα .
j
jk
k
ϕ
ϕ 2F 2 ϕ
Next, using 2.18 in 3.3b and keeping 1.9 in view, we find
a
Kαβ
◦a
j
◦a
K αβ 2 ϕ ϕ − ϕ2 i ◦ ◦
ϕ
j
i
gjk y y yj yk Bia Bα Bβk ,
ϕ 2F 2 ϕ
ϕ ϕ 2F 2 ϕ
3.6
◦i
where K αβ Bia Bα Čjβ . Thus, we have the following.
Theorem 3.1. The following Gauss-Weingarten formulae for the subspace Ľm of an APL space hold:
∇Bαi Bai Πaα ,
β
∇Bai −Bβi Πa ,
3.7
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International Journal of Mathematics and Mathematical Sciences
where
a
Hαβ
β
a
a
duβ Kαβ
δvβ ,
Πaα Hαβ
◦ a
H αβ γ p j i
Bia Bp Vj Bβ Bαγ
i
Vkr Cjr
Πa g βγ δab Πbγ ,
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
j
Bia Bα Bβk
ϕ ◦ i
ϕ
j
i
y
δ
g
y
Bbk Hβb Bia Bα ,
j k
jk
ϕ
ϕ 2F 2 ϕ
◦a
2 ϕ ϕ − ϕ2 i ◦ ◦
ϕ
j
a
i
Kαβ K αβ gjk y y yj yk Bia Bα Bβk .
ϕ 2F 2 ϕ
ϕ ϕ 2F 2 ϕ
3.8
a
a
Remark 3.2. Hαβ
and Kαβ
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 Ľm of an APL space, we have the following:
a ∇aαβ 0,
3.9
b ∇Bαi 0,
if and only if
◦ a
H αβ −
γ p j i
Bia Bp Vj Bβ Bαγ
i
Vkr Cjr
i
Vjr Ckr
Vpr aip Crkj
−
◦i
p
Cjr Bbr Bpb Vk
ϕ ◦ i
ϕ
i
k b a j
yj δk gjk y Bb Hβ Bi Bα ,
ϕ
ϕ 2F 2 ϕ
⎞
⎛
2 ϕ ϕ − ϕ 2
◦a
ϕ
◦
◦
j
⎟
⎜
K αβ −⎝ gjk yi yi yj yk ⎠Bia Bα Bβk .
ϕ 2F 2 ϕ
ϕ ϕ 2F 2 ϕ
j
Bia Bα Bβk
3.10
4. The Gauss-Codazzi Equations
The Gauss-Codazzi Equations for the subspace Ľm M̌, Ľu, v of a Lagrange space Ln are
given by cf. 8
j
Bαi Bβ Ω̌ij − Ωαβ Πβa ∧ Πaα ,
j
γ
Bai Bb Ω̌ij − Ωab Πγb ∧ Πa ,
j
β
−Bαi Ba Ω̌ij δab dΠbα Πbβ ∧ ωα − Πcα ∧ ωcb ,
4.1
4.2
4.3
International Journal of Mathematics and Mathematical Sciences
13
where
β
a Παa gαβ Πa ,
b Πγb δbc Πcγ .
4.4
◦ ◦
4.5
Using 1.3 in 2.41a, we find that
Ω̌ij ϕ Ω̌hi ghj 2ϕ Ω̌hi yh yj .
j
j
γ
Applying aγβ Bγi Bβ aij in 2.41b, we have Ωαβ Bγi Bβ Ωα aij , which in view of 1.3
becomes
j
◦ ◦
γ
j
γ
Ωαβ ϕ gij Bγi Bβ Ωα 2ϕ yi yj Bγi Bβ Ωα ,
4.6
that is,
◦ ◦
γ
j
γ
Ωαβ ϕ gγβ Ωα 2ϕ yi yj Bγi Bβ Ωα .
4.7
β
For the subspace Ľm of an APL space, 4.4a is of the form Παa aαβ Πa , which in view of
j
j
β
◦ ◦
j
β
aαβ Bαi Bβ aij and 1.3 becomes Παa ϕ Bαi Bβ aij Πa 2ϕ yi y j Bαi Bβ Πa , that is,
β
◦ ◦
j
β
Παa ϕ gαβ Πa 2ϕ yi yj Bαi Bβ Πa .
4.8
Thus, we have the following.
Theorem 4.1. The Gauss-Codazzi equations for a Lagrange subspace Ľm of an APL space are given
by 4.1–4.3 with Παa , Πγb , Ω̌ij , 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.
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 Geometry, vol. 46 of Colloquia Mathematica Societatis János 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.
14
International Journal of Mathematics and Mathematical Sciences
6 P. L. Antonelli and D. Hrimiuc, “On the theory of ϕ-Lagrange manifolds with applications in biology
and 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 ID
505161, 16 pages, 2011.
8 P. L. Antonelli, Ed., Handbook of Finsler Geometry, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
9 R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Academic
Publishers, 1994.
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