Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 857953, 10 pages
doi:10.1155/2011/857953
Research Article
On CR-Lightlike Product of an Indefinite
Kaehler Manifold
Rakesh Kumar,1 Jasleen Kaur,2 and R. K. Nagaich2
1
2
University College of Engineering, Punjabi University, Patiala, India
Department of Mathematics, Punjabi University, Patiala, India
Correspondence should be addressed to Rakesh Kumar, dr rk37c@yahoo.co.in
Received 20 December 2010; Accepted 3 March 2011
Academic Editor: B. N. Mandal
Copyright q 2011 Rakesh Kumar et al. 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 have studied mixed foliate CR-lightlike submanifolds and CR-lightlike product of an indefinite
Kaehler manifold and also obtained relationship between them. Mixed foliate CR-lightlike
submanifold of indefinite complex space form has also been discussed and showed that the
indefinite Kaehler manifold becomes the complex semi-Euclidean space.
1. Introduction
The geometry of CR-submanifolds of Kaehler manifolds was initiated by Bejancu 1 and has
been developed by 2–5 and others. They studied the geometry of CR-submanifolds with
positive definite metric. Thus, this geometry may not be applicable to the other branches of
mathematics and physics, where the metric is not necessarily definite. Moreover, because of
growing importance of lightlike submanifolds and hypersurfaces in mathematical physics,
especially in relativity, make the geometry of lightlike submanifolds and hypersurfaces a
topic of chief discussion in the present scenario. In the establishment of the general theory of
lightlike submanifolds and hypersurfaces, Kupeli 6 and Duggal and Bejancu 7 played a
very crucial role. The objective of this paper is to study CR-lightlike submanifolds of an indefinite Kaehler manifold. This motivated us to study CR-lightlike submanifolds extensively.
2. Lightlike Submanifolds
We recall notations and fundamental equations for lightlike submanifolds, which are due to
7 by Duggal and Bejancu.
Let M, g be a real m n-dimensional semi-Riemannian manifold of constant index
q such that m, n ≥ 1, 1 ≤ q ≤ m n − 1, M, g an m-dimensional submanifold of M, and g
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the induced metric of g on M. If g is degenerate on the tangent bundle T M of M, then M is
called a lightlike submanifold of M. For a degenerate metric g on M,
T M⊥ ∪ u ∈ Tx M : gu, v 0, ∀v ∈ Tx M, x ∈ M
2.1
is a degenerate n-dimensional subspace of Tx M. Thus, both Tx M and Tx M⊥ are degenerate
orthogonal subspaces but no longer complementary. In this case, there exists a subspace
Rad Tx M Tx M ∩ Tx M⊥ which is known as radical null subspace. If the mapping
Rad T M : x ∈ M −→ Rad Tx M
2.2
defines a smooth distribution on M of rank r > 0, then the submanifold M of M is called
r-lightlike submanifold and Rad T M is called the radical distribution on M.
Let ST M be a screen distribution which is a semi-Riemannian complementary
distribution of RadT M in T M, that is,
T M Rad T M ⊥ ST M,
2.3
where ST M⊥ is a complementary vector subbundle to Rad T M in T M⊥ . Let trT M and
ltrT M be complementary but not orthogonal vector bundles to T M in T M|M and to
Rad T M in ST M⊥ ⊥ , respectively. Then, we have
trT M ltrT M ⊥ S T M⊥ ,
2.4
T M|M T M ⊕ trT M Rad T M ⊕ ltrT M ⊥ ST M ⊥ S T M⊥ .
2.5
Let u be a local coordinate neighborhood of M and consider the local quasiorthonormal
fields of frames of M along M, on u as {ξ1 , . . . , ξr , Wr1 , . . . , Wn , N1 , . . . , Nr , Xr1 , . . . , Xm },
where {ξ1 , . . . , ξr }, {N1 , . . . , Nr } are local lightlike bases of ΓRad T M|u , ΓltrT M|u and
{Wr1 , . . . , Wn }, {Xr1 , . . . , Xm } are local orthonormal bases of ΓST M⊥ |u and ΓST M|u ,
respectively. For this quasi-orthonormal fields of frames, we have the following.
Theorem 2.1 see 7. Let M, g, ST M, ST M⊥ be an r-lightlike submanifold of a semiRiemannian manifold M, g. Then, there exists a complementary vector bundle ltrT M of Rad T M
in ST M⊥ ⊥ and a basis of ΓltrT M|u consisting of smooth section {Ni } of ST M⊥ ⊥ |u , where u
is a coordinate neighborhood of M, such that
g Ni , ξj δij ,
g Ni , Nj 0,
where {ξ1 , . . . , ξr } is a lightlike basis of ΓRadT M.
2.6
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3
Let ∇ be the Levi-Civita connection on M. Then, according to the decomposition 2.5,
the Gauss and Weingarten formulas are given by
∀X, Y ∈ ΓT M,
2.7
∀X ∈ ΓT M, U ∈ ΓtrT M,
2.8
∇X Y ∇X Y hX, Y ,
∇X U −AU X ∇⊥X U,
where {∇X Y, AU X} and {hX, Y , ∇⊥X U} belong to ΓT M and ΓtrT M, respectively. Here,
∇ is a torsion-free linear connection on M, h is a symmetric bilinear form on ΓT M which
is called second fundamental form, and AU is a linear operator on M and known as shape
operator.
According to 2.4, considering the projection morphisms L and S of trT M on
ltrT M and ST M⊥ , respectively, 2.7 and 2.8 give
∇X Y ∇X Y hl X, Y hs X, Y ,
l
s
∇X U −AU X DX
U DX
U,
2.9
2.10
s
l
U L∇⊥X U,and DX
U where we put hl X, Y LhX, Y , hs X, Y ShX, Y , DX
⊥
S∇X U.
As hl and hs are ΓltrT M-valued and ΓST M⊥ -valued, respectively, therefore,
they are called the lightlike second fundamental form and the screen second fundamental
form on M. In particular,
∇X N −AN X ∇lX N Ds X, N,
2.11
∇X W −AW X ∇sX W Dl X, W,
2.12
where X ∈ ΓT M, N ∈ ΓltrT M, and W ∈ ΓST M⊥ .
Using 2.4-2.5 and 2.9–2.12, we obtain
ghs X, Y , W g Y, Dl X, W gAW X, Y ,
g hl X, Y , ξ g Y, hl X, ξ gY, ∇X ξ 0,
2.13
g AN X, N gN, AN X 0,
for any ξ ∈ ΓRad T M, W ∈ ΓST M⊥ , and N, N ∈ ΓltrT M.
Let P is a projection of T M on ST M. Now, considering the decomposition 2.3, we
can write
∇X P Y ∇∗X P Y h∗ X, P Y ,
∇X ξ −A∗ξ X ∇∗t
X ξ,
2.14
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for any X, Y ∈ ΓT M and ξ ∈ ΓRad T M, where {∇∗X P Y, A∗ξ X} and {h∗ X, P Y , ∇∗t
X ξ} belong
∗t
∗
to ΓST M and ΓRad T M, respectively. Here, ∇ and ∇X are linear connections on ST M
and Rad T M, respectively. By using 2.9-2.10 and 2.14, we obtain
g hl X, P Y , ξ g A∗ξ X, P Y ,
2.15
g h∗ X, P Y , N g AN X, P Y .
Definition 2.2. Let M, J, g be a real 2m-dimensional indefinite Kaehler manifold and M an
n-dimensional submanifold of M. Then, M is said to be a CR-lightlike submanifold if the
following two conditions are fulfilled:
A JRad T M is distribution on M such that
Rad T M ∩ JRad T M 0,
2.16
B there exist vector bundles ST M, ST M⊥ , ltrT M, D0 and D over M such that
ST M JRad T M ⊕ D ⊥ D0 ,
JD0 D0 ,
J D L1 ⊥ L 2 ,
2.17
where ΓD0 is a nondegenerate distribution on M, ΓL1 and ΓL2 are vector sub-bundles
of ΓltrT M and ΓST M⊥ , respectively.
Clearly, the tangent bundle of a CR-lightlike submanifold is decomposed as
T M D ⊕ D ,
2.18
D Rad T M ⊥ JRad T M ⊥ D0 .
2.19
where
Theorem 2.3. Let M be a 1-lightlike submanifold of codimension 2 of a real 2 m-dimensional
indefinite almost Hermitian manifold M, J, g such that JRad T M is a distribution on M. Then,
M is a CR-lightlike submanifold.
Proof. Since gJξ, ξ 0, therefore, Jξ is tangent to M. Moreover, Rad T M and JRad T M
are distributions of rank 1 on M, therefore, Rad T M ∩ JRad T M {0}. This enables
one to choose a screen distribution ST M such that it contains JRad T M. For any W ∈
ΓST M⊥ , we have gJW, ξ −gW, Jξ 0 and gJW, W 0. As ST M⊥ is of rank 1,
therefore, JST M⊥ is also a distribution on M such that JST M⊥ ∩ {T M⊥ ⊥ JT M⊥ } {0} and ST M {JRad T M ⊥ JST M⊥ } ⊕ E, where E is a distribution of rank 2 m–
5. Now, for any N ∈ ΓltrT M, we have gJN, ξ −gN, Jξ 0 and gJN, W 0,
therefore, JN is tangent to M and gJN, N gJN, JN gJN, JW 0. This implies
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5
that JN has no component in Rad T M, JRad T M, and JST M⊥ . Thus, JN lies in ΓE
and we have
⊥ D0 ,
2.20
ST M JRad T M ⊕ JltrT M ⊥ J S T M⊥
where D0 is a nondegenerate distribution; otherwise ST M would be degenerate.
Now, denote D JltrT M ⊥ JST M⊥ , then condition B is satisfied, where
JD L1 ⊥ L2 , L1 ltr T M, and L2 ST M⊥ . Hence the result is present.
3. Mixed Geodesic and CR-Lightlike Product
Definition 3.1. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is
called mixed geodesic CR-lightlike submanifolds if the second fundamental form h satisfies
hX, Y 0, for any X ∈ ΓD and Y ∈ ΓD .
Definition 3.2. A CR-lightlike submanifold of an indefinite almost Hermitian manifold is
called D-geodesic resp., D -geodesic CR-lightlike submanifolds if its second fundamental
form h, satisfies hX, Y 0, for any X, Y ∈ ΓD resp., X, Y ∈ ΓD .
Definition 3.3 see 8. A CR-lightlike submanifold M in a Kaehler manifold is said to be a
mixed foliate if the distribution D is integrable and M is mixed totally geodesic CR-lightlike
submanifold.
Now, suppose {N1 , N2 , . . . , Nr } be a basis of ΓltrT M with respect to the basis
{ξ1 , ξ2 , . . . , ξr } of ΓRad T M, such that {N1 , N2 , . . . , Np } is a basis of ΓL1 . Also, consider
an orthonormal basis {W1 , W2 , . . . , Ws } of ΓL2 .
Theorem 3.4. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. Then, the
distribution D defines a totally geodesic foliation if M is D-geodesic.
Proof. Distribution D defines a totally geodesic foliation, if and only if,
∇X Y ∈ ΓD,
∀X, Y ∈ ΓD.
3.1
Since D JL1 ⊥ L2 , therefore 3.1 holds, if and only if, g∇X Y, Jξi 0,
g∇X Y, JWα 0, for all i ∈ {1, . . . , p}, α ∈ {1, . . . , s}.
Now let X, Y ∈ ΓD, then g∇X Y, Jξi g∇X Y, Jξi −ghX, JY , ξi , similarly
g∇X Y, JWα −ghX, JY , Wα . Then, the result follows directly by using hypothesis.
Corollary 3.5. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. If the
distribution D defines a totally geodesic foliation, then hs X, JY 0, for all X, Y ∈ ΓD.
Theorem 3.6 See 7. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M.
Then, the almost complex distribution D is integrable, if and only if, the second fundamental form of
M satisfies
3.2
h X, JY h JX, Y ,
for any X, Y ∈ ΓD.
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International Journal of Mathematics and Mathematical Sciences
Now, let S and Q be the projections on D and D , respectively. Then, one has
JX fX wX,
∀X ∈ ΓT M,
3.3
where fX JSX and wX JQX. Clearly, f is a tensor field of type 1, 1 and w is ΓL1 ⊥ L2 valued 1-form on M. Clearly, X ∈ ΓD if and only if, wX 0. On the other hand, one sets
JV BV CV,
∀V ∈ ΓtrT M,
3.4
where BV and CV are sections of T M and trT M, respectively.
By using Kaehlerian property of ∇ with 2.7 and 2.8, one has the following lemmas.
Lemma 3.7. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. Then, one
has
∇X f Y AwY X BhX, Y ,
t ∇X w Y ChX, Y − h X, fY ,
3.5
3.6
for any X, Y ∈ ΓT M, where
∇X f Y ∇X fY − f∇X Y ,
t ∇X w Y ∇tX wY − w∇X Y .
3.7
3.8
Lemma 3.8. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. Then, one
has
∇X BV −fAV X ACV X,
3.9
∇X CV −wAV X − hX, BV ,
3.10
for any X ∈ ΓT M and V ∈ ΓtrT M, where
∇X BV ∇X BV − B∇tX V,
3.11
∇X CV ∇tX CV − C∇tX V.
3.12
Theorem 3.9. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. Then,
distribution D defines totally geodesic foliation, if and only if, D is integrable.
Proof. From 3.6, we obtain
h X, JY ChX, Y w∇X Y ,
3.13
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7
for all X, Y ∈ ΓD. Then, taking into account that h is symmetric and ∇ is torsion free, we
obtain
h X, JY − h Y, JX w∇X Y − w∇Y X,
3.14
which proves the assertion.
Corollary 3.10. If CR -lightlike submanifold M in an indefinite Kaehler manifold is mixed foliate,
then, using Theorem 3.9 and Corollary 3.5, it is clear that hX, Y 0, for any X ∈ ΓD, Y ∈ ΓD ,
and hs X, JY 0, for any X, Y ∈ ΓD.
Theorem 3.11. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. Then, M
is mixed geodesic, if and only if, AwZ X ∈ ΓD, ∇tX wZ ∈ ΓL1 ⊥ L2 , for any X ∈ ΓD, Z ∈ ΓD .
Proof. Since M is a Kaehler manifold, therefore hX, Y −J ∇X JZ − ∇X Z −J{−AJZ X ∇tX JZ} − ∇X Z. Since Z ∈ ΓD , this implies that JZ wZ, fZ 0, therefore, hX, Y JAJZ X−J∇tX JZ−∇X Z. This gives hX, Y fAwZ XwAwZ X−B∇tX wZ−C∇tX wZ−∇X Z;
comparing transversal parts, we obtain hX, Y wAwZ X − C∇tX wZ. Thus, M is mixed
geodesic, if and only if, wAwZ X 0, C∇tX wZ 0. This implies AwZ X ∈ ΓD, ∇tX wZ ∈
ΓL1 ⊥ L2 .
Lemma 3.12. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. Then,
∇X Z −fAW X B∇sX W BDl X, W,
3.15
for any X ∈ ΓT M, Z ∈ ΓJL2 and W ∈ ΓL2 .
Proof. Let W ∈ ΓL2 such that Z JW. Since M is a Kaehler manifold, therefore, ∇X Z J ∇XW, for any X ∈ ΓT M. Then, ∇X Z hX, Z J−AW X ∇sX W Dl X, W −fAW X −
wAW XB∇sX W C∇sX W BDl X, WCDl X, W. Comparing the tangential parts, we obtain
3.9.
Let M be an indefinite Kaehler manifold of constant holomorphic sectional curvature
c, then curvature tensor is given by
RX, Y Z c
gY, ZX − gX, ZY g JY, Z JX
4
−g JX, Z JY 2g X, JY JZ .
3.16
Theorem 3.13. Let M be a mixed foliate CR -lightlike submanifold of an indefinite complex space
form Mc. Then, M is a complex semi-Euclidean space.
Proof. From 3.16, we have
c
g R X, JX Z, JZ − gX, XgZ, Z,
2
3.17
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International Journal of Mathematics and Mathematical Sciences
for any X ∈ ΓD0 and Z ∈ ΓJL2 . Since, M is a mixed foliate, therefore, for X ∈ ΓD0 and
Z ∈ ΓJL2 , we have
g R X, JX Z, JZ g ∇X hs JX, Z − ∇JX hs X, Z, JZ ,
3.18
where ∇X hs JX, Z ∇sX hs JX, Z − hs ∇X JX, Z − hs JX, ∇X Z and ∇JX hs X, Z ∇s hs X, Z − hs ∇JX X, Z − hs X, ∇JX Z. By hypothesis, M is mixed geodesic, thereJX
fore, ∇X hs JX, Z − ∇JX hs X, Z
hs ∇JX X, Z hs X, ∇JX Z − hs ∇X JX, Z −
hs JX, ∇X Z. Again, by using hypothesis and Theorem 3.9, the distribution D defines
totally geodesic foliation. Hence, ∇JX X, ∇X JX ∈ ΓD. Therefore, we have ∇X hs JX, Z −
∇JX hs X, Z
hs X, ∇JX Z − hs JX, ∇X Z; using Lemma 3.12 here, we obtain
∇X hs JX, Z − ∇JX hs X, Z −hs X, fAW JX hs X, B∇s W hs X, BDl JX, W JX
hs JX, fAW X − hs JX, B∇sX W − hs JX, BDl X, W.
By using, M is mixed geodesic and Corollary 3.10, we obtain
∇X hs JX, Z − ∇JX hs X, Z 0.
3.19
Therefore, from 3.17–3.19, we get c/2 gX, XgZ, Z 0. Since D0 and JL2 are nondegenerate, therefore, we have c 0, which completes the proof.
Definition 3.14 see 7. A CR-lightlike submanifold M of a Kaehler manifold M is called a
CR-lightlike product if both the distribution D and D define totally geodesic foliations on M.
Theorem 3.15 see 7. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M.
Then, D defines a totally geodesic foliation on M, if and only if, for any X, Y ∈ ΓD, hX, Y has no
component in ΓL1 ⊥ L2 .
Theorem 3.16 see 7. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M.
Then, D defines a totally geodesic foliation on M, if and only if, for any X, Y ∈ ΓD , one has
h∗ X, Y 0,
3.20
AJY X has no component in D0 ,
3.21
AJY X BhX, Y has no component in Rad T M.
3.22
Theorem 3.17 see 3. A CR -submanifold of a Kaehler manifold M is a CR -product, if and only
if, P is parallel, that is, ∇P 0, where JX P X FX.
Now, we will give the characterization of CR-lightlike product in the following form.
Theorem 3.18. A CR -lightlike submanifold of an indefinite Kaehler manifold M is a CR -lightlike
product, if and only if, f is parallel, that is, ∇f 0.
International Journal of Mathematics and Mathematical Sciences
9
Proof. If f is parallel, then 3.5, gives
BhX, Y −AwY X,
3.23
for any X, Y ∈ ΓT M. In particular, if X ∈ ΓD, then wX 0. Hence, 3.23 implies
BhX, Y 0, thus, Theorem 3.15 follows.
Let X, Y ∈ ΓD , then JY wY , then 3.23 implies AJY X BhX, Y 0, and,
hence, 3.22 follows. For any Z ∈ ΓD0 , gAJY X, Z gAwY X, Z −gBhX, Y , Z ghX, Y , JZ 0. Thus, 3.21 holds well. Since ∇X fY 0 for any X, Y ∈ ΓT M, let
X, Y ∈ ΓD , then we have f∇X Y 0. Hence, ∇X Y ∈ ΓD . Therefore, g∇X Y, N 0, for
any X, Y ∈ ΓD and N ∈ ΓltrT M and consequently h∗ X, Y 0. Hence, 3.20 holds
well. Thus, M is a CR-lightlike product in M.
Conversely, if M is a CR-lightlike product, then by definition, D and D both defines
totally geodesic foliation, that is, ∇X Y ∈ ΓD, for any X, Y ∈ ΓD and ∇X Y ∈ ΓD , for any
X, Y ∈ ΓD . If ∇X Y ∈ ΓD for any X, Y ∈ ΓD, then by comparing the transversal parts in
the simplification of Kaehlerian property of ∇, we get
h X, JY JhX, Y .
3.24
Therefore, from 2.7, 3.7, and 3.24, we may prove that ∇X fY 0 or ∇f 0.
Let ∇X Y ∈ ΓD , for any X, Y ∈ ΓD , then by comparing the tangential parts in the
simplification of Kaehlerian property of ∇, we get
BhX, Y AwY X 0.
3.25
Thus, from 3.5 and 3.25, we obtain ∇X fY 0 or ∇f 0. Hence, the proof is complete.
Theorem 3.19. Let M be a CR -lightlike submanifold of an indefinite Kaehler manifold M. If M is a
CR -lightlike product, then it is a mixed foliate CR-lightlike submanifold.
Proof. Since M is a CR-lightlike product then by Theorem 3.9, it is clear that the distribution
D is integrable. Also, by using Theorem 3.18, for CR-lightlike product, ∇f 0, then 3.5
gives
BhX, Y −AwY X,
3.26
for any X, Y ∈ ΓT M. Then, in particular, if X ∈ ΓD, then wX 0. Hence, 3.26 implies
BhX, Y 0, that is,
AJZ X 0,
3.27
for any Z ∈ ΓD and X ∈ ΓD.
Next, by the definition of CR-lightlike submanifold M is mixed geodesic, if and only if,
ghX, Y , ξ 0 and ghX, Y , W 0, for any X ∈ ΓD, Y ∈ ΓD , and W ∈ ΓST M⊥ .
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International Journal of Mathematics and Mathematical Sciences
By hypothesis, we have ∇X fY 0 for any X, Y ∈ ΓT M; let X ∈ ΓD and Y ∈ ΓD ,
then we have f∇X Y 0. Hence, ∇X Y ∈ ΓD . Therefore, ghX, Y , ξ g∇X Y, ξ, since
Y ∈ ΓD , therefore, there exists V ∈ ΓL1 ⊥ L2 such that Y JV ; this gives ghX, Y , ξ g∇X JV, ξ −g∇X V, Jξ gAV X, Jξ −gAJY X, Jξ 0, by virtue of 3.27.
Similarly, ghX, Y , W gJ ∇X V, W −gC∇tX V, W. Now, from 3.10 and 3.12,
we have C∇tX V ∇tX CV wAV X hX, BV ; since V ∈ ΓL1 ⊥ L2 , therefore CV 0, so
we have C∇tX V wAV X hX, BV . Therefore, by using 3.27, we have ghX, Y , W −ghX, BV , W −ghX, JY , W; this gives ghX, Y , W 0, for all X ∈ ΓD, Y ∈
ΓD . Hence, the proof is complete.
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4 B.-Y. Chen, “CR-submanifolds of a Kaehler manifold. II,” Journal of Differential Geometry, vol. 16, no. 3,
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5 K. Yano and M. Kon, “CR submanifolds of a complex projective space,” Journal of Differential Geometry,
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Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014