CONTACT CR-DOUBLY WARPED PRODUCT
SUBMANIFOLDS IN KENMOTSU SPACE FORMS
CR-doubly Warped Product
Submanifolds
ANDREEA OLTEANU
Faculty of Mathematics and Computer Science
University of Bucharest
Str. Academiei 14
011014 Bucharest, Romania
Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
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EMail: andreea_d_olteanu@yahoo.com
Received:
26 February, 2009
Accepted:
06 November, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 53C40; Secondary 53C25.
Key words:
Doubly warped product, contact CR-doubly warped product, invariant submanifold, anti-invariant submanifold, Laplacian, mean curvature, Kenmotsu space
form.
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Abstract:
Recently, the author established general inequalities for CR-doubly warped products isometrically immersed in Sasakian space forms.
In the present paper, we obtain sharp estimates for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping functions
(intrinsic invariants) for contact CR-doubly warped products isometrically immersed in Kenmotsu space forms. The equality case is considered. Some applications are derived.
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Contents
1
Introduction
3
2
Preliminaries
4
3
Contact CR-doubly Warped Product Submanifolds
7
4
Another Inequality
11
CR-doubly Warped Product
Submanifolds
Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
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1.
Introduction
In 1978, A. Bejancu introduced the notion of CR-submanifolds which is a generalization of holomorphic and totally real submanifolds in an almost Hermitian
manifold ([2]). Following this, many papers and books on the topic were published. The first main result on CR-submanifolds was obtained by Chen [4]: any
CR-submanifold of a Kaehler manifold is foliated by totally real submanifolds. As
non-trivial examples of CR-submanifolds, we can mention the (real) hypersurfaces
of Hermitian manifolds.
Recently, Chen [5] introduced the notion of a CR-warped product submanifold in
a Kaehler manifold and proved a number of interesting results on such submanifolds.
In particular, he established a sharp relationship between the warping function f of
f and the
a warped product CR-submanifold M1 ×f M2 of a Kaehler manifold M
2
squared norm of the second fundamental form ||h|| .
On the other hand, there are only a handful of papers about doubly warped product Riemannian manifolds which are the generalization of a warped product Riemannian manifold.
Recently, the author obtained a general inequality for CR-doubly warped products
isometrically immersed in Sasakian space forms ([12]).
In the present paper, we study contact CR-doubly warped product submanifolds
in Kenmotsu space forms.
We prove estimates of the squared norm of the second fundamental form in terms
of the warping function. Equality cases are investigated. Obstructions to the existence of contact CR-doubly warped product submanifolds in Kenmotsu space forms
are derived.
CR-doubly Warped Product
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Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
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2.
Preliminaries
f, g) is said to be a Kenmotsu
A (2m + 1)-dimensional Riemannian manifold (M
f, a vector field ξ
manifold if it admits an endomorphism φ of its tangent bundle T M
and a 1-form η satisfying:
φ2 = −Id + η ⊗ ξ,
(2.1)
η (ξ) = 1,
φξ = 0,
η ◦ φ = 0,
g (φX, φY ) = g (X, Y ) − η (X) η (Y ) , η (X) = g (X, ξ) ,
e
e X ξ = X − η (X) ξ,
∇X φ Y = −g (X, φY ) ξ − η (Y ) φX, ∇
f, where ∇
e denotes the Riemannian connection with
for any vector fields X, Y on M
respect to g.
f, i.e.,
We denote by ω the fundamental 2-form of M
f .
(2.2)
ω (X, Y ) = g (φX, Y ) ,
∀X, Y ∈ Γ T M
It was proved that the pairing (ω, η) defines a locally conformal cosymplectic structure, i.e.,
dω = 2ω ∧ η,
dη = 0.
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vol. 10, iss. 4, art. 119, 2009
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f is called a φ-section if it is spanned by X and φX,
A plane section π in Tp M
where X is a unit tangent vector orthogonal to ξ. The sectional curvature of a φsection is called a φ-sectional curvature. A Kenmotsu manifold with constant φholomorphic sectional curvature c is said to be a Kenmotsu space form and is denoted
f (c).
by M
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e of a Kenmotsu space form is given by [8]
The curvature tensor R
e (X, Y ) Z
(2.3) R
c−3
c+1
=
{g (Y, Z) X − g(X, Z)Y } +
{[η (X) Y − η (Y ) X] η (Z)
4
4
+ [g (X, Z) η (Y ) − g (Y, Z) η (X)] ξ + ω (Y, Z) φX
− ω (X, Z) φY − 2ω (X, Y ) φZ}.
CR-doubly Warped Product
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Andreea Olteanu
f be a Kenmotsu manifold and M an n-dimensional submanifold tangent to
Let M
ξ. For any vector field X tangent to M , we put
vol. 10, iss. 4, art. 119, 2009
φX = P X + F X,
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where P X (resp. F X) denotes the tangential (resp. normal) component of φX.
Then P is an endomorphism of the tangent bundle T M and F is a normal bundle
valued 1-form on T M .
The equation of Gauss is given by
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(2.4)
e (X, Y, Z, W )
(2.5) R
= R (X, Y, Z, W ) + g (h (X, W ) , h (Y, Z)) − g (h (X, Z) , h (Y, W ))
for any vectors X, Y , Z, W tangent to M .
Let p ∈ M and {e1 , ..., en , en+1 , ..., e2m+1 } be an orthonormal basis of the tanf, such that e1 , ..., en are tangent to M at p. We denote by H the
gent space Tp M
mean curvature vector, that is
n
(2.6)
1X
H (p) =
h (ei , ei ) .
n i=1
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As is known, M is said to be minimal if H vanishes identically.
Also, we set
(2.7)
hrij = g (h (ei , ej ) , er ) ,
i, j ∈ {1, ..., n}, r ∈ {n + 1, ..., 2m + 1}
as the coefficients of the second fundamental form h with respect to {e1 , ..., en ,
en+1 , ..., e2m+1 }, and
||h||2 =
(2.8)
n
X
g (h (ei , ej ) , h (ei , ej )) .
i,j=1
By analogy with submanifolds in a Kaehler manifold, different classes of submanifolds in a Kenmotsu manifold were considered (see, for example, [13]).
A submanifold M tangent to ξ is called an invariant (resp. anti-invariant) submanifold if φ (Tp M ) ⊂ Tp M , ∀ p ∈ M (resp. φ (Tp M ) ⊂ Tp⊥ M , ∀ p ∈ M ).
A submanifold M tangent to ξ is called a contact CR-submanifold ([13]) if there
exists a pair of orthogonal differentiable distributions D and D⊥ on M , such that:
CR-doubly Warped Product
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Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
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⊥
1. T M = D ⊕ D ⊕ {ξ}, where {ξ} is the 1-dimensional distribution spanned by
ξ;
2. D is invariant by φ, i. e., φ (Dp ) ⊂ Dp , ∀p ∈ M ;
3. D⊥ is anti-invariant by φ, i. e., φ Dp⊥ ⊂ Dp⊥ , ∀p ∈ M .
⊥
In particular, if D = {0} (resp. D ={0}), M is an invariant (resp. anti-invariant)
submanifold.
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3.
Contact CR-doubly Warped Product Submanifolds
Singly warped products or simply warped products were first defined by Bishop and
O’Neill in [3] in order to construct Riemannian manifolds with negative sectional
curvature.
In general, doubly warped products can be considered as generalizations of singly
warped products.
Let (M1 , g1 ) and (M2 , g2 ) be two Riemannian manifolds and let f1 : M1 →
(0, ∞) and f2 : M2 → (0, ∞) be differentiable functions.
The doubly warped product M =f2 M1 ×f1 M2 is the product manifold M1 × M2
endowed with the metric
(3.1)
g = f22 g1 + f12 g2 .
CR-doubly Warped Product
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Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
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More precisely, if π1 : M1 × M2 → M1 and π2 : M1 × M2 → M2 are natural
projections, the metric g is defined by
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g = (f2 ◦ π2 )2 π1∗ g1 + (f1 ◦ π1 )2 π2∗ g2 .
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(3.2)
The functions f1 and f2 are called warping functions. If either f1 ≡ 1 or f2 ≡ 1,
but not both, then we obtain a warped product. If both f1 ≡ 1 and f2 ≡ 1, then we
have a Riemannian product manifold. If neither f1 nor f2 is constant, then we have
a non-trivial doubly warped product.
We recall that on a doubly warped product one has
(3.3)
∇X Z = Z (ln f2 ) X + X (ln f1 ) Z,
for any vector fields X tangent to M1 and Z tangent to M2 .
If X and Z are unit vector fields, it follows that the sectional curvature K (X ∧ Z)
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of the plane section spanned by X and Z is given by
K (X ∧ Z) =
(3.4)
1
1
{ ∇1X X f1 − X 2 f1 } + { ∇2Z Z f2 − Z 2 f2 },
f1
f2
where ∇1 , ∇2 are the Riemannian connections of the Riemannian metrics g1 and g2
respectively.
By reference to [12], a doubly warped product submanifold M =f2 M1 ×f1 M2
f, with M1 a (2α + 1)-dimensional invariant submanifold
of a Kenmotsu manifold M
f is said to be a
tangent to ξ and M2 a β-dimensional anti-invariant submanifold of M
contact CR-doubly warped product submanifold.
We state the following estimate of the squared norm of the second fundamental
form for contact CR-doubly warped products in Kenmotsu manifolds.
f (c) be a (2m + 1)-dimensional Kenmotsu manifold and M =f2
Theorem 3.1. Let M
M1 ×f1 M2 an n-dimensional contact CR-doubly warped product submanifold, such
that M1 is a (2α + 1)-dimensional invariant submanifold tangent to ξ and M2 is a
f (c). Then:
β-dimensional anti-invariant submanifold of M
CR-doubly Warped Product
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(i) The squared norm of the second fundamental form of M satisfies
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(3.5)
||h||2 ≥ 2β||∇ (ln f1 ) ||2 − 1],
where ∇ (ln f1 ) is the gradient of ln f1 .
(ii) If the equality sign of (3.5) holds identically, then M1 is a totally geodesic
f. Moreover, M is
submanifold and M2 is a totally umbilical submanifold of M
f.
a minimal submanifold of M
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Proof. Let M =f2 M1 ×f1 M2 be a doubly warped product submanifold of a Kenf, such that M1 is an invariant submanifold tangent to ξ and M2 is
motsu manifold M
f.
an anti-invariant submanifold of M
For any unit vector fields X tangent to M1 and Z, W tangent to M2 respectively,
we have:
e
e
(3.6)
g (h (φX, Z) , φZ) = g ∇Z φX, φZ = g φ∇Z X, φZ
e Z X, Z = g (∇Z X, Z) = X ln f1 ,
=g ∇
CR-doubly Warped Product
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Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
g (h (φX, Z) , φW ) = (X ln f1 ) g (Z, W ) .
f is a Kenmotsu manifold, it is
On the other hand, since the ambient manifold M
easily seen that
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h (ξ, Z) = 0.
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Obviously, (3.3) implies ξ ln f1 = 1. Therefore, by (3.6) and (3.7), the inequality
(3.5) is immediately obtained.
Denote by h00 the second fundamental form of M2 in M . Then, we get
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(3.7)
g (h00 (Z, W ) , X) = g (∇Z W, X) = − (X ln f1 ) g (Z, W, )
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or equivalently
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(3.8)
h” (Z, W ) = −g (Z, W ) ∇ (ln f1 ) .
If the equality sign of (3.5) identically holds, then we obtain
(3.9)
h (D,D) = 0, h D⊥ ,D⊥ = 0, h D,D⊥ ⊂ φD⊥ .
The first condition (3.9) implies that M1 is totally geodesic in M . On the other hand,
one has
e
g (h (X, φY ) , φZ) = g ∇X φY, φZ = g (∇X Y, Z) = 0.
f.
Thus M1 is totally geodesic in M
The second condition in (3.9) and (3.8) imply that M2 is a totally umbilical subf.
manifold in M
f.
Moreover, by (3.9), it follows that M is a minimal submanifold of M
CR-doubly Warped Product
Submanifolds
Andreea Olteanu
In particular, if the ambient space is a Kenmotsu space form, one has the following.
f (c) be a (2m + 1)-dimensional Kenmotsu space form of conCorollary 3.2. Let M
stant φ-sectional curvature c and M =f2 M1 ×f1 M2 an n-dimensional non-trivial
contact CR-doubly warped product submanifold, satisfying
||h||2 = 2β ||∇ (ln f1 ) ||2 − 1 .
Then, we have
f (c). Hence M1 is a Ken(a) M1 is a totally geodesic invariant submanifold of M
motsu space form of constant φ-sectional curvature c.
f (c). Hence M2 is a
(b) M2 is a totally umbilical anti-invariant submanifold of M
real space form of sectional curvature ε > c−3
.
4
Proof. Statement (a) follows from Theorem 3.1.
f (c). The Gauss
Also, we know that M2 is a totally umbilical submanifold of M
equation implies that M2 is a real space form of sectional curvature ε ≥ c−3
.
4
c−3
Moreover, by (3.3), we see that ε = 4 if and only if the warping function f1 is
constant.
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4.
Another Inequality
In the present section, we will improve the inequality (3.5) for contact CR-doubly
warped product submanifolds in Kenmotsu space forms. Equality case is characterized.
f (c) be a (2m + 1)-dimensional Kenmotsu space form of conTheorem 4.1. Let M
stant φ-sectional curvature c and M =f2 M1 ×f1 M2 an n-dimensional contact
CR-doubly warped product submanifold, such that M1 is a (2α + 1)-dimensional
invariant submanifold tangent to ξ and M2 is a β-dimensional anti-invariant subf (c). Then:
manifold of M
(i) The squared norm of the second fundamental form of M satisfies
(4.1)
||h||2 ≥ 2β ||∇ (ln f1 ) ||2 − ∆1 (ln f1 ) − 1 + αβ (c + 1) ,
where ∆1 denotes the Laplace operator on M1 .
(ii) The equality sign of (4.1) holds identically if and only if we have:
f (c). Hence M1 is a
(a) M1 is a totally geodesic invariant submanifold of M
Kenmotsu space form of constant φ-sectional curvature c.
f (c). Hence M2
(b) M2 is a totally umbilical anti-invariant submanifold of M
c−3
is a real space form of sectional curvature ε ≥ 4 .
Proof. Let M =f2 M1 ×f1 M2 be a contact CR-doubly warped product submanifold
f (c), such that M1 is an invariant
of a (2m+1)-dimensional Kenmotsu space form M
f (c).
submanifold tangent to ξ and M2 is an anti-invariant submanifold of M
We denote by ν be the normal subbundle orthogonal to φ (T M2 ). Obviously, we
have
T ⊥ M = φ (T M2 ) ⊕ ν,
φν = ν.
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vol. 10, iss. 4, art. 119, 2009
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For any vector fields X tangent to M1 and orthogonal to ξ and Z tangent to M2 ,
equation (2.3) gives
e (X, φX, Z, φZ) = c + 1 g(X, X)g(Z, Z).
R
2
On the other hand, by the Codazzi equation, we have
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e (X, φX, Z, φZ)
(4.2) R
= −g ∇⊥
X h (φX, Z) − h (∇X φX, Z) − h (φX, ∇X Z) , φZ
+g
∇⊥
φX h (X, Z)
Andreea Olteanu
vol. 10, iss. 4, art. 119, 2009
− h (∇φX X, Z) − h (X, ∇φX Z) , φZ .
By using the equation (3.3) and structure equations of a Kenmotsu manifold, we get
g ∇⊥
X h (φX, Z) , φZ
= Xg (h (φX, Z) , φZ) − g h (φX, Z) , ∇⊥
X φZ
e
= Xg (∇Z X, Z) − g h (φX, Z) , φ∇X Z
= X ((X ln f1 ) g (Z, Z)) − (X ln f1 ) g (h (φX, Z) , φZ) − g (h (φX, Z) , φhν (X, Z))
= X 2 ln f1 g (Z, Z) + (X ln f1 )2 g (Z, Z) − ||hν (X, Z) ||2 ,
where we denote by hν (X, Z) the ν-component of h (X, Z).
Also, by (3.6) and (3.3), we obtain respectively
g (h (∇X φX, Z) , φZ) = ((∇X X) ln f1 ) g (Z, Z) ,
g (h (φX, ∇X Z) , φZ) = (X ln f1 )g (h (φX, Z) , φZ) = (X ln f1 )2 g (Z, Z) .
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Substituting the above relations in (4.2), we find
e (X, φX, Z, φZ)
(4.3) R
= 2||hν (X, Z) ||2 − X 2 ln f1 g (Z, Z) + ((∇X X) ln f1 ) g (Z, Z)
− (φX)2 ln f1 g (Z, Z) + ((∇φX φX) ln f1 ) g (Z, Z) .
Then the equation (4.3) becomes
c+1
2
g(X, X) + X 2 ln f1 − ((∇X X) ln f1 )
(4.4) 2||hν (X, Z) || =
2
i
2
+ (φX) ln f1 − ((∇φX φX) ln f1 ) g(Z, Z).
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vol. 10, iss. 4, art. 119, 2009
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Let
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{X0 = ξ, X1 , ..., Xα , Xα+1 = φX1 , ..., X2α = φXα , Z1 , ..., Zβ }
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be a local orthonormal frame on M such that X0 , ..., X2α are tangent to M1 and
Z1 , ..., Zβ are tangent to M2 .
Therefore
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(4.5)
2
β
2α X
X
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||hν (Xj , Zt ) ||2 = αβ (c + 1) − 2β∆1 (ln f1 ).
j=1 t=1
Combining (3.5) and (4.5), we obtain the inequality (4.1).
The equality case can be solved similarly to Corollary 3.2.
f (c) be a Kenmotsu space form with c < −1. Then there do not
Corollary 4.2. Let M
f (c) such
exist contact CR-doubly warped product submanifolds f2 M1 ×f1 M2 in M
that ln f1 is a harmonic function on M1 .
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Proof. Assume that there exists a contact CR-doubly warped product submanifold
f
f2 M1 ×f1 M2 in a Kenmotsu space form M (c) such that ln f1 is a harmonic function
on M1 .Then (4.5) implies c ≥ −1.
f (c) be a Kenmotsu space form with c ≤ −1. Then there do not
Corollary 4.3. Let M
f (c) such
exist contact CR-doubly warped product submanifolds f2 M1 ×f1 M2 in M
that ln f1 is a non-negative eigenfunction of the Laplacian on M1 corresponding to
an eigenvalue λ > 0.
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References
[1] K. ARSLAN, R. EZENTAS, I. MIHAI AND C. MURATHAN, Contact CRwarped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc.,
42(5) (2005), 1101–1110.
[2] A. BEJANCU, CR-submanifolds of a Kaehler manifold I, Proc. Amer. Math.
Soc., 69(1) (1978), 135–142.
CR-doubly Warped Product
Submanifolds
[3] R.L. BISHOP AND B. O’NEILL, Manifolds of negative curvature, Trans. Amer.
Math. Soc., 145 (1969), 1–49.
vol. 10, iss. 4, art. 119, 2009
[4] B.Y. CHEN, CR-submanifolds of a Kaehler manifold, J. Differential Geom., 16
(1981), 305–323.
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[5] B.Y. CHEN, Geometry of warped product CR-submanifolds in Kaehler Manifolds, Monatsh. Math., 133 (2001), 177–195.
[6] B.Y. CHEN, On isometric minimal immersions from warped products into real
space forms, Proc. Edinburgh Math. Soc., 45 (2002), 579–587.
[7] I. HASEGAWA AND I. MIHAI, Contact CR-warped product submanifolds in
Sasakian manifolds, Geom. Dedicata, 102 (2003), 143–150.
[8] K. KENMOTSU, A class of almost contact Riemannian Manifolds, Tohoku
Math. J., 24 (1972), 93–103.
[9] K. MATSUMOTO AND V. BONANZINGA, Doubly warped product CRsubmanifolds in a locally conformal Kaehler space form, Acta Mathematica
Academiae Paedagogiace Nyíregyháziensis, 24 (2008), 93–102.
[10] I. MIHAI, Contact CR-warped product submanifolds in Sasakian space forms,
Geom. Dedicata, 109 (2004), 165–173.
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[11] M.I. MUNTEANU, Doubly warped product CR-submanifolds in locally conformal Kaehler manifolds, Monatsh. Math., 150 (2007), 333–342.
[12] A. OLTEANU, CR-doubly warped product submanifolds in Sasakian space
forms, Bulletin of the Transilvania University of Brasov, 1 (50), III-2008, 269–
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[13] K. YANO AND M. KON, Structures on Manifolds, World Scientific, Singapore,
1984.
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